aa r X i v : . [ m a t h . DG ] M a y The Gaussian Multi-Bubble Conjecture
Emanuel Milman and Joe Neeman May 29, 2018
Abstract
We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perime-ter way to decompose R n into q cells of prescribed (positive) Gaussian measure when2 ≤ q ≤ n +
1, is to use a “simplicial cluster”, obtained from the Voronoi cells of q equidistantpoints. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers(up to null-sets). The case q = q = q >
3, numerous new ideas andtools are required: we employ higher codimension regularity of minimizing clusters andknowledge of the combinatorial incidence structure of its interfaces, establish integrabilityproperties of the interface curvature for stationary regular clusters, derive new formulaefor the second variation of weighted volume and perimeter of such clusters, and constructa family of (approximate) inward vector-fields, having (almost) constant normal compo-nent on the interfaces of a given cell. We use all of this information to show that when2 ≤ q ≤ n +
1, stable regular clusters must have convex polyhedral cells, which is the mainnew ingredient in this work (this was not known even in the q = Let γ = γ n denote the standard Gaussian probability measure on Euclidean space ( R n , ∣ ⋅ ∣) : γ n ∶ = ( π ) n e − ∣ x ∣ dx = ∶ e − W ( x ) dx. More generally, if H k denotes the k -dimensional Hausdorff measure, let γ k denote itsGaussian-weighted counterpart: γ k ∶ = e − W ( x ) H k . The Gaussian-weighted (Euclidean) perimeter of a Borel set U ⊂ R n is defined as: P γ ( U ) ∶ = sup { ∫ U ( div X − ⟨∇ W, X ⟩) dγ ∶ X ∈ C ∞ c ( R n ; T R n ) , ∣ X ∣ ≤ } . Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel. Email: [email protected]. Department of Mathematics, University of Texas at Austin. Email: [email protected] research leading to these results is part of a project that has received funding from the European ResearchCouncil (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreementNo 637851). This material is also based upon work supported by the National Science Foundation under GrantNo. 1440140, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley,California, during the Fall semester of 2017.2010 Mathematics Subject Classification: 49Q20, 53A10. or nice sets (e.g. open sets with piecewise smooth boundary), P γ ( U ) is known to agreewith γ n − ( ∂U ) (see e.g. [37]). The weighted perimeter P γ ( U ) has the advantage of beinglower semi-continuous with respect to L ( γ ) convergence, and thus fits well with the directmethod of calculus-of-variations.The classical Gaussian isoperimetric inequality, established independently by Sudakov–Tsirelson [57] and Borell [14] in 1975, asserts that among all Borel sets U in R n havingprescribed Gaussian measure γ ( U ) = v ∈ [ , ] , halfspaces minimize Gaussian-weightedperimeter P γ ( U ) (see also [23, 36, 5, 11, 7, 12, 15, 43]). Later on, it was shown by Carlen–Kerce [18] (see also [24, 43, 39]), that up to null-sets, halfspaces are in fact the unique min-imizers for the Gaussian isoperimetric inequality. These results are the Gaussian analogueof the classical unweighted isoperimetric inequality in Euclidean space ( R n , ∣ ⋅ ∣) , which statesthat Euclidean balls uniquely minimizes (up to null-sets) unweighted perimeter among allsets of prescribed Lebesgue measure [16, 37].In this work, we extend these classical results for the Gaussian measure to the case ofclusters. A q -cluster Ω = ( Ω , . . . , Ω q ) is a q -tuple of Borel subsets Ω i ⊂ R n called cells,such that { Ω i } are pairwise disjoint, P γ ( Ω i ) < ∞ for each i , and γ ( R n ∖ ⋃ qi = Ω i ) =
0. Notethat the cells are not required to be connected. The total Gaussian perimeter of a clusterΩ is defined as: P γ ( Ω ) ∶ = q ∑ i = P γ ( Ω i ) . The Gaussian measure of a cluster is defined as: γ ( Ω ) ∶ = ( γ ( Ω ) , . . . , γ ( Ω q )) ∈ ∆ ( q − ) , where ∆ ( q − ) ∶ = { v ∈ R q ∶ v i ≥ , ∑ qi = v i = } denotes the ( q − ) -dimensional probabilitysimplex. The isoperimetric problem for q -clusters consists of identifying those clustersΩ of prescribed Gaussian measure γ ( Ω ) = v ∈ ∆ ( q − ) which minimize the total Gaussianperimeter P γ ( Ω ) .Note that easy properties of the perimeter ensure that for a 2-cluster Ω, P γ ( Ω ) = P γ ( Ω ) = P γ ( Ω ) , and so the case q = R n , we will refer to the case q = and having complement Ω ). Accordingly, thecase q = q is referred toas the “multi-bubble” problem. See below for further motivation behind this terminologyand related results.A natural conjecture is then the following: Conjecture (Gaussian Multi-Bubble Conjecture) . For all ≤ q ≤ n + , the least Gaussian-weighted perimeter way to decompose R n into q cells of prescribed Gaussian measure v ∈ int ∆ ( q − ) is by using “simplicial clusters”, obtained as the Voronoi cells of q equidistantpoints in R n . Recall that the Voronoi cells of { x , . . . , x q } ⊂ R n are defined as:Ω i = int { x ∈ R n ∶ min j = ,...,q ∣ x − x j ∣ = ∣ x − x i ∣} i = , . . . , q , where int denotes the interior operation (to obtain pairwise disjoint cells).Indeed, when q =
2, the cells of a simplicial cluster are precisely halfspaces, and thesingle-bubble conjecture holds by the classical Gaussian isoperimetric inequality. The sim-plest case beyond q = q = v ∈ int ∆ ( ) ), when the interfaces R (intersectedwith a Euclidean ball). of a simplicial cluster’s cells are three half-hyperplanes meeting along an ( n − ) -dimensionalplane at 120 ○ angles (forming a tripod or “Y” shape in the plane). Simplicial clusters arethe naturally conjectured minimizers in the range 3 ≤ q ≤ n + ○ angles in threes, both ofwhich are necessary conditions for any extremizer of Gaussian perimeter under a Gaussianmeasure constraint – see Section 4. Equal measure simplicial clusters also naturally appearin other related problems (see e.g. [29] and the references therein). The Gaussian Double-Bubble Conjecture (case q =
3) was recently established in our previous work [40], afterprior confirmation in the range max i = , , ∣ v i − / ∣ ≤ .
04 by Corneli-et-al [20]. To the bestof our knowledge, no prior results on the Gaussian Multi-Bubble Conjecture in the range q > Theorem 1.1 (Gaussian Multi-Bubble Theorem) . The Gaussian Multi-Bubble Conjectureholds true for all ≤ q ≤ n + . Theorem 1.2 (Uniqueness of Minimizing Clusters) . Up to null-sets, simplicial q -clustersare the unique minimizers of Gaussian perimeter among all q -clusters of prescribed Gaus-sian measure v ∈ int ∆ ( q − ) , for all ≤ q ≤ n + . The Gaussian Multi-Bubble Conjecture is known to experts. Presumably, its origins maybe traced to an analogous problem of J. Sullivan from 1995 in the unweighted
Euclideansetting [58, Problem 2], where the conjectured uniquely minimizing q -cluster (up to null-sets) for q ≤ n + ( q − ) -bubble – spherical caps bounding connected cells { Ω i } qi = which are obtained by taking the Voronoi cells of q equidistant points in S n ⊂ R n + , and applying all stereographic projections to R n . That the standard ( q − ) -bubblein unweighted Euclidean space exists and is unique (up to isometries) for all volumeswas proved by Montesinos-Amilibia [41]. Similarly, on the (unweighted) sphere S n , wherespherical caps are known to uniquely minimize perimeter (up to null-sets) [16], the SphericalMulti-Bubble Conjecture asserts that the uniquely perimeter minimizing q -cluster (up tonull-sets) for q ≤ n + ( q − ) -bubble – a stereographic projection ofa standard ( q − ) -bubble in R n . See also Schechtman [53] for a formulation of the equalvolumes case of the Gaussian Multi-Bubble Conjecture. n the double-bubble case q =
3, various prior results have been established in a varietyof settings. As the main focus of this work is the case q >
3, we refer the reader toF. Morgan’s excellent book [44, Chapters 13,14,18,19] and to our previous work [40] for amore detailed account of the double-bubble case, and only mention here a few key results.In the unweighted Euclidean setting R n , despite being long believed to be true, the double-bubble case first appeared explicitly as a conjecture in an undergraduate thesis by J. Foisyin 1991 [26]. It was considered in the 1990’s by various authors [27, 28], culminatingin the work of Hutchings–Morgan–Ritor´e–Ros [31], who proved that up to null-sets, thestandard double-bubble is uniquely perimeter minimizing in R ; this was later extendedto R n in [50, 49]. The Spherical Double-Bubble Conjecture was resolved on S by Masters[38], but on S n for n ≥ q = n ≥
2. Theorem 1.1 for q = n ≥ i = , , ∣ v i − / ∣ ≤ .
04 by Corneli-et-al [20], who simultaneously obtained an analogousresult (along with uniqueness) on S n .In the triple-bubble case q =
4, Wichiramala proved in [62] that the standard triple-bubble is uniquely perimeter minimizing in the unweighted Euclidean plane R . We arenot aware of any other (even partial) isoperimetric results on multi-bubbles when q > prior to this work , not in any of the above settings, nor in any other.It is therefore not surprising that we do not employ in this work the traditional ingre-dients used in the above mentioned results, such a B. White’s symmetrization argument(see [26, 30]), or Hutchings’ theory [30] of bounds on the number of connected componentscomprising each cell of a minimizing cluster. Contrary to previous approaches, we do notidentify the minimizing clusters by systematically ruling out competitors, and in fact we donot characterize them at all in our proof of Theorem 1.1 (the characterization is obtainedonly later in Theorem 1.2). We do not know how to use other tools from the Gaussiansingle-bubble setting such as tensorization [11], semi-group methods [5], martingale meth-ods [7] or parabolic techniques [15], even after a possible reduction to the case n = q − directly obtain asharp lower bound on the perimeter of a cluster having prescribed measure v ∈ ∆ ( q − ) .The starting point of our approach is the idea from our previous work [40] of obtaininga matrix-valued partial differential inequality (PDI) on the associated isoperimetric profile;in the next subsection, we explain why in the double-bubble setting of [40] the desiredPDI was obtained with relative ease, and why this cannot be extended to handle the case q >
3. We then describe some of the new ideas and ingredients we need for obtaining theaforementioned PDI for general 2 ≤ q ≤ n +
1. In particular, we need to first show that (inthe latter range for q ) stable regular q -clusters necessarily have flat interfaces, a difficultresult of independent interest, which we could (and did) bypass in [40]. Let I ( q − ) ∶ ∆ ( q − ) → R + denote the Gaussian isoperimetric profile for q -clusters, definedas: I ( q − ) ( v ) ∶ = inf { P γ ( Ω ) ∶ Ω is a q -cluster with γ ( Ω ) = v } . Assume for the sake of this sketch that I ( q − ) is twice continuously differentiable onint ∆ ( q − ) . Given an isoperimetric minimizing cluster Ω with γ ( Ω ) = v ∈ int ∆ ( q − ) , let A ij ( v ) ∶ = γ n − ( ∂ ∗ Ω i ∩ ∂ ∗ Ω j ) , i < j , denote the weighted areas of the cluster’s interfaces,where ∂ ∗ U denotes the reduced boundary of a Borel set U having finite perimeter (see ection 3). Assume again for simplicity that A ij ( v ) are well-defined, that is, depend onlyon v .Let E = E ( q − ) denote the tangent space to ∆ ( q − ) , which we identify with { x ∈ R q ∶∑ qi = x i = } . Given A = { A ij } ≤ i < j ≤ q with A ij ≥
0, we consider the following q × q positivesemi-definite matrix: L A ∶ = ∑ ≤ i < j ≤ q A ij ( e i − e j )( e i − e j ) T . (1.1)In fact, as a quadratic form on E , it is easy to show that L A ( v ) for A ( v ) = { A ij ( v )} isstrictly positive-definite whenever v ∈ int ∆ ( q − ) (see Lemma 7.11).Following our approach from [40], our goal in this work is to show that the followingmatrix-valued differential inequality holds: ∇ I ( q − ) ( v ) ≤ − L − A ( v ) on int ∆ ( q − ) , (1.2)as quadratic forms on E (differentiation and inversion are both carried out on E as well).Once this is established, Theorem 1.1 will follow exactly as in [40]; let us briefly recall why.Let I ( q − ) m ∶ int ∆ ( q − ) → R + denote the Gaussian multi-bubble model profile: I ( q − ) m ( v ) ∶ = P γ ( Ω m ) where Ω m is a simplicial q -cluster with γ ( Ω m ) = v (see Section 2 for why this is well-defined). It is not too hard to show that I ( q − ) m maybe extended continuously to the entire ∆ ( q − ) by setting I ( q − ) m ( v ) ∶ = I ( k ) m ( v J ) for v ∈ J ,where J is a k -dimensional face of ∂ ∆ ( q − ) , and v J is the natural restriction of v to thecoordinates defined by J . Our goal is to show that I ( q − ) = I ( q − ) m on int ∆ ( q − ) ; we clearlyhave I ( q − ) ≤ I ( q − ) m on ∆ ( q − ) , and we may assume that equality occurs on the boundaryby induction on q . A direct calculation verifies that: ∇ I ( q − ) m ( v ) = − L − A m ( v ) on int ∆ ( q − ) , (1.3)where A m ( v ) = { A mij ( v )} denote the weighted areas of the model cluster Ω m ’s interfaces.Consequently, (1.2) and (1.3) imply that: − tr [( ∇ I ( q − ) ( v )) − ] ≤ tr ( L A ( v ) ) = ∑ i < j A ij ( v ) = I ( q − ) ( v ) on int ∆ ( q − ) (1.4)with equality when I ( q − ) and A are replaced by I ( q − ) m and A m , respectively. Since I ( q − ) ≤ I ( q − ) m on ∆ ( q − ) with equality on the boundary, an application of the maximum principle forthe (fully non-linear) second-order elliptic PDE (1.4) will yield the desired I ( q − ) = I ( q − ) m .The bulk of this work is thus aimed at establishing a rigorous (approximate) version of(1.2). To this end, we consider an isoperimetric minimizing cluster Ω, and perturb it usinga flow F t along an (admissible) vector-field X . Since: I ( q − ) ( γ ( F t ( Ω ))) ≤ P γ ( F t ( Ω )) (1.5)with equality at t =
0, we deduce (at least, conceptually) that the first variations mustcoincide and that the second variations must obey the inequality (this idea is well-knownin the single-bubble setting, see e.g. [56, 45, 34, 51, 9, 33, 8]). Contrary to the single-bubblesetting, the interfaces Σ ij = ∂ ∗ Ω i ∩ ∂ ∗ Ω j will meet each other at common boundaries, whichmay contribute to these variations. Fortunately, for the first variation of perimeter, thesecontributions cancel out thanks to the isoperimetric stationarity of the cluster. However,for the second variation, the boundary of the interfaces will in general have a contribution,involving moreover the boundary curvature. e thus arrive to the crucial difference between the case q > q = ( q − ) -dimensional inequality (1.2), we need to use q − X (with non-trivial action) in (1.5). If we a-priori know that for some k ≥ q − q -clusterΩ in R n is effectively k -dimensional, meaning that the normals to its interfaces span a k -dimensional space N , the idea in [40] is to simply use all constant vector-fields X ≡ w with w ∈ N , amounting to translation of the cluster Ω in the non-trivial directions; in thatcase, the second variation formulae are especially simple, the curvatures of the interfacesand their boundary do not play any role, and the desired (1.5) is fairly easily deduced in[40] by an application of a matrix Cauchy-Schwarz inequality. Unfortunately, we do notknow how to a-priori rule out that the minimizing q -cluster might be effectively lowerdimensional (even though a-posteriori , by Theorem 1.2, we know that this is impossible).When q =
3, the only other possibility is that the cluster is effectively one-dimensional,in which case (1.5) was established in [40] by a direct one-dimensional construction of atwo-parameter family of (no longer constant) vector-fields. However, when q >
3, we needto also consider the possibility that the cluster is effectively k -dimensional for 1 < k < q − k degrees of information, whereas we require q − k =
2, and becomesa genuine challenge for k = k ≥
4, as we are required to use the fullcodimension-1 and 2 regularity theory for the interface strata of a minimizing cluster in R k to construct our ( q − ) -parameter family of k -dimensional vector-fields (in practice,to avoid referring to k , we perform the construction on R n ). It is precisely this a-prioripossibility which prevents us from repeating the argument from [40], and forces us todevelop an alternative and much more complicated strategy for deriving (1.5). The addedvalue of this alternative strategy is that we are forced to gain a deeper understanding ofthe structure of stable regular clusters. Let us detail some of the new ingredients which we employ to establish (1.5): • Classical results from Geometric Measure Theory due to Almgren [3] (see also Maggi’sexcellent book [37]) ensure the existence of isoperimetric minimizing clusters and C ∞ regularity of their interfaces. However, we will in addition require regularity andstructural information on the codimension-1 and 2 boundary (as a manifold) of theinterfaces Σ = ⋃ i < j Σ ij , as well as information on the negligibility of the residualtopological boundary. Such results have been obtained by various authors: F. Morganwhen n = n = , n ≥
4. Together with ellipticregularity results of Kinderlehrer, Nirenberg and Spruck [32], these results implythat the codimension-1 part of the boundary of the interfaces (denoted Σ ) consistsof ( n − ) -dimensional C ∞ manifolds where three of the cells locally meet (like thecells of a simplicial 3-cluster); the codimension-2 part of the boundary (denoted Σ )consists of ( n − ) -dimensional C ,α manifolds where four of the cells locally meet (likethe cells of a simplicial 4-cluster); and the residual topological boundary Σ satisfies H n − ( Σ ) =
0. Being slightly inaccurate in this introductory section, we call suchclusters “regular”. • For any stationary regular cluster with respect to a measure with smooth positivedensity on R n , we establish new integrability properties of the curvature of Σ , whichmay a-priori be blowing up near Σ . We show in Proposition 4.14 that the curvature isin L ( Σ ) ∩ L ( Σ ) . This is achieved using Schauder estimates for elliptic systemsof PDEs, following the reflection technique of Kinderlehrer–Nirenberg–Spruck [32]. The formula for the second variation of weighted area under a volume constraint for ageneral cluster and admissible vector-field X involves integration over Σ ij of 7 terms, 4of which are divergences of some vector-fields Y ij . In order to justify integrating thesedivergence terms by parts on Σ ij , replacing them with integrals on its codimension-1 boundary ∂ Σ ij , we establish a version of Stokes’ theorem on the manifold-with-boundary M ij ∶ = Σ ij ∪ ∂ Σ ij for vector-fields which are non-compactly supported andwhich may be blowing up near M ij ∖ M ij . The fact that the latter set has locally-finite H n − -measure, enables us to handle vector-fields in L ( Σ ij , γ n − ) ∩ L ( ∂ Σ ij , γ n − ) whose divergence is in L ( Σ ij , γ n − ) ; it turns out that whenever X is compactly-supported and avoids the singular part Σ , the Y ij ’s will indeed belong to this classthanks to the integrability properties of our curvatures. The isoperimetric stationarityof our minimizing cluster ensures that the contribution of 3 out of the 4 divergenceterms vanishes, yielding an elegant final formula for the second variation of weightedarea under a volume constraint: ∑ i < j [ ∫ Σ ij (∣ ∇ t X n ij ∣ − ( X n ij ) ∥ II ij ∥ − ( X n ij ) ) dγ n − − ∫ ∂ Σ ij X n ij X n ∂ij II ij∂,∂ dγ n − ] (1.6)(see Theorem 5.2 for the precise assumptions on the cluster Ω, the vector-field X ,and the notation used above). • Our ( q − ) -parameter family of vector-fields is constructed as follows. For each ofthe q cells Ω i , we would like to construct an “inward field”: a vector-field X i on R n so that X i ’s normal component X n i is constant 1 on Σ ∩ ∂ Ω i with respect to theinward normal to Ω i , and constant 0 on Σ ∖ ∂ Ω i . The family is obtained by takinglinear combinations of the X i ’s, and one degree of freedom is obviously lost because ∑ qi = X n i ≡ . The idea behind this particular choice of vector-fields is thatif X = ∑ qi = a i X i , F t is the flow along X , and n ij denotes the unit-normal to Σ ij pointing from Ω i to Ω j , then: δ X V ( Ω ) ∶ = ddt ∣ t = γ ( F t ( Ω )) = ⎛⎝ ∑ j ≠ i ∫ Σ ij X n ij dγ n − ⎞⎠ i = ( ∑ j ≠ i A ij ( a j − a i )) i = − L A a, which already shows that the family is always ( q − ) -dimensional (as L A is positivedefinite and hence full-rank on E ( q − ) ). In addition, when the interfaces have vanish-ing curvature (such as for a model simplicial cluster), the expression in (1.6) for thesecond variation of weighted area under a volume constraint reduces to: ∑ i < j ∫ Σ ij (∣ ∇ t X n ij ∣ − ( X n ij ) ) dγ n − = − ∑ i < j A ij ( a j − a i ) = − a T L A a = − ( δ X V ) T L − A δ X V, (1.7)revealing the relation to (1.2). The construction of our inward fields makes heavy useof the structure of Σ and Σ where the different cells meet in threes and in fours, aswell as of the negligibility of Σ . In practice, we can only ensure the above propertyof the X i ’s approximately (in a variety of senses detailed in Proposition 6.1) • All of the above components are then used in the proof of Theorem 7.2, which isthe main new ingredient in this work and of independent interest. In particular, itverifies:
Theorem 1.3 (Stable Regular Clusters) . A stable regular q -cluster in R n ( ≤ q ≤ n + ) with respect to γ has convex polyhedral cells (up to null-sets), and effectivedimension at most q − . ote that in the single-bubble case q =
2, this implies that the cells must be com-plementary halfspaces. If instead of regularity one makes the stronger assumptionthat the entire topological boundary of the single-bubble is smooth, the case q = q ≥ q = X which violatesstability, i.e. decreases area in second-order while maintaining the volume constraint.As a first step, we take a linear combination X = ∑ qi = a i X i of our inward fields, andshow that there is some linear combination so that the contribution of the curvatureterms in formula (1.6) for the second variation is strictly negative; the other termswork in our favor. The problem is to ensure that the volume constraint is preservedto first order: δ X V =
0. When the cluster is full-dimensional (or just effectively ( q − ) -dimensional), then the map R n ∋ w ↦ δ w V ∈ E ( q − ) is surjective (where wethink of w as a constant vector-field on R n ), and we can make sure that δ X V = X by an appropriate w ∈ R n . However, as already explained, we do nota-priori know that a minimizing cluster will be effectively ( q − ) -dimensional, andmoreover, a general stable cluster may very well be effectively k -dimensional with k < q −
1. To handle the case that the cluster Ω is dimension-deficient, we write Ω as˜Ω × R , and perform the above construction of ˜ X for ˜Ω in R n − ; we then define X to bethe product of ˜ X and a linear function on the one-dimensional fiber, yielding a flow F t which skews the cluster out of its product state. By oddness of the linear function,we are ensured that δ X V =
0, and it turns out that, by the Poincar´e inequality forthe one-dimensional Gaussian measure (which is saturated by linear functions), wehave just enough room to still get a negative overall second variation. This yields acontradiction to stability, and shows that a stable regular cluster must be flat.Stability similarly implies connectedness of the cells; convexity follows since the flatinterfaces meet on their codimension-1 boundary at 120 ○ angles while their lower-dimensional boundary satisfies H n − ( Σ ∪ Σ ) = • Once we know that a stable regular cluster (and in particular, a minimizing cluster) isflat, the formula (1.6) simplifies to (1.7), and we can use the entire ( q − ) -dimensionalfamily of inward fields to deduce a rigorous (approximate) version of the ( q − ) -dimensional PDI (1.2) – see Proposition 8.1.Once Theorem 1.1 is proved, uniqueness of minimizers is established as in [40]: weobserve that all of our inequalities in the derivation of (1.2) must have been equalities,and this already provides enough information for characterizing simplicial-clusters; theconvexity of the cells allows for a simplification of the argument from [40]. An alternativeapproach for establishing uniqueness is presented in Section 10: with any stable regular q -cluster on R n (2 ≤ q ≤ n +
1) we associate a two-dimensional simplicial complex andshow that its first (simplicial) homology must be trivial; from this we deduce that such acluster must be the pull-back of a shifted canonical model cluster on E ( q − ) by a linear-maphaving certain properties (see Theorem 10.1); for a minimizing cluster, we show that thismap must be conjugate to an isometry, and hence the cluster must itself be a simplicialcluster.The rest of this work is organized as follows. In Section 2, we construct the model implicial-clusters and associated model isoperimetric profile, and establish their proper-ties. In Section 3 we recall relevant definitions and provide some preliminaries for theensuing calculations. In Section 4 we collect the results we require regarding isoperimetricminimizing clusters; the proof of the curvature integrability is deferred to Appendix B.In Section 5 we derive a generalized version of Stokes’ theorem; we then apply it to thegeneral formula for the second variation of weighted area under a volume constraint, whosecalculation is deferred to Appendix A, and obtain the simplified formula (1.6) for regularstationary clusters. In Section 6 we construct our inward fields and record some of theiruseful properties. In Section 7 we prove that stable regular q -clusters in R n have convexpolyhedral cells when 2 ≤ q ≤ n +
1. In Section 8 we establish an (approximate) rigorousversion of (1.2) and prove Theorem 1.1. In Section 9 we prove Theorem 1.2. In Section 10we provide some concluding remarks and highlight some questions which remain open.
Acknowledgement.
We thank Francesco Maggi for his continued interest and support,Frank Morgan for many helpful references, and Brian White for informing us of the referenceto the recent [19]. We also acknowledge the hospitality of MSRI where part of this workwas conducted.
In this section, we construct the model q -clusters which are conjectured to be optimalon R n , for all q ≥ n ≥ q −
1. It will be enough to construct them on R q − , since bytaking Cartesian product with R n − q + and employing the product structure of the Gaussianmeasure, these clusters extend to R n for all n ≥ q −
1. Actually, it will be convenient toconstruct them on E ( q − ) , rather than on R q − , where recall that E = E ( q − ) denotes thetangent space to ∆ = ∆ ( q − ) , which we identify with { x ∈ R q ∶ ∑ qi = x i = } . Strictlyspeaking, we will construct them on the dual space E ∗ , but we will freely identify between E and E ∗ via the standard Euclidean structure. Consequently, in this section, let γ = γ q − denote the standard ( q − ) -dimensional Gaussian measure on E , and if ϕ = e − W denotesits (smooth, Gaussian) density on E , set γ q − = γ q − q − ∶ = ϕ H q − .We will frequently use the fact that if Z ∈ E ( q − ) is distributed according to γ q − , then: √ qq − Z i and Z i − Z j √ i ≠ j ) are distributed according to γ . (2.1)(for the former statement, note that the orthogonal projection of e i onto E ( q − ) is e i − q ∑ qj = e j which has norm √ q − q ). We use Φ to denote the one-dimensional Gaussiancumulative distribution function Φ ( t ) = γ ( −∞ , t ] .Define Ω mi = int { x ∈ E ∶ max j x j = x i } (“ m ” stands for “model”). For any x ∈ E , x + Ω m = ( x + Ω m , . . . , x + Ω mq ) is a q -cluster, which we call a canonical model simplicial cluster, asit corresponds to the Voronoi cells of the q -equidistant points { x + e i − q ∑ j e j } i = ,...,q ⊂ E .Let Σ mij ∶ = ∂ Ω mi ∩ ∂ Ω mj denote the codimension-1 interfaces of Ω m . Observe that x + Σ mij ,the interfaces of x + Ω m , are flat, and meet at 120 ○ angles along codimension-2 flat surfaces.We denote: A mij ( x ) ∶ = γ q − ( x + Σ mij ) . Lemma 2.1.
The map: E ∋ x ↦ Ψ ( x ) ∶ = γ ( x + Ω m ) ∈ int ∆ s a diffeomorphism between E and int ∆ . Its differential is given by: D Ψ ( x ) = − √ L A m ( x ) , (2.2) where, recall, L A was defined in (1.1).Proof. Clearly Ψ ( x ) is C ∞ , since the Gaussian density is C ∞ and all of its derivativesvanish rapidly at infinity.To see that Ψ is injective, simply note that if y ≠ x , then there exists i ∈ { , . . . , q } suchthat y ∈ x + Ω mi , and as Ω mi is a convex cone, it follows that y + Ω mi ⊊ x + Ω mi and hence γ ( y + Ω mi ) < γ ( x + Ω mi ) .We now compute: ∇ v Ψ i ( x ) = ∫ x + Ω mi ∇ v e − W ( y ) dy = ∫ x + ∂ Ω mi ⟨ v, n ⟩ e − W ( y ) d H q − , where n denotes the outward unit-normal. Note that ∂ Ω mi = ⋃ j ≠ i Σ mij (the union is disjointup to H q − -null sets), and that the outward unit-normal on Σ mij is the constant vector-field ( e j − e i )/√
2. Therefore, ∇ v Ψ i ( x ) = √ ∑ j ≠ i ⟨ v, e j − e i ⟩ A mij ( x ) = − √ v T L A m ( x ) e i , thereby confirming (2.2). Since each of the A mij ( x ) is strictly positive, L A m ( x ) is non-singular (in fact, strictly positive-definite) as a quadratic form on E , and hence D Ψ ( x ) isnon-singular as well. It follows that Ψ is a diffeomorphism onto its image.Finally, to show that Ψ is surjective, fix R > K R = { x ∈ E ∶ max i x i < R } . For any x ∈ ∂K R , there is some i with x i = R , and hence x + Ω mi = { z ∈ E ∶ z i − x i ≥ max j ( z j − x j )} = { z ∈ E ∶ z i − R ≥ max j ( z j − x j )} ⊂ { z ∈ E ∶ z i − R ≥ } , since z − x ∈ E implies that max j ( z j − x j ) ≥
0. It follows by (2.1) that for all x ∈ ∂K R , γ ( x + Ω mi ) ≤ − Φ (√ q /( q − ) R ) < − Φ ( R ) . In other words, if we consider the shrunkensimplex ∆ R = { v ∈ ∆ ∶ min i v i ≥ − Φ ( R )} , then Ψ ( ∂K R ) ∩ ∆ R = ∅ . On the other hand,Ψ ∶ K R → Ψ ( K R ) is continuous and injective, and is therefore a homeomorphism by theInvariance of Domain theorem [46]. Since ∂ Ψ ( K R ) = Ψ ( ∂K R ) is disjoint from ∆ R , itfollows that ∆ R ∩ Ψ ( K R ) and ∆ R ∖ Ψ ( K R ) are two complementing relatively-open subsetsof the connected set ∆ R . Since Ψ ( ) = ( q , . . . , q ) ∈ ∆ R for R > R ⊂ Ψ ( K R ) . Taking R → ∞ , we see that Ψ ∶ E → int ∆ is surjective.We can now give the following definition, which is well-posed according to the bijectionestablished in Lemma 2.1. Adopting the jargon from the Euclidean setting, note that acluster with q cells corresponds to q − Definition 2.2.
The model isoperimetric ( q − ) -bubble profile I m ∶ int ∆ → R is definedas: I m ( v ) ∶ = P γ ( x + Ω m ) = ∑ i < j A mij ( x ) for x such that Ψ ( x ) = γ ( x + Ω m ) = v. Ω Ω Ω Ω Figure 2: Given the configuration on the left, we delete the smallchamber and extend the large chambers, as seen on the right.
When we need to explicitly refer to the number of bubbles involved, we will write I ( q − ) m and ∆ ( q − ) instead of I m and ∆, respectively. For completeness, we define I ( ) m ( ) = R n (instead of canonical ones on E ).Thanks to the next lemma, we may (and do) extend I m by continuity to the entire ∆.Given J ⊂ { , . . . , q } with 0 < ∣ J ∣ < q , let ∆ J denote the face of ∆ consisting of all v ∈ ∆such that { i ∶ v i > } = J . Given v ∈ ∆ J , let v J denote its natural projection to int ∆ (∣ J ∣− ) obtained by only keeping the coordinates in J . Lemma 2.3. I ( q − ) m is C ∞ on int ∆ ( q − ) , and continuous up to ∂ ∆ ( q − ) . Moreover, thecontinuous extension of I ( q − ) m satisfies I ( q − ) m ( v ) = I (∣ J ∣− ) m ( v J ) whenever v ∈ ∆ ( q − ) J , for all J ⊂ { , . . . , q } with < ∣ J ∣ < q . The main idea in the proof of Lemma 2.3 is to take a non-degenerate q -simplicial clusterwith a few small cells, and compare it to the lower-dimensional simplicial cluster obtainedby deleting the small cells and “extending” the large cells in the natural way (see Figure 2for an illustration). By comparing the measure and perimeter of these two clusters, we cansee how I m behaves near ∂ ∆. Proof of Lemma 2.3.
Since I m ( v ) = P γ ( Ψ − ( v ) + Ω m ) and both Ψ − and the map x ↦ P γ ( x + Ω m ) are C ∞ on their respective domains, it follows that I m is C ∞ on int ∆.For the rest of this proof, we write Ω = Ω q for the model q -cluster Ω m on E = E ( q − ) , andrecall the definition of Ψ = Ψ ( q ) from Lemma 2.1. Given 0 < k < q , consider the embedding E ( k − ) ⊂ E ( q − ) by padding with zeros the last q − k coordinates. Let Π k ∶ E ( q − ) → E ( k − ) denote the orthogonal projection onto E ( k − ) , defined by ( Π k x ) i = x i − k ∑ kj = x j for i = , . . . , k .We now define the q -cluster Ω q ← k on E ( q − ) by taking Ω q ← ki empty for i > k , andΩ q ← ki = int { z ∈ E ( q − ) ∶ max ≤ j ≤ k z j = z i } otherwise. In other words, Ω q ← k = Π − k ( Ω k ) . Observe that Ω i ⊂ Ω q ← ki if i ≤ k , and so by theproduct structure of the Gaussian measure, γ ( x + Ω i ) ≤ Ψ ( k ) i ( Π k x ) ∀ i ≤ k. (2.3)Since the functions I ( q − ) are symmetric in their arguments for every q , it suffices toprove that if { v m } m ⊂ int ∆ ( q − ) is a convergent sequence with ( v m, , . . . , v m,k ) → v ∈ nt ∆ ( k − ) then I ( q − ) ( v m ) → I ( k − ) ( v ) . So, fix such a sequence, and define: x m ∶ = Ψ − ( v m ) ∈ E ( q − ) , y ∶ = ( Ψ ( k ) ) − ( v ) ∈ E ( k − ) , y m ∶ = Π k x m ∈ E ( k − ) . First, we observe that y m → y . Indeed, (2.3) implies that for i ≤ k , v i = lim inf m → ∞ Ψ i ( x m ) ≤ lim inf m → ∞ Ψ ( k ) i ( y m ) . But the inequality cannot be strict, since ∑ ki = v i = k ∑ i = lim inf m → ∞ Ψ ( k ) i ( y m ) ≤ lim inf m → ∞ k ∑ i = Ψ ( k ) i ( y m ) = . It follows that Ψ ( k ) ( y m ) → v, (2.4)and since v = Ψ ( k ) ( y ) and Ψ ( k ) ∶ E ( k − ) → ∆ ( k − ) is a diffeomorphism, we must have y m → y .Next, we claim that for every i ≤ k and j > k , x m,j − x m,i → ∞ . Assume in thecontrapositive this is not the case. Then we can choose C > i ≤ k , and j > k such that(after passing to a subsequence) x m,j = min ℓ > k x m,ℓ and x m,j − x m,i ≤ C for all m . Since y m = Π k x m converges, it follows that the first k coordinates of x m differ by at most aconstant independent of m . Hence, x m,j ≤ min ℓ ≤ k x m,ℓ + C ′ (for a different constant C ′ ).Since x m,j ≤ x m,ℓ for ℓ > k , we have x m,j ≤ min ℓ = ,...,q x m,ℓ + C ′ . But then x m + Ω j = { z ∈ E ∶ z j − x m,j > min ℓ ≠ j ( z ℓ − x m,ℓ )} ⊃ { z ∈ E ∶ z j − x m,j > min ℓ ≠ j ( z ℓ + C ′ − x m,j )} = { z ∈ E ∶ z j > min ℓ ≠ j z ℓ − C ′ } . The Gaussian measure of the latter set is strictly positive, in contradiction to the assump-tion that v m = Ψ ( x m ) = γ ( x m + Ω ) satisfies ( v m, , . . . , v m,k ) → v ∈ int ∆ ( k − ) , which inparticular implies that v m,j → j > k .Having understood something about x m = Ψ − ( v m ) , we turn to the interfaces x m + Σ ij .Recall that the interfaces Σ ij of Ω are given by Σ ij = { z ∈ E ( q − ) ∶ z i = z j = max ℓ z ℓ } , whilethe interfaces Σ q ← kij of Ω q ← k are given byΣ q ← kij = ⎧⎪⎪⎨⎪⎪⎩{ z ∈ E ( q − ) ∶ z i = z j = max ℓ ≤ k z ℓ } if i, j ≤ k ∅ otherwise . In particular, Σ q ← kij ⊃ Σ ij as long as i, j ≤ k . On the other hand,Σ q ← kij ⊂ Σ ij ∪ ⋃ ℓ > k { z ∈ E ( q − ) ∶ z i = z j and z ℓ > z i } , meaning that x + Σ q ← kij ⊂ ( x + Σ ij ) ∪ ⋃ ℓ > k { z ∈ E ( q − ) ∶ z i − z j = x i − x j and z ℓ − z i > x ℓ − x i } . Recalling that i ≤ k and ℓ > k implies that x m,ℓ − x m,i → ∞ , we have γ q − ({ z ∈ E ( q − ) ∶ z i − z j = x m,i − x m,j and z ℓ − z i > x m,ℓ − x m,i }) → or every ℓ > k , and so it follows that ∣ γ q − ( x m + Σ ij ) − γ q − ( x m + Σ q ← kij )∣ → m → ∞ for every i, j ≤ k . By the product structure of the Gaussian measure, (2.4), andcontinuity of I ( k − ) on int ∆ ( k − ) , we know that: ∑ ≤ i < j ≤ k γ q − ( x m + Σ q ← kij ) = ∑ ≤ i < j ≤ k γ k − k − ( y m + Σ kij ) = I ( k − ) ( Ψ ( k ) ( y m )) → I ( k − ) ( v ) , and hence we deduce by (2.5) that: ∑ ≤ i < j ≤ k γ q − ( x m + Σ ij ) → I ( k − ) ( v ) . To conclude that I ( q − ) ( v m ) → I ( k − ) ( v ) , we will show that the remaining terms of I ( q − ) ( v m ) = P γ ( x m + Ω ) = ∑ ≤ i < j ≤ q γ q − ( x m + Σ ij ) are negligible: γ q − ( x m + Σ ij ) → i > k (and by symmetry, we will reach the same conclusion whenever either i > k or j > k ). For j ≤ k , this follows from the inclusion x m + Σ ij = { z ∈ E ∶ z i − x m,i = z j − x m,j = max ℓ ( z ℓ − x m,ℓ )} ⊂ { z ∈ E ∶ z i − z j = x m,i − x m,j } , and the fact that x m,i − x m,j → ∞ , which implies by (2.1) that: γ q − ( x m + Σ ij ) ≤ γ { x m,i − x m,j √ } → . For j > k , it follows similarly by fixing a single ℓ ≤ k and using the inclusion x m + Σ ij ⊂ { z ∈ E ∶ z i − x m,i = z j − x m,j ≥ z ℓ − x m,ℓ } = { z ∈ E ∶ z i − z j = x m,i − x m,j and z i − z ℓ ≥ x m,i − x m,ℓ } , noting again that x m,i − x m,ℓ → ∞ .We have constructed the canonical model simplicial ( k − ) -bubble clusters Ω in E ( k − ) with γ k − ( Ω ) = v for all v in the interior int ∆ ( k − ) and for all k ≥
2. For k =
1, the trivial0-bubble (1-cell) cluster on E ( ) = { } is Ω = { } . Clearly, by taking Cartesian productswith E ( q − k + ) and employing the product structure of the Gaussian measure, we obtainthe canonical model ( q − ) -bubble clusters Ω in E ( q − ) with γ q − ( Ω ) = v for all v ∈ ∆ ( q − ) .Similarly, these clusters extend to R n for all n ≥ q −
1. Consequently, I ( v ) ≤ I ( q − ) m ( v ) forall v ∈ ∆ ( q − ) and n ≥ q −
1, and our main goal in this work is to establish the converseinequality.To this end, we observe that the model profile I m = I ( q − ) m satisfies a remarkable differen-tial equation (in the double-bubble case q =
3, this was already derived in [40, Proposition2.6]).
Proposition 2.4.
At any point in int ∆ , ∇ I m = √ − and ∇ I m = − ( L A m ○ Ψ − ) − as tensors on E , where, recall, L A was defined in (1.1). roof. Recalling that A mij ( x ) = ∫ x + Σ mij e − W d H q − , we have ∇ v A mij ( x ) = ∫ x + Σ mij ∇ v e − W d H q − . Decomposing v into its normal and tangential components and applying Stokes’ theoremto the tangential part (which is justified since the Gaussian density decays faster than anypolynomial) ∇ v A mij ( x ) = − ∫ x + Σ mij ⟨ v, n ij ⟩⟨ n ij , y ⟩ e − W ( y ) d H q − + ∫ x + ∂ Σ mij ⟨ v, n ∂ij ⟩ e − W ( y ) d H q − , where n ij = √ e j − e i is the unit-normal to Σ mij , ∂ Σ mij denotes the ( q − ) -dimensionalboundary of Σ mij in the manifold sense, and n ∂ij is the boundary outer unit-normal. Now, ⟨ n ij , y ⟩ = ⟨ n ij , x ⟩ for all y ∈ x + Σ mij . Note also that the ( q − ) -dimensional interfaces meetat their ( q − ) -dimensional boundaries in threes, and that the boundary outer normals atthese points sum to zero. Hence, summing over all i < j , the total contribution of the lastintegral above vanishes, and we are left with: ∇ v ∑ i < j A mij ( x ) = − ∑ i < j ⟨ v, n ij ⟩⟨ n ij , x ⟩ A mij ( x ) = − ∑ i < j ⟨ v, e j − e i ⟩⟨ e j − e i , x ⟩ A mij ( x ) = − v T L A m x. Finally, recall that I m = ∑ i < j A mij ○ Ψ − . By (2.2), D ( Ψ − ) = ( D Ψ ) − ○ Ψ − = − √ ( L A m ○ Ψ − ) − ,and so the chain rule implies that ∇ I m = √ − , thus establishing the first claim. The second claim follows from differentiating the firstclaim. We will be working in Euclidean space ( R n , ∣ ⋅ ∣) endowed with a measure µ = µ n having C ∞ -smooth and strictly-positive density e − W with respect to Lebesgue measure; we willnot restrict ourselves to the Gaussian measure γ until reaching Section 7. Recall that thecells { Ω i } i = ,...,q of a cluster Ω are assumed to be pairwise disjoint Borel subsets of R n ,and satisfy µ ( R n ∖ ∪ qi = Ω i ) =
0. In addition, they are assumed to have finite µ -weightedperimeter P µ ( Ω i ) < ∞ , to be defined below. Given distinct i, j, k ∈ { , . . . , q } , we define theset of cyclically ordered pairs in { i, j, k } : C ( i, j, k ) ∶ = {( i, j ) , ( j, k ) , ( k, i )} . We write div X to denote divergence of a smooth vector-field X , and div µ X to denote itsweighted divergence: div µ X ∶ = div ( Xe − W ) e + W = div X − ∇ X W. (3.1) or a smooth hypersurface Σ ⊂ R n co-oriented by a unit-normal field n , let H Σ ∶ Σ → R denote its mean-curvature, defined as the trace of its second fundamental form II Σ . Theweighted mean-curvature H Σ ,µ is defined as: H Σ ,µ ∶ = H Σ − ∇ n W. We write div Σ X for the surface divergence of a vector-field X defined on Σ, i.e. ∑ n − i = ⟨ t i , ∇ t i X ⟩ where { t i } is a local orthonormal frame on Σ; this coincides with div X − ⟨ n , ∇ n X ⟩ for anysmooth extension of X to a neighborhood of Σ. The weighted surface divergence div Σ ,µ isdefined as: div Σ ,µ X = div Σ X − ∇ X W, so that div Σ ,µ X = div Σ ( Xe − W ) e + W if X is tangential to Σ. Note that div Σ n = H Σ anddiv Σ ,µ n = H Σ ,µ . We will also abbreviate ⟨ X, n ⟩ by X n , and we will write X t for thetangential part of X , i.e. X − X n n .We will frequently use that:div Σ ,µ X = div Σ ,µ ( X n n ) + div Σ ,µ X t = H Σ ,µ X n + div Σ ,µ X t . Note that the above definitions ensure the following weighted version of Stokes’ theorem(which we interchangeably refer to in this work as the Gauss–Green or divergence theorem):if Σ is a smooth manifold with C boundary, denoted ∂ Σ, (completeness of Σ ∪ ∂ Σ is notrequired), and X is a smooth vector-field on Σ, continuous up to ∂ Σ, with compact supportin Σ ∪ ∂ Σ, then: ∫ Σ div Σ ,µ Xdµ n − = ∫ Σ H Σ ,µ X n dµ n − + ∫ ∂ Σ X n ∂ dµ n − , (3.2)where n ∂ denotes the exterior unit-normal to ∂ Σ, and: µ k ∶ = e − W H k . Given a Borel set U ⊂ R n with locally-finite perimeter, its reduced boundary ∂ ∗ U is defined(see e.g. [37, Chapter 15]) as the subset of ∂U for which there is a uniquely defined outerunit normal vector to U in a measure theoretic sense. While the precise definition will notplay a crucial role in this work, we provide it for completeness. The set U is said to havelocally-finite (unweighted) perimeter, if for any compact subset K ⊂ R n we have:sup { ∫ U div X dx ∶ X ∈ C ∞ c ( R n ; T R n ) , supp ( X ) ⊂ K , ∣ X ∣ ≤ } < ∞ . With any Borel set with locally-finite perimeter one may associate a vector-valued Radonmeasure µ U on R n , called the Gauss–Green measure, so that: ∫ U div X dx = ∫ R n ⟨ X, dµ U ⟩ ∀ X ∈ C ∞ c ( R n ; T R n ) . The reduced boundary ∂ ∗ U of a set U with locally-finite perimeter is defined as the collec-tion of x ∈ supp µ U so that the vector limit: n U ∶ = lim ǫ → + µ U ( B ( x, ǫ ))∣ µ U ∣ ( B ( x, ǫ )) exists and has length 1 (here ∣ µ U ∣ denotes the total-variation of µ U and B ( x, ǫ ) is the openEuclidean ball of radius ǫ centered at x ). When the context is clear, we will abbreviate n U y n . Note that modifying U on a null-set does not alter µ U nor ∂ ∗ U . It is known that ∂ ∗ U is a Borel subset of ∂U and that ∣ µ U ∣ ( R n ∖ ∂ ∗ U ) =
0. If U is an open set with C smooth boundary, it is known (e.g. [37, Remark 15.1]) that ∂ ∗ U = ∂U .Recall that the µ -weighted perimeter of U was defined in the Introduction as: P µ ( U ) ∶ = sup { ∫ U div µ X dµ ∶ X ∈ C ∞ c ( R n ; T R n ) , ∣ X ∣ ≤ } . Clearly, if U has finite weighted-perimeter P µ ( U ) < ∞ , it has locally-finite (unweighted)perimeter. It is known [37, Theorem 15.9] that in that case: P µ ( U ) = µ n − ( ∂ ∗ U ) . In addition, by the Gauss–Green–De Giorgi theorem, the following integration by partsformula holds for any C c vector-field X on R n , and Borel subset U ⊂ R n with locally finiteperimeter (see [37, Theorem 15.9] and recall (3.1)): ∫ U div µ X dµ n = ∫ ∂ ∗ U X n dµ n − . (3.3)Given a cluster Ω = ( Ω , . . . , Ω q ) , we define the interface between cells i and j (for i ≠ j )as: Σ ij = Σ ij ( Ω ) ∶ = ∂ ∗ Ω i ∩ ∂ ∗ Ω j , and we define: A ij = A ij ( Ω ) ∶ = µ n − ( Σ ij ) . It is standard to show (see [37, Exercise 29.7, (29.8)]) that for any S ⊂ { , . . . , q } : H n − ( ∂ ∗ ( ∪ i ∈ S Ω i ) ∖ ∪ i ∈ S,j ∉ S Σ ij ) = . (3.4)In particular: H n − ( ∂ ∗ Ω i ∖ ∪ j ≠ i Σ ij ) = ∀ i = , . . . , q, (3.5)and hence: P µ ( Ω i ) = ∑ j ≠ i A ij ( Ω ) , and: P µ ( Ω ) = q ∑ i = P µ ( Ω i ) = ∑ i < j A ij ( Ω ) . In addition, if follows that: ∀ i ∂ ∗ Ω i = ∪ j ≠ i Σ ij . (3.6)Indeed, since H n − ∣ ∂ ∗ Ω i = H n − ∣ ∪ j ≠ i Σ ij they must have the same support. But the supportof H n − ∣ ∪ j ≠ i Σ ij is clearly contained in the right-hand-side of (3.6), whereas H n − ∣ ∂ ∗ Ω i = ∣ µ Ω i ∣ by [37, Theorem 15.9], and hence its support is (e.g. [37, Remark 15.3]) the left-hand-sideof (3.6); the converse inclusion is trivial. Definition 3.1 (Admissible Vector-Fields) . A vector-field X on R n is called admissible ifit is C ∞ -smooth and satisfies ∀ i ≥ x ∈ R n ∥ ∇ i X ( x )∥ ≤ C i < ∞ . ny smooth, compactly supported vector-field is clearly admissible, and so is the sumof a constant vector field and a smooth, compactly supported vector field (these will bemain kinds of vector fields that we will use).Let F t denote the associated flow along an admissible vector-field X , defined as thefamily of maps { F t ∶ R n → R n } solving the following ODE: ddt F t ( x ) = X ○ F t ( x ) , F ( x ) = x. (3.7)It is well-known that a unique smooth solution in t ∈ R exists for all x ∈ R n , and that theresulting maps F t ∶ R n → R n are C ∞ diffeomorphisms, so that the partial derivatives in t and x of any fixed order are uniformly bounded in ( x, t ) ∈ R n × [ − T, T ] , for any fixed T > = ( Ω , . . . , Ω q ) is a cluster then so is its image F t ( Ω ) = ( F t ( Ω ) , . . . , F t ( Ω q )) :obviously, its cells remain Borel and pairwise disjoint; the fact that they have finite µ -weighted perimeter and satisfy µ ( R n ∖ ∪ i F t ( Ω i )) = µ ( F t ( R n ∖ ∪ i Ω i )) = F t is a Lipschitz map.We define the r -th variations of weighted volume and perimeter of Ω as: δ rX V ( Ω ) ∶ = ( ddt ) r ∣ t = µ ( F t ( Ω )) ,δ rX A ( Ω ) ∶ = ( ddt ) r ∣ t = P µ ( F t ( Ω )) , whenever the right-hand sides exist. When Ω is clear from the context, we will simply write δ rX V and δ rX A ; when r =
1, we will write δ X V and δ X A .It will be of crucial importance for us in this work to calculate the first and especiallysecond variations of weighted volume and perimeter for non- compactly supported vector-fields, for which even the existence of δ rX V ( Ω ) and especially δ rX A ( Ω ) is not immediatelyclear. Indeed, even for the case of the standard Gaussian measure, the derivatives of its den-sity are asymptotically larger at infinity than the Gaussian density itself. We consequentlyintroduce the following: Definition 3.2 (Volume / Perimeter Regular Set) . A Borel set U is said to be volumeregular with respect to the measure µ = e − W ( x ) dx if: ∀ i, j ≥ ∃ δ > ∫ U sup z ∈ B ( x,δ ) ∥ ∇ i W ( z )∥ j e − W ( z ) dx < ∞ , It is said to be perimeter regular with respect to the measure µ if: ∀ i, j ≥ ∃ δ > ∫ ∂ ∗ U sup z ∈ B ( x,δ ) ∥ ∇ i W ( z )∥ j e − W ( z ) d H n − ( x ) < ∞ . If δ > i, j ≥ U is called uniformly volume /perimeter regular.Here and throughout this work ∥ ⋅ ∥ = ∥ ⋅ ∥ denotes the Hilbert-Schmidt norm of a tensor,defined as the square-root of the sum of squares of its coordinates in any local orthonormalframe. Note that volume (perimeter) regular sets clearly have finite weighted volume(perimeter).Write JF t = det ( dF t ) for the Jacobian of F t , and observe that by the change-of-variablesformula for smooth injective functions: µ ( F t ( U )) = ∫ U JF t e − W ○ F t dx, (3.8) or any Borel set U . Similarly, if U is in addition of locally finite-perimeter, let Φ t = F t ∣ ∂ ∗ U and write J Φ t = det (( d n ⊥ U F t ) T d n ⊥ U F t ) / for the Jacobian of Φ t on ∂ ∗ U . Since ∂ ∗ U islocally H n − -rectifiable, [37, Theorem 11.6] implies: µ n − ( F t ( ∂ ∗ U )) = ∫ ∂ ∗ U J Φ t e − W ○ F t d H n − . (3.9) Lemma 3.3.
1. If U is uniformly volume regular with respect to µ and X is admissible, or alter-natively, if X is C ∞ c , then t ↦ µ ( F t ( U )) is C ∞ in an open neighborhood of t = ,and: δ rX V ( U ) = ∫ U d r ( dt ) r ∣ t = ( JF t e − W ○ F t ) dx.
2. If U is uniformly perimeter regular with respect to µ and X is admissible, or alter-natively, if X is C ∞ c , then t ↦ P µ ( F t ( U )) is C ∞ in an open neighborhood of t = ,and: δ rX A ( U ) = ∫ ∂ ∗ U d r ( dt ) r ∣ t = ( J Φ t e − W ○ F t ) d H n − . The proof of Lemma 3.3 for admissible vector-fields and volume/perimeter regular sets U may be found in [40, Lemma 3.3]; an inspection of the proof verifies that whenever X is C ∞ c , no regularity assumption on U is needed. A cluster Ω is called an isoperimetric minimizer with respect to µ (or simply minimizing )if P µ ( Ω ′ ) ≥ P µ ( Ω ) for every other cluster Ω ′ satisfying µ ( Ω ′ ) = µ ( Ω ) . We will frequentlyinvoke the following additional assumption: µ is a probability measure for which all cells of any isoperimetricminimizing cluster are uniformly volume and perimeter regular. (4.1)It was shown in [40, Corollary 4.4] that the Gaussian measure γ satisfies (4.1). regularity The following theorem is due to Almgren [3] (see also [44, Chapter 13] and [37, Chapters29-30]); for an adaptation to the weighted setting, see [40, Section 4].
Theorem 4.1 (Almgren) . Let µ = e − W dx with W ∈ C ∞ ( R n ) .(i) If µ is a probability measure, then for any prescribed v ∈ ∆ ( q − ) = { v ∈ R q ∶ v i ≥ , ∑ qi = v i = } , an isoperimetric minimizing cluster Ω satisfying µ ( Ω ) = v exists.For every isoperimetric minimizing cluster Ω :(ii) Ω may and will be modified on a µ -null set (thereby not altering { ∂ ∗ Ω i } ) so that allof its cells are open, and so that for every i , ∂ ∗ Ω i = ∂ Ω i and µ n − ( ∂ Ω i ∖ ∂ ∗ Ω i ) = .(iii) For all i ≠ j the interfaces Σ ij = Σ ij ( Ω ) are C ∞ -smooth ( n − ) -dimensional manifolds,relatively open in Σ ∶ = ⋃ k ∂ Ω k , and for every x ∈ Σ ij there exists ǫ > such that B ( x, ǫ ) ∩ Ω k = ∅ for all k ≠ i, j .(iv) For any compact set K in R n , there exist constants Λ K , r K > so that: µ n − ( Σ ∩ B ( x, r )) ≤ Λ K r n − ∀ x ∈ Σ ∩ K ∀ r ∈ ( , r K ) . (4.2) v) For any open bounded set U , Σ ∩ U is an ( M , ǫ ( r ) = Λ U r, δ U ) -minimizing set in U . We refer to [3, 44, 19] for Almgren’s definition of an ( M , ǫ, δ ) -minimizing set, which wewill not directly require for the purposes of this work. Whenever referring to the cells ofa minimizing cluster or their topological boundary in this work, we will always choose arepresentative such as in Theorem 4.1 (ii). Proof of Theorem 4.1.
For a proof of part (i), part (ii) without the assertion that ∂ ∗ Ω i = ∂ Ω i and part (iii) with the relative openness asserted only in ∂ Ω i ∩ ∂ Ω j , see [40, The-orem 4.1]. To justify why in addition ∂ ∗ Ω i = ∂ Ω i in part (ii), see [37, Theorem 12.19].To justify the relative openness of Σ ij in ∪ k ∂ Ω k in part (iii), note that ∂ Ω i ⊂ ⋃ k ≠ i ∂ Ω k ,and hence ∂ Ω i ∖ ∂ Ω j ⊂ ∪ k ≠ i,j ∂ Ω k . It follows that given x ∈ Σ ij and ǫ > B ( x, ǫ ) ∩ Ω k = ∅ for all k ≠ i, j , we have: B ( x, ǫ / ) ∩ ∪ k ∂ Ω k = B ( x, ǫ / ) ∩ ( ∂ Ω i ∪ ∂ Ω j ) = B ( x, ǫ / ) ∩ ∂ Ω i ∩ ∂ Ω j , and so the relative openness in ∪ k ∂ Ω k is equivalent to that in ∂ Ω i ∩ ∂ Ω j , which was alreadyestablished in [40, Theorem 4.1].For a proof of part (iv) in the unweighted setting see [37, Lemma 30.5] or [37, Example21.3 and Theorem 21.11]; the proof easily transfers to the weighted setting on any compactset K , where the density is bounded between two positive constants (depending on K ),only resulting in modified constants.As for part (v) – it is well known in the unweighted setting that Σ is ( M , ǫ ( r ) = Λ r, δ ) -minimizing for any minimizing cluster Ω (see [19, Theorem 3.8] and the references therein),and by inserting the effect of the smooth positive density into the excess function, it followsthat the same holds in the weighted setting inside any bounded open set U .Let n ij be the (smooth) unit normal field along Σ ij that points from Ω i to Ω j . We use n ij to co-orient Σ ij , and since n ij = − n ji , note that Σ ij and Σ ji have opposite orientations.When i and j are clear from the context, we will simply write n . We will typically abbreviate H Σ ij and H Σ ij ,µ by H ij and H ij,µ , respectively. Using Theorem 4.1 and Lemma 3.3, a standard first variation argument (see [40, Theorem4.9]) gives necessary first-order conditions for the minimality of a cluster.
Lemma 4.2 (Stationarity) . For any minimizing cluster Ω :(i) On each Σ ij , H ij,µ is constant.(ii) There is a unique λ ∈ E ∗ such that H ij,µ = λ i − λ j .(iii) For every C ∞ c vector-field X : ∑ i < j ∫ Σ ij div Σ ,µ X t dµ n − = . Moreover, assuming (4.1), this holds for any admissible vector-field X . Definition 4.3 (Stationary Cluster) . A cluster Ω satisfying parts (ii) and (iii) of Theorem4.1 and parts (i), (ii) and (iii) of Lemma 4.2 is called stationary (with respect to µ andwith Lagrange multiplier λ ∈ E ∗ ).Indeed, the following lemma provides an insightful interpretation of λ ∈ E ∗ as a Lagrangemultiplier for the isoperimetric constrained minimization problem (see again the proofof [40, Theorem 4.9]): emma 4.4 (Lagrange Multiplier) . Let Ω be stationary cluster with respect to µ and withLagrange multiplier λ ∈ E ∗ . Then for every C ∞ c vector-field X : δ X V ( Ω ) i = ∑ j ≠ i ∫ Σ ij X n ij dµ n − ∀ i, (4.3) δ X A ( Ω ) = ∑ i < j H ij,µ ∫ Σ ij X n ij dµ n − . In particular, δ X A = ⟨ λ, δ X V ⟩ . Moreover, assuming the cells of Ω are volume and perimeter regular, the above holds forany admissible vector-field X . Similarly, a standard second variation argument (see [40, Lemma 4.12]) gives necessarysecond-order conditions for the local minimality of a cluster.
Definition 4.5 (Index Form Q ) . The Index Form Q associated to a stationary clusterwith respect to µ and with Lagrange multiplier λ ∈ E ∗ is defined as the following quadraticform: Q ( X ) ∶ = δ X A − ⟨ λ, δ X V ⟩ , defined on vector-fields X for which the right-hand-side is well defined. Lemma 4.6 (Stability) . For any minimizing cluster Ω and C ∞ c vector-field X : δ X V = ⇒ Q ( X ) ≥ . (4.4) Moreover, assuming (4.1), then (4.4) holds for any admissible vector-field X . Definition 4.7 (Stable Cluster) . A stationary cluster satisfying (4.4) for any admissiblevector-field X is called stable (with respect to µ ). Given a minimizing cluster Ω, denote:Σ ∶ = ∪ i ∂ Ω i , Σ ∶ = ∪ i < j Σ ij , and observe that Σ = Σ by (3.6) and our convention from Theorem 4.1 (ii). We will alsorequire additional information on the lower-dimensional structure of Σ. To this end, definetwo special cones: Y = { x ∈ E ( ) ∶ there exist i ≠ j ∈ { , , } with x i = x j = max k ∈{ , , } x k } , T = { x ∈ E ( ) ∶ there exist i ≠ j ∈ { , , , } with x i = x j = max k ∈{ , , , } x k } . In other words, Y is the boundary of a model 3-cluster in E ( ) and T is the boundary of amodel 4-cluster in E ( ) . Note that Y consists of 3 half-lines meeting at the origin in 120 ○ angles, and that T consists of 6 two-dimensional sectors meeting in threes at 120 ○ anglesalong 4 half-lines, which in turn all meet at the origin in cos − ( − / ) ≃ ○ angles. Itturns out that on the codimension-2 and codimension-3 parts of a minimizing cluster, Σlocally looks like Y × R n − and T × R n − , respectively. Theorem 4.8 (Taylor, White, Colombo–Edelen–Spolaor) . Let Ω be a minimizing clusterfor the measure µ = exp ( − W ) dx in R n with W ∈ C ∞ ( R n ) . Then there exist α > and sets Σ , Σ , Σ ⊂ Σ such that: i) Σ is the disjoint union of Σ , Σ , Σ , Σ ;(ii) Σ is a locally-finite union of embedded ( n − ) -dimensional C ,α manifolds, and forevery p ∈ Σ there is a C ,α diffeomorphism mapping a neighborhood of p in R n to aneighborhood of the origin in E ( ) × R n − , so that p is mapped to the origin and Σ islocally mapped to Y × R n − ;(iii) Σ is a locally-finite union of embedded ( n − ) -dimensional C ,α manifolds, and forevery p ∈ Σ there is a C ,α diffeomorphism mapping a neighborhood of p in R n to aneighborhood of the origin in E ( ) × R n − , so that p is mapped to the origin and Σ islocally mapped to T × R n − ;(iv) Σ is closed and dim H ( Σ ) ≤ n − . Remark 4.9.
Clearly, when n =
2, necessarily Σ is discrete and Σ = Σ = ∅ , and when n =
3, necessarily Σ is discrete and Σ = ∅ . The C ,α regularity in (ii) will be improvedto C ∞ regularity in Corollary 4.13 using an argument of Kinderlehrer–Nirenberg–Spruck[32]. The work of Naber and Valtorta [47] implies that Σ is actually H n − -rectifiable andhas locally-finite H n − measure, but we will not require this here. Proof of Theorem 4.8.
Let us first give references in the classical unweighted setting (whenΩ is a minimizing cluster with respect to the Lebesgue measure in R n ). The case n = n = ( M , ǫ, δ ) sets in the sense of Almgren. When n ≥
4, Theorem 4.8 was announcedby B. White [60, 61] for general ( M , ǫ, δ ) sets. Theorem 4.8 with part (iii) replaced bydim H ( Σ ) ≤ n − U ⊂ R n having associated cycle structure,no boundary in U , bounded mean-curvature and whose support is ( M , ǫ, δ ) minimizing,was very recently established by M. Colombo, N. Edelen and L. Spolaor [19, Theorem 1.3,Remark 1.4, Theorem 3.10]; in particular, this applies to isoperimetric minimizing clustersin R n [19, Theorem 3.8], yielding Theorem 4.8 when µ is the Lebesgue measure.All the the above regularity results equally apply in the weighted setting. Let us explainhow to deduce Theorem 4.8 from the results of [19]. Given an open set U ⊂ R n and a setΣ U ⊂ U satisfying a certain requirement, to be described next, [19, Theorem 3.10] assertsa stratification of Σ U into a disjoint union of Σ U , Σ U , Σ U , Σ U , where Σ iU for i = , , C ,α ( n − i ) -dimensional manifolds for some α > n ,with Σ U diffeomorphic to R n − , Y × R n − and T × R n − near Σ U , Σ U and Σ U , respectively,and with Σ U satisfying (iv) in U . The stratification of Σ U into Σ iU may be describedexplicitly using symmetry properties of the possible tangent cones of Σ U at a given x ∈ Σ U ,but we will not require this here; it suffices to note that the stratification is determinedby the local structure of Σ U , so that if Σ U ⊂ Σ V with U ⊂ V open, then Σ iU ⊂ Σ iV for all i = , , , U ∶ = Σ ∩ U , where U = B R is an open ball of arbitrary radius R >
0. Indeed, setting ˜Σ i ∶ = ∪ R > Σ iB R , it wouldfollow that { ˜Σ i } i = , , , satisfy all the asserted properties (i) through (iv) of Theorem 4.8.Moreover, recalling that Σ = ∪ ij Σ ij , clearly Σ is disjoint from ˜Σ and ˜Σ since Σ isdiffeomorphic to R n − near Σ , and therefore Σ ⊂ ˜Σ ∪ ˜Σ . On the other hand, we claimthat ˜Σ ⊂ Σ . To see this, let p ∈ ˜Σ , and let N p be a neighborhood of p where Σ islocally diffeomorphic to { x n = } ⊂ R n . Applying our local diffeomorphism, we concludethat there exist j ≠ k so that p ∈ ∂ Ω j ∩ ∂ Ω k , and on N p , Ω j is mapped to { x n < } andΩ k is mapped to { x n > } . Pulling back via our diffeomorphism (and since for open setswith C smooth boundary, the reduced boundary coincides with the topological one), itfollows that p ∈ ∂ ∗ Ω j ∩ ∂ ∗ Ω k = Σ jk ⊂ Σ , as asserted. Hence, defining Σ ∶ = ˜Σ , Σ ∶ = ˜Σ nd Σ ∶ = Σ ∖ ( Σ ∪ Σ ∪ Σ ) ⊂ ˜Σ , all the asserted properties (i) through (iv) would holdfor { Σ i } i = , , , , thereby concluding the proof.As for the aforementioned requirement on Σ U in [19, Theorem 3.10] – it holds, inparticular, if Σ U is ( M , ǫ, δ ) -minimizing in U in the sense of Almgren, and in addition itis the support of a multiplicity-one integral ( n − ) -current with bounded mean curvatureand no boundary in U . Let us verify this requirement for our Σ U = Σ ∩ U for an arbitraryopen bounded set U . Theorem 4.1 (v) already asserts that Σ U is ( M , ǫ, δ ) -minimizing, so itremains to check that Σ U is the support of an integral ( n − ) -current with bounded meancurvature and no boundary in U .Indeed, consider Σ as the ( n − ) -dimensional smooth manifold ⊔ i < j Σ ij , which is inaddition oriented (recall that the co-orientation is given by n ij ), and define the associatedrectifiable ( n − ) -current T on U acting by integration: T ( α ) = ∫ Σ ∩ U α, for any smooth differential ( n − ) -form α compactly supported in U . Since Σ is a smoothmanifold, which is relatively open in its closure Σ, the support of T in U is easily seen tobe Σ ∩ U = Σ ∩ U = Σ U .Next, note that Σ ∩ U and hence T has bounded mean-curvature, because the weightedmean-curvature H Σ ij ,µ is constant by Lemma 4.2, and it differs from H Σ ij by ∇ n ij W , whichis a bounded quantity on any bounded set U . In addition, we claim that T has no boundaryas a current, i.e. T ( dβ ) = ( n − ) -form β compactly supportedin U . Indeed, on Σ , write β = β + η , where η is identically zero when acting on Λ n − T Σ ,the space of ( n − ) -vectorfields in the tangent bundle T Σ , and β is an ( n − ) -formon Λ n − T Σ . It is then easy to check that dη remains identically zero when acting onΛ n − T Σ , and that β = i Y vol Σ for some smooth vector-field Y tangential to Σ andcompactly supported in U (where i denotes interior product). It follows that as forms onΛ n − T Σ : dβ = d ( i Y vol Σ ) = div Σ Y vol Σ . Hence: T ( dβ ) = ∫ Σ div Σ Y vol Σ = ∫ Σ div Σ ,µ ( Y e + W ) e − W vol Σ = ∑ i < j ∫ Σ ij div Σ ij ,µ ( Y e + W ) dµ n − . But since
Y e + W is tangential to Σ and compactly supported, Lemma 4.2 (iii) asserts thatthe latter integral is zero, confirming that T has no boundary, and concluding the proof. Definition 4.10 (Regular Cluster) . A cluster satisfying parts (ii), (iii) and (iv) of Theorem4.1 and the conclusion of Theorem 4.8, and whose cells are volume and perimeter regularwith respect to µ , is called regular (with respect to µ ). Remark 4.11.
For the results of this work, we can slightly relax the above definition ofregularity: we do not need the local-finiteness in parts (ii) and (iii) of Theorem 4.8, andinstead of dim H ( Σ ) ≤ n − H n − ( Σ ) =
0. In addition,it is worthwhile noting that parts (ii), (iii) and (iv) of Theorem 4.1 are a consequence ofΣ being a locally ( M , ǫ, δ ) -minimizing set (part (v) of Theorem 4.1), and that the latterproperty together with stationarity are the only properties which were used in the proofof Theorem 4.8; as we will only consider regularity in conjunction with stationarity in thiswork, one could therefore replace “satisfying parts (ii), (iii) and (iv) of Theorem 4.1 and theconclusion of Theorem 4.8” in the above definition by (the a-priori stronger, but simplerto state) “satisfying part (v) of Theorem 4.1”. n summary, Theorem 4.1, Lemmas 4.2 and 4.6 and Theorem 4.8 assert that, assumingthat µ satisfies (4.1), an isoperimetric minimizing cluster Ω is necessarily stationary, stableand regular (with respect to µ ). We proceed to describe additional properties of Σ, whichhold for any stationary and regular cluster Ω. C ∞ regularity Theorem 4.8 implies that every point in Σ , which we will call the triple-point set , belongsto the closure of exactly three cells, as well as to the closure of exactly three interfaces.Given distinct i, j, k , we will write Σ ijk for the subset of Σ which belongs to the closureof Ω i , Ω j and Ω k , or equivalently, to the closure of Σ ij , Σ jk and Σ ki . Similarly, we willcall Σ the quadruple-point set , and given distinct i, j, k, l , denote by Σ ijkl the subset of Σ which belongs to the closure of Ω i , Ω j , Ω k and Ω l , or equivalently, to the closure of all sixΣ ab for distinct a, b ∈ { i, j, k, l } .We will extend the normal fields n ij to Σ ijk and Σ ijkl by continuity (thanks to C regularity). We will also define n ∂ij on Σ ijk to be the outward-pointing unit boundary-normal to Σ ij . When i and j are clear from the context, we will write n ∂ for n ∂ij .Let us also denote: ∂ Σ ij ∶ = ⋃ k ≠ i,j Σ ijk . Note that Σ ij ∪ ∂ Σ ij is a (possibly incomplete) C ∞ manifold with C ,α boundary (theboundary ∂ Σ ij will be shown to actually be C ∞ in Corollary 4.13). Corollary 4.12.
For any stationary regular cluster, the following holds at every point in Σ ijk : ∑ ( ℓ,m )∈ C ( i,j,k ) n ∂ℓm = and ∑ ( ℓ,m )∈ C ( i,j,k ) n ℓm = . In other words, Σ ij , Σ jk and Σ ki meet at Σ ijk in ○ angles.Proof. Let X be any C ∞ c vector field whose support intersects Σ only in Σ ijk ∪ Σ ij ∪ Σ jk ∪ Σ ki .According to Lemma 4.2 (iii), we have: ∑ ( ℓ,m )∈ C ( i,j,k ) ∫ Σ ℓm div Σ ,µ X t dµ n − = . On the other hand, since Σ is closed, note that X has compact support in Σ ℓm ∪ ∂ Σ ℓm forall ( ℓ, m ) ∈ C ( i, j, k ) , and so applying Stokes’ theorem (3.2), we see that the above quantityis equal to ∑ ( ℓ,m )∈ C ( i,j,k ) ∫ Σ ijk ⟨ X, n ∂ℓm ⟩ dµ n − . It follows that for any such vector field X , ∫ Σ ijk ⟨ X, ∑ ( ℓ,m )∈ C ( i,j,k ) n ∂ℓm ⟩ dµ n − = n ∂ℓm is a continuous vector-field on Σ ijk and Σ ijk ∪ Σ ij ∪ Σ jk ∪ Σ ki is relatively openin Σ (by Theorems 4.1 and 4.8), the first assertion follows. For the second assertion, let W denote the span of { n ∂ij , n ∂jk , n ∂ki } , which we already know is two-dimensional. Wewill orient W so that n ∂ij , n ∂jk , n ∂ki are in counterclockwise order. Then n ℓm = R n ∂ℓm for all ( ℓ, m ) ∈ C ( i, j, k ) , where R is a 90 ○ clockwise rotation in W , and the second assertionfollows from the first. orollary 4.13 (Kinderlehrer–Nirenberg–Spruck) . For any stationary regular cluster,Theorem 4.8 (ii) holds with C ∞ regularity (instead of just C ,α ) for Σ and its associ-ated local diffeomorphisms.Proof. It was shown by Nitsche [48] that three minimal surfaces in R meeting at equal 120 ○ angles must do so on a smooth curve. Kinderlehrer, Nirenberg and Spruck [32, Theorem 5.2]extended this to the case that three hypersurfaces with analytic mean-curvature in R n meetat arbitrary constant transversal angles on a C ,α ( n − ) -dimensional manifold Σ , andshowed that Σ must in fact be analytic. As we shall essentially reproduce in Appendix B fora different purpose, the key step in their argument is to use elliptic methods to show that the C k,α regularity of Σ (as well as of the local diffeomorphism as in Theorem 4.8 (ii)) may beimproved to C k + ,α regularity if the mean-curvature of the three hypersurfaces is a smooth-enough function of the first and zeroth order derivatives of the surfaces’ parametrization.By Corollary 4.12, Σ ij , Σ jk and Σ ki meet at equal 120 ○ angles on the C ,α manifoldΣ ijk . Since the weighted mean-curvature of Σ ℓm is constant H ℓm,µ , its unweighted mean-curvature is of the form H ℓm,µ + ⟨ ∇ W, n ℓm ⟩ , which is a C ∞ function of the zeroth andfirst order terms since W ∈ C ∞ ( R n ) . The assertion therefore follows by applying theKinderlehrer–Nirenberg–Spruck argument. We will also crucially require the following local integrability properties of our variouscurvatures. Let II ij denotes the second fundamental form on Σ ij , which may be extendedby continuity to ∂ Σ ij . When i and j are clear from the context, we will write II for II ij .Recall that ∥ II ij ∥ denotes (say) the Hilbert-Schmidt norm of II ij . Proposition 4.14.
Let Ω be a stationary regular cluster. For any compact set K ⊂ R n which is disjoint from Σ :(1) ∑ i,j ∫ Σ ij ∩ K ∥ II ij ∥ dµ n − < ∞ .(2) ∑ i,j,k ∫ Σ ijk ∩ K ∥ II ij ∥ dµ n − < ∞ . By compactness and the fact that Σ is closed, it is enough to verify the above integrabil-ity locally, in arbitrarily-small relatively open subsets of Σ and Σ , respectively, which aredisjoint from Σ . Since Σ ij is C ∞ smooth all the way up to ∂ Σ ij ⊂ Σ , it remains to verifythe integrability around small neighborhoods of quadruple points in Σ . Proposition 4.14follows from combining the locally uniform C ,α regularity of Σ around quadruple points,together with the elliptic regularity method used by Kinderlehrer–Nirenberg–Spruck [32] toself-improve the C ,α regularity around triple points Σ to C ∞ smoothness. The idea is touse Schauder estimates to show that the curvature (or C semi-norm of the graph of Σ ij )at a distance of r from Σ blows up at a rate of at most 1 / r − α as r →
0, and hence is locallyin L on Σ and locally in L on Σ . As the tools we require for establishing Proposition4.14 will not be used in any other part of this work, we defer its proof to Appendix B.Note that the same compactness argument as above, together with the C ,α regularityof Σ around quadruple-points in Σ , immediately yields the following simple corollary ofTheorem 4.8: Corollary 4.15.
For any compact set K ⊂ R n which is disjoint from Σ , µ n − ( Σ ∩ K ) < ∞ . In fact, by the local finiteness statement in Theorem 4.8 (ii), the same holds withoutassuming that K is disjoint from Σ , but we will not require this here. The Second Variation
Throughout this section, unless otherwise stated, Ω is assumed to be a stationary regularcluster. Recall that ∂ Σ ij denotes ⋃ k ≠ i,j Σ ijk and that Σ ij ∪ ∂ Σ ij is a C ∞ manifold withboundary. Recall also that II ij denotes the second fundamental form of Σ ij , which weextend by continuity to ∂ Σ ij . We will abbreviate II ij∂,∂ for II ij ( n ∂ij , n ∂ij ) on ∂ Σ ij . When i and j are clear from the context, we use II for II ij . Finally, recall the definition of theindex-form Q ( X ) = δ X A − ⟨ λ, δ X V ⟩ . Definition 5.1 (Tame Field) . The vector-field X is called tame for a regular cluster Ω ifit is C ∞ c and supported in R n ∖ Σ .In this section, we establish the following key formula for Q ( X ) . We denote by ∇ t thetangential component of the derivative. Theorem 5.2 (Index-Form Formula for Tame Fields) . If Ω is a stationary regular clusterand X is a tame vector-field, then Q ( X ) is given by ∑ i < j [ ∫ Σ ij (∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ − ( X n ) ∇ n , n W ) dµ n − − ∫ ∂ Σ ij X n X n ∂ II ∂,∂ dµ n − ] . (5.1)We will also require the following variant of Theorem 5.2, which will require additionalwork: Theorem 5.3 (Index-Form Formula for Tame + Constant Fields) . With the same as-sumptions as in Theorem 5.2, if Y = X + w where w is a constant vector-field, then Q ( Y ) is given by ∑ i < j ⎡⎢⎢⎢⎢⎣ ∫ Σ ij (∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ − ( Y n ) ∇ n , n W − ( X + Y ) n ∇ w t , n W ) dγ n − − ∫ ∂ Σ ij X n X n ∂ II ∂,∂ dγ n − ⎤⎥⎥⎥⎥⎦ . The elementary case X ≡ µ = γ n ,when ∇ n , n W = ∇ w t , n W = Q ( X ) ,which may be derived starting from Lemma 3.3; its proof consists of a long computation,which is deferred to Appendix A. Lemma 5.4.
For any stationary cluster Ω and vector-field X , so that either X is C ∞ c , oralternatively, X is admissible and the cells of Ω are volume and perimeter regular, Q ( X ) is given by ∑ i < j ∫ Σ ij ⎡⎢⎢⎢⎢⎣∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ − ( X n ) ∇ n , n W − div Σ ,µ ( X n ∇ X t n ) + div Σ ,µ ( X n ∇ n X ) + div Σ ,µ ( X div Σ ,µ X ) − H ij,µ X n div µ X ⎤⎥⎥⎥⎥⎦ dµ n − . In order to simplify the expression in Lemma 5.4, we will want to apply Stokes’ theoremto the various divergence terms. Even if we assume that X has compact support in R n , thisrequires some justification; each M ij = Σ ij ∪ ∂ Σ ij is a smooth manifold with boundary, but he vector-fields Y ij whose divergence is integrated may not have compact support inside M ij , as their support may intersect Σ ∪ Σ . Furthemore, Y ij may be blowing up near Σ ∪ Σ , precluding us from invoking a version of Stokes’ theorem for vector-fields with boundedmagnitude and divergence which only requires that H n − ( M ij ∖ M ij ) =
0, see e.g. [35, XXIII,Section 6]. However, in our setting, we actually know that M ij ∖ M ij has locally-finite H n − -measure, which enables us to handle vector-fields in L ( Σ ij , µ n − ) ∩ L ( ∂ Σ ij , µ n − ) whosedivergence is in L ( Σ ij , µ n − ) ; it turns out that whenever X is tame, our Y ij ’s will indeedbelong to this class thanks to the integrability properties of our curvatures established inProposition 4.14. The idea is to use a well-known truncation argument to cut out thelow-dimensional sets (e.g. as in [35, 56]). A function on R n which is C ∞ smooth and takes values in [ , ] will be called a cutofffunction. Lemma 5.5.
For every ǫ > , there is a compactly supported cutoff function η such that η ≡ on a neighborhood of Σ , ∫ Σ ∣ ∇ η ∣ dµ n − ≤ ǫ, and µ n − { p ∈ Σ ∶ η ( p ) < } ≤ ǫ. Proof.
Fix ǫ >
0, and choose R ≥ µ n − ( Σ ∖ B R ) ≤ ǫ . Let ζ be a C ∞ c function that is identically one on B R + , identically zero outside B R − , and satisfies ∣ ∇ ζ ∣ ≤
1. By the upper density estimate (4.2), there exist Λ > r ∈ ( , ) so that: µ n − ( Σ ∩ B ( x, r )) ≤ Λ r n − ∀ x ∈ Σ ∩ B R ∀ r ∈ ( , r ) . (5.2)Since H n − ( Σ ) = ∩ B R is compact, we may find finite sequences x i ∈ Σ ∩ B R and δ i ∈ ( , r ) , i = , . . . , N , such that { B ( x i , δ i )} i = ,...,N cover Σ ∩ B R and ∑ i δ n − i ≤ ǫ / Λ.For each i , choose a cutoff function η i such that η i ≡ B ( x i , δ i ) , η i ≡ B ( x i , δ i ) , and ∣ ∇ η i ∣ ≤ / δ i . Now define ˜ η ( x ) = min { ζ, min i = ,...,N η i ( x )} ; ˜ η iscompactly supported, vanishes on ( R n ∖ B R − ) ⋃ ∪ i B ( x i , δ i ) , and is identically 1 on B R + ∖ ∪ i B ( x i , δ i ) . Since it is only piecewise C ∞ , let us mollify ˜ η with a smooth mollifiersupported in B ( , δ ) , for δ = min i = ,...,N δ i ∈ ( , ) - this will be our desired function η . Thecutoff function η is compactly supported and vanishes on ( R n ∖ B R ) ⋃ ∪ i B ( x i , δ i ) , an openneighborhood of Σ and ∞ . Since δ i < r <
1, it follows by (5.2) that: µ n − { p ∈ Σ ∶ η ( p ) < } ≤ µ n − ( Σ ∖ B R ) + ∑ i µ n − ( Σ ∩ B ( x i , δ i )) ≤ ǫ + Λ ∑ i δ n − i ≤ ǫ + Λ ∑ i δ n − i ≤ ǫ. Finally, we have for a.e. x ∈ R n : ∣ ∇ ˜ η ( x )∣ ≤ max {∣ ∇ ζ ( x )∣ , max i ∣ ∇ η i ( x )∣ } ≤ ∣ ∇ ζ ( x )∣ + ∑ i ∣ ∇ η i ( x )∣ , and hence after mollification it follows that for all x ∈ R n : ∣ ∇ η ( x )∣ ≤ B R ∖ B R ( x ) + ∑ i ( / δ i ) B ( x i , δ i )∖ B ( x i ,δ i ) ( x ) . onsequently: ∫ Σ ∣ ∇ η ∣ dµ n − ≤ µ ( Σ ∖ B R ) + ∑ i ( / δ i ) µ ( Σ ∩ B ( x i , δ i )) ≤ ǫ + ∑ i δ n − i ≤ ǫ. Appropriately modifying the value of ǫ , the assertion follows.A slight variation on the proof of Lemma 5.5 implies that we can cut off Σ also, whileonly paying a bounded amount in the W , norm. Lemma 5.6.
Let U be an open neighborhood of Σ and infinity. There is a constant C U depending on the cluster Ω and U , such that for every δ > , there is a cutoff function ξ such that ξ ≡ on an open neighborhood of Σ ∖ U , ∫ Σ ∣ ∇ ξ ∣ dµ n − ≤ C U , and µ n − { p ∈ Σ ∶ ξ ( p ) < } ≤ δ. Proof.
First, note that Σ ⊂ Σ ∪ Σ by the relative openness of Σ ∪ Σ in the closed Σ. Itfollows that Σ ∖ U is compact, since it is clearly bounded, but also closed, as it coincideswith Σ ∖ U . Since Σ has finite H n − -measure in a neighborhood of every point in Σ (byTheorem 4.8), compactness implies that H n − ( Σ ∖ U ) is finite. In addition, by the upperdensity estimate (4.2), there exist Λ U , r U > µ n − ( Σ ∩ B ( x, r )) ≤ Λ U r n − ∀ x ∈ Σ ∖ U ∀ r ∈ ( , r U ) . (5.3)For any δ ∈ ( , r U ) , the definition of Hausdorff measure and compactness imply theexistence of finite sequences x i ∈ Σ ∖ U and δ i ∈ ( , δ ) , i = , . . . , N , such that the sets { B ( x i , δ i )} cover Σ ∖ U and ∑ i δ n − i ≤ C n H n − ( Σ ∖ U ) , where C n > n . The rest of the proof is identical to that of Lemma 5.5. For each i ,we select a cutoff function ξ i which is identically zero on B ( x i , δ i ) , identically one outsideof B ( x i , δ i ) , and satisfies ∣ ∇ ξ i ∣ ≤ / δ i . We then set ˜ ξ ( x ) = min i = ,...,N ξ i ( x ) and define ξ to be the mollification of ˜ ξ using a smooth mollifier compactly supported in B δ , where δ = min i = ,...,N δ i . The cutoff function ξ vanishes on ∪ i B ( x i , δ i ) , an open neighborhood ofΣ ∖ U . It follows that: ∫ Σ ∣ ∇ ξ ∣ dµ n − ≤ ∑ i ( / δ i ) µ n − ( B ( x i , δ i )) ≤ U ∑ i δ n − i ≤ U C n H n − ( Σ ∖ U ) , which proves the claimed gradient bound. On the other hand, µ n − { p ∈ Σ ∶ ξ ( p ) < } ≤ ∑ i µ n − ( Σ ∩ B ( x i , δ i )) ≤ Λ U ∑ i δ n − i ≤ Λ U C n δ H n − ( Σ ∖ U ) , which can be made smaller than δ by an appropriate choice of δ . Lemma 5.7 (Stokes’ theorem on Σ ij ∪ ∂ Σ ij ) . Let Ω be a stationary regular cluster. Supposethat Y ij is a vector-field which is C ∞ on Σ ij and continuous up to ∂ Σ ij . Suppose, moreover,that ∫ Σ ij ∣ Y ij ∣ dµ n − , ∫ Σ ij ∣ div Σ ,µ Y ij ∣ dµ n − , ∫ ∂ Σ ij ∣ Y n ∂ ij ∣ dµ n − < ∞ . (5.4) hen: ∫ Σ ij div Σ ,µ Y ij dµ n − = ∫ Σ ij H ij,µ Y n ij dµ n − + ∫ ∂ Σ ij Y n ∂ ij dµ n − . Remark 5.8.
Observe that whenever Y ij is the restriction to Σ ij ∪ ∂ Σ ij of a tame vector-field on R n , all of the integrability requirements in (5.4) are automatically satisfied, since µ n − ( Σ ij ) < ∞ and µ n − ( ∂ Σ ij ∩ K ) < ∞ for any compact K disjoint from Σ by Corollary4.15.While we will only apply Lemma 5.7 to vector-fields which are supported outside aneighborhood of Σ and infinity, we provide a proof for general vector-fields as above, asthis comes at almost no extra cost. Proof of Lemma 5.7.
Fix ǫ > η = η ǫ as in Lemma 5.5. Let U = U η be a neighborhood of Σ and infinity on which η vanishes, and let C η denote the constant C U from Lemma 5.6. Fix δ > ξ = ξ ǫ,δ as in Lemma 5.6. Note that ηξ vanishes on an open neighborhood of Σ ∪ Σ and infinity, and therefore ηξY ij has compactsupport in Σ ij ∪ ∂ Σ ij . Applying Stokes’ theorem (3.2) to ηξY ij , we obtain: ∫ Σ ij div Σ ,µ ( ηξY ij ) dµ n − = ∫ Σ ij H ij,µ ηξY n ij dµ n − + ∫ ∂ Σ ij ηξY n ∂ ij dµ n − . (5.5)Next, we check what happens when we send δ → ǫ →
0. By the DominatedConvergence theorem, the right-hand-side of (5.5) converges to ∫ Σ ij H ij,µ Y n ij dµ n − + ∫ ∂ Σ ij Y n ∂ ij dµ n − . Indeed, the second term is absolutely integrable directly by assumption, and the first oneis too since H ij,µ is constant and: ( ∫ Σ ij ∣ Y n ij ∣ dµ n − ) ≤ µ n − ( Σ ij ) ∫ Σ ij ∣ Y ij ∣ dµ n − < ∞ . As for the left-hand-side of (5.5), we split it as ∫ Σ ij div Σ ,µ ( ηξY ij ) dµ n − = ∫ Σ ij ηξ div Σ ,µ Y ij dµ n − + ∫ Σ ij ∇ Y t ij ( ηξ ) dµ n − . Taking δ → ǫ → ∫ Σ ij ηξ div Σ ,µ Y ij dµ n − → ∫ Σ ij div Σ ,µ Y ij dµ n − because weassumed div Σ ,µ Y ij to be absolutely integrable. On the other hand, ∣ ∫ Σ ij ∇ Y t ij ( ηξ ) dµ n − ∣ ≤ ∫ Σ ij (∣ ∇ η ∣ + ∣ ∇ ξ ∣)∣ Y ij ∣ dµ n − . Applying the Cauchy-Schwarz inequality, ∫ Σ ij ∣ ∇ η ∣∣ Y ij ∣ dµ n − ≤ ( ǫ ∫ Σ ij ∣ Y ij ∣ dµ n − ) / , which converges to zero as ǫ →
0, and ∫ Σ ij ∣ ∇ ξ ∣∣ Y ij ∣ dµ n − ≤ ( C η ∫ Σ ij { ξ < } ∣ Y ij ∣ dµ n − ) / because ∇ ξ = { ξ = } . For any fixed ǫ and η , this last integral converges tozero as δ → δ → ǫ →
0, itfollows that ∫ Σ ij ∇ Y t ij ( ηξ ) dµ n − → ∫ Σ ij div Σ ,µ ( ηξY ij ) dµ n − → ∫ Σ ij div Σ ,µ Y ij dµ n − . Plugging this back into (5.5) proves the claim. .3 Cancellation identities To see that the boundary integrals simplify or even vanish after an application of Stokes’theorem on the divergence terms appearing in Lemma 5.4, we will require a couple of usefulidentities. Recall that by Corollary 4.12, for any distinct i, j, k , the three interfaces Σ ij ,Σ jk and Σ ki meet on Σ ijk at 120 ○ angles: ∑ ( ℓ,m )∈ C ( i,j,k ) n ℓm = . (5.6) Lemma 5.9.
At every point of Σ ijk and for every x ∈ R n , ∑ ( ℓ,m )∈ C ( i,j,k ) ⟨ x, n ℓm ⟩⟨ ∇ x t ℓm n ℓm , n ∂ℓm ⟩ = ∑ ( ℓ,m )∈ C ( i,j,k ) ⟨ x, n ℓm ⟩⟨ x, n ∂ℓm ⟩ II ℓm∂,∂ . Proof.
To simplify notation, we will assume that { i, j, k } = { , , } and fix p ∈ Σ . Let y be the component of x which is tangent to Σ at p , so that for any distinct i, j, k ∈ { , , } , x t ij = y + n ∂ij ⟨ x, n ∂ij ⟩ . Hence, ⟨ ∇ x t ij n ij , n ∂ij ⟩ = ⟨ x, n ∂ij ⟩⟨ ∇ n ∂ij n ij , n ∂ij ⟩ + ⟨ ∇ y n ij , n ∂ij ⟩ = ⟨ x, n ∂ij ⟩ II ij∂,∂ + ⟨ ∇ y n ij , n ∂ij ⟩ . (5.7)Next, we observe that the second term is independent of ( i, j ) ∈ C ( , , ) . Indeed, by (5.6),we know that n jk = √ n ∂ij − n ij and n ∂jk = − √ n ij − n ∂ij . Clearly, ⟨ ∇ y n ij , n ij ⟩ = ∇ y ⟨ n ij , n ij ⟩ = ⟨ ∇ y n ∂ij , n ∂ij ⟩ = ⟨ ∇ y n ∂ij , n ij ⟩ = − ⟨ n ∂ij , ∇ y n ij ⟩ .Hence, ⟨ ∇ y n jk , n ∂jk ⟩ = − ⟨ ∇ y n ∂ij , n ij ⟩ + ⟨ ∇ y n ij , n ∂ij ⟩ = ⟨ ∇ y n ij , n ∂ij ⟩ . It follows by (5.6) again that: ∑ ( i,j )∈ C ( , , ) ⟨ x, n ij ⟩⟨ ∇ y n ij , n ∂ij ⟩ = ⟨ x, ∑ ( i,j )∈ C ( , , ) n ij ⟩ ⟨ ∇ y n , n ∂ ⟩ = . Multiplying (5.7) by ⟨ x, n ij ⟩ and summing over ( i, j ) ∈ C ( , , ) completes the proof. Lemma 5.10.
At every point of Σ ijk , the following -tensor is identically zero: T αβγ = ∑ ( ℓ,m )∈ C ( i,j,k ) ( n αℓm n βℓm n γ∂ℓm − n α∂ℓm n βℓm n γℓm ) . Proof.
Consider a two-dimensional Euclidean space W and let R ∶ W → W denote a 90 ○ clockwise rotation. Note that for any unit vector u ∈ W , u ( Ru ) T − ( Ru ) u T is independentof u and is equal to − R , as immediately seen by expressing this operator in the orthogonalbasis { u, Ru } .Now assume, without loss of generality as before, that { i, j, k } = { , , } , and fix p ∈ Σ . Let W be the span of { n , n , n } , which by (5.6) is two-dimensional. We willorient W so that n , n , n are in clockwise order. It follows that for every ( ℓ, m ) ∈ C ( , , ) , n ∂ℓm is a 90 ○ clockwise rotation of n ℓm . By the previous remarks, n αℓm n γ∂ℓm − n α∂ℓm n γℓm = − R αγ is independent of ( ℓ, m ) ∈ C ( , , ) . Hence, T αβγ = − R αγ ∑ ( ℓ,m )∈ C ( , , ) n βℓm , which vanished identically by (5.6), as asserted. .4 Proof of Theorem 5.2 The first step is to apply Stokes’ theorem to show that the second line in the formula ofLemma 5.4 vanishes. Observe that on Σ ij :div Σ ,µ ( X n ∇ n X ) + div Σ ,µ ( X div Σ ,µ X ) − H Σ ,µ X n div µ X = div Σ ,µ Y ij , (5.8)where Y ij is the following vector-field on Σ ij ∪ ∂ Σ ij : Y ij ∶ = X n ij ∇ n ij X − X ⟨ ∇ n ij X, n ij ⟩ + X t ij div µ X. (5.9)Note that Y ij is C ∞ on Σ ij ∪ ∂ Σ ij thanks to Corollary 4.13, and satisfies ∣ Y ij ∣ ≤ ( + n )∣ X ∣∥ ∇ X ∥ + ∣ X ∣ ∣ ∇ W ∣ . Since X is tame, ∣ Y ij ∣ is uniformly bounded and has boundedsupport disjoint from Σ , and hence satisfies the first integrability condition of Lemma 5.7,as well as the third one by Corollary 4.15. In addition, one sees that div Σ ,µ Y ij is absolutelyintegrable on Σ ij by inspecting (5.8); each of the terms on the left-hand-side is bounded by ∥ II ∥ times a polynomial in ∣ X ∣ , ∥ ∇ X ∥ , and ∥ ∇ X ∥ , and is therefore in L ( Σ ij , µ n − ) thanksto tameness of X and Proposition 4.14 (1), and hence in L ( Σ ij , µ n − ) since µ n − ( Σ ij ) < ∞ .It follows that we may apply Lemma 5.7 (Stokes’ theorem) to each Y ij on Σ ij ∪ ∂ Σ ij . Notethat Y ij is tangential to Σ ij since ⟨ Y ij , n ij ⟩ =
0, and so ∑ i < j ∫ Σ ij div Σ ,µ Y ij dµ n − = ∑ i < j ∫ ∂ Σ ij ⟨ Y ij , n ∂ij ⟩ dµ n − = ∑ i < j < k ∫ Σ ijk ∑ ( ℓ,m )∈ C ( i,j,k ) ⟨ Y ℓm , n ∂ℓm ⟩ dµ n − . (5.10)In order to see that this vanishes, we will show that for every distinct i, j, k , ∑ ( ℓ,m )∈ C ( i,j,k ) ⟨ Y ℓm , n ∂ℓm ⟩ = ijk . Indeed, for the last term in (5.9), note that ⟨ X t ℓm , n ∂ℓm ⟩ div µ X = ⟨ X, n ∂ℓm ⟩ div µ X, which vanishes when summed over ( ℓ, m ) ∈ C ( i, j, k ) by (5.6). The other two terms in (5.9)vanish after taking inner product with n ∂ℓm and summing, because by Lemma 5.10: ∑ ( ℓ,m )∈ C ( i,j,k ) ⟨ X, n ℓm ⟩⟨ ∇ n ℓm X, n ∂ℓm ⟩ − ⟨ X, n ∂ℓm ⟩⟨ ∇ n ℓm X, n ℓm ⟩ = T αβγ X α ∇ β X γ = . Summarizing, we see that the entire second line of the formula in Lemma 5.4 vanishes,and we are left with the formula ∑ i < j ∫ Σ ij ⎡⎢⎢⎢⎢⎣∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ − ( X n ) ∇ n , n W − div Σ ,µ ( X n ∇ X t n )⎤⎥⎥⎥⎥⎦ dµ n − (5.11)for Q ( X ) . Note that the integrand in Lemma 5.4 as a whole is integrable by Lemma 3.3,and hence so is the integrand in (5.11) (since the terms we have already removed wereintegrable and integrated to zero). On the other hand, all of the terms in (5.11) besides thelast are individually integrable: the third term is integrable because ∥ ∇ W ∥ and ∣ X ∣ arebounded on X ’s compact support, the second term is integrable by Proposition 4.14 (1),and the first term is integrable because ∣ ∇ t X n ∣ ≤ ∥ ∇ X ∥ + ∣ X ∣∥ II ∥ , which is square integrableagain by Proposition 4.14 (1). It follows that the remaining term, div Σ ,µ ( X n ∇ X t n ) , isintegrable on Σ ij . t remains to establish that: ∑ i < j ∫ Σ ij div Σ ,µ ( X n ∇ X t n ) dµ n − = ∑ i < j ∫ ∂ Σ ij X n X n ∂ II ∂,∂ dµ n − . (5.12)This will follow by applying Lemma 5.7 (Stokes’ theorem) to the C ∞ vector-field Z ij ∶ = X n ij ∇ X t ij n ij , defined on Σ ij ∪ ∂ Σ ij for each i < j . Let us first verify the integrability conditions ofLemma 5.7: div Σ ,µ ( Z ij ) is integrable by the preceding paragraph, and as ∣ Z ij ∣ ≤ ∣ X ∣ ∥ II ∥ and X is tame, we see that Z ij is square-integrable on Σ ij and integrable on Σ ijk byProposition 4.14. Since Z ij is clearly tangential to Σ ij , Lemma 5.7 yields: ∑ i < j ∫ Σ ij div Σ ,µ ( Z ij ) dµ n − = ∑ i < j ∫ ∂ Σ ij X n ij ⟨ ∇ X t ij n ij , n ∂ij ⟩ dµ n − . An application of Lemma 5.9 then establishes (5.12), and completes the proof of Theorem5.2.
To establish Theorem 5.3, we will need the following polarization formula with respect toconstant vector fields, whose proof is deferred to Appendix A.
Lemma 5.11.
For any stationary cluster Ω whose cells are volume and perimeter regular,and for any admissible vector-field X : Q ( X + w ) = Q ( X ) + Q ( w ) + ∑ i < j ∫ Σ ij [ div Σ ,µ ( ∇ w X ) − Σ ,µ ( ∇ w W ⋅ X ) − X n ∇ w, n W − H ij,µ ( w n div µ X − X n ∇ w W ) ] dµ n − . Recall that Ω is assumed to be stationary and regular, and X is assumed to be tame. Tosimplify the integral in Lemma 5.11, we start by applying Stokes’ theorem (Lemma 5.7) to Z ij , the restriction of the tame vector-field Z = ∇ w X − X ∇ w W to Σ ij ∪ ∂ Σ ij ; by Remark5.8, the integrability assumptions of Lemma 5.7 are in force. It follows that: ∑ i < j ∫ Σ ij [ div Σ ,µ ( ∇ w X ) − Σ ,µ ( ∇ w W ⋅ X )] dµ n − = ∑ i < j ∫ Σ ij H ij,µ (⟨ ∇ w X, n ⟩ − X n ∇ w W ) dµ n − + ∑ i < j ∫ ∂ Σ ij ⟨ Z, n ∂ij ⟩ dµ n − . But since ∑ ( ℓ,m )∈ C ( i,j,k ) n ∂ℓm =
0, the total contribution of the boundary integral is zero.Plugging this into the formula of Lemma 5.11, we deduce that: Q ( X + w ) = Q ( X ) + Q ( w ) (5.13) + ∑ i < j ∫ Σ ij [ H ij,µ (⟨ ∇ w X, n ⟩ − X n ∇ w W − w n div µ X ) − X n ∇ w, n W ] dµ n − . We now claim that the first three terms of the above integrand are equal to H ij,µ div Σ ,µ ( X n w − w n X ) . Indeed, using the Leibniz rule div Σ ,µ ( f X ) = f div Σ ,µ X + ∇ X t f , the facts that iv Σ ,µ w = −∇ w W and that ∇ w =
0, and the symmetry of the second fundamental-form:div Σ ,µ ( X n w − w n X ) = X n div Σ ,µ w − w n div Σ ,µ X + ∇ w t ( X n ) − ∇ X t ( w n ) = − X n ∇ w W − w n div Σ ,µ X + ⟨ ∇ w t X, n ⟩ = − X n ∇ w W − w n div Σ ,µ X + ⟨ ∇ w X, n ⟩ − w n ⟨ ∇ n X, n ⟩ = − X n ∇ w W − w n div µ X + ⟨ ∇ w X, n ⟩ . Setting Y ij = X n ij w − w n ij X on Σ ij ∪ ∂ Σ ij , we would like to apply Stokes’ theorem(Lemma 5.7) again. The first and third integrability assumptions from (5.4) are satisfiedbecause X is tame (as in Remark 5.8). The second assumption is also satisfied because, asusual, div Σ ,µ Y ij is bounded by ∥ II ij ∥ times a polynomial in ∣ w ∣ , ∣ X ∣ and ∥ ∇ X ∥ , and hence isintegrable on Σ ij by tameness, Proposition 4.14 (1) and µ ( Σ ij ) < ∞ ; alternatively, div Σ ,µ Y ij must be integrable by Lemma 3.3, as all of the other terms in (5.13) are integrable.Applying Lemma 5.7, since Y ij is clearly tangential, we obtain as in (5.10): ∑ i < j ∫ Σ ij H ij,µ div Σ ,µ Y ij dµ n − = ∑ i < j < k ∫ Σ ijk ∑ ( ℓ,m )∈ C ( i,j,k ) H ℓm,µ ⟨ Y ℓm , n ∂ℓm ⟩ dµ n − . We now claim that the above integrand is pointwise zero. Indeed, introducing the followingvector-field on Σ ijk : Λ ∶ = − √ ( λ i n ∂jk + λ j n ∂ki + λ k n ∂ij ) , it is immediate to check that ⟨ Λ , n ℓm ⟩ = λ ℓ − λ m = H ℓm,µ . Hence, H ℓm,µ ⟨ Y ℓm , n ∂ℓm ⟩ = X n ℓm Λ n ℓm w n ∂ℓm − w n ℓm Λ n ℓm X n ∂ℓm , and so summing over all ( ℓ, m ) ∈ C ( i, j, k ) we obtain T αβγ X α Λ β w γ =
0, where T αβγ is the3-tensor from Lemma 5.10 which vanishes identically.We conclude from the above discussion that: Q ( X + w ) = Q ( X ) + Q ( w ) − ∑ i < j ∫ Σ ij X n ∇ w, n W dµ n − . (5.14)In particular, we formally deduce by setting X = w and using Q ( w ) = Q ( w ) , that: Q ( w ) = − ∑ i < j ∫ Σ ij w n ∇ w, n W dµ n − . (5.15)This is only formal, since w is not a tame vector-field, but can easily be made rigorous; infact, Q ( w ) was already computed in [40, Theorem 4.1] for µ = γ , the Gaussian measure,and the general case (5.15) is immediately obtained from [40, Lemma 5.2 and 5.3] (forany stationary cluster with volume and perimeter regular cells). Plugging (5.15) and theexpression for Q ( X ) from Theorem 5.2 into (5.14), the assertion of Theorem 5.3 readilyfollows. An additional crucial consequence of Theorem 4.8 is that we can define a consistent familyof (approximate) “inward” vector-fields: for each cell Ω i , we will try to define a smoothvector-field X i such that ⟨ X i , n Ω i ⟩ ≡ − n Ω i is the outward unit normal toΩ i , defined on ∂ ∗ Ω i ). This family of approximate inward fields will be crucial for ourvariational arguments. heorem 4.1 allows us to construct the inward fields on Σ = ⋃ ij Σ ij , since X i ∶ = − n ij is smooth on Σ ij . The issue is to extend these vector fields smoothly to all of R n , and thisis where Theorem 4.8 comes in: it will allow us to extend our inward fields smoothly toΣ (the triple points), and to construct approximate inward fields near Σ (the quadruplepoints).As usual, let δ ij denote the Kronecker delta (i.e., 1 if i = j and 0 otherwise), and let ∇ t denote the tangential component of the derivative. Proposition 6.1 (Existence of Approximate Inward Fields) . Let Ω be a stationary regularcluster with respect to µ . For every ǫ > , there is a subset K ⊂ R n , such that for every ǫ > , there is a family of vector-fields X , . . . , X q with the following properties:(1) K is compact, disjoint from Σ , satisfies µ n − ( Σ ∖ K ) ≤ ǫ , and each X k is C ∞ c andsupported inside K ;(2) for every k , ∫ Σ ∣ ∇ t X n k ∣ dµ n − ≤ ǫ ; (3) ∑ i < j µ n − { p ∈ Σ ij ∶ ∃ k ∈ { , . . . , q } X n ij k ( p ) ≠ δ kj − δ ki } ≤ ǫ + ǫ ; (4) For every i ≠ j , for every p ∈ Σ ij , there is some α ∈ [ , ] such that for every k , ∣ X n ij k ( p ) − α ( δ kj − δ ki )∣ ≤ αǫ ; (5) for every k , and at every point in R n , ∣ X k ∣ ≤ √ / . Definition 6.2 (Approximate Inward Fields) . A family X , . . . , X q of vector-fields satis-fying properties (1)-(5) above is called a family of ( ǫ , ǫ ) -approximate inward fields.The distinction between ǫ and ǫ and their order of quantification will be important inonly a single instance in this work (Lemma 7.5), but in all other applications of Proposition6.1 we will simply use ǫ = ǫ = ǫ . To construct our inward fields we will require a bit more information on the conformalproperties of the diffeomorphisms appearing in Theorem 4.8. Below, Ω is assumed to be astationary regular cluster.
Lemma 6.3.
Suppose that φ is a local C diffeomorphism defined on a neighborhood N p of p ∈ Σ ijk that maps p to the origin and Σ to Y × R n − . Let E = E ( ) , and write P E for the or-thogonal projection onto E . For r ∈ N p ∩ Σ ijk , let W ijk ( r ) = span { n ∂ij ( r ) , n ∂jk ( r ) , n ∂ki ( r )} .Then for any r ∈ N p ∩ Σ ijk , d r φ maps W ijk ( r ) ⊥ into { } × R n − , and P E ○ d r φ is aconformal (i.e. angle-preserving) transformation from W ijk ( r ) to E .Proof. First, note that if A ∶ E → E is any linear transformation that preserves Y then A is conformal. Indeed, up to relabelling coordinates we may assume that A preserves each“arm” of the Y . For { i, j, k } = { , , } , let w i = e j + e k − e i , so that each arm of the Y has the form { λw i ∶ λ ≥ } . Since A preserves the arms, Aw i is a positive multiple of w i ;say, Aw i = λ i w i . On the other hand, ∑ i λ i w i = A ∑ i w i =
0, and since the w i ’s are affinelyindependent, it follows that the λ i ’s are all equal, and so A is a multiple of the identity.Now consider r ∈ N p ∩ Σ (without loss of generality, { i, j, k } = { , , } ). Up torelabeling the coordinates, we may assume that for distinct i, j, k ∈ { , , } , φ maps Σ ij into Σ mij ∶ = Y ij × R n − , where Y ij ∶ = { x ∈ E ∶ x i = x j > x k } . To prove the first claim, notethat if v ∈ W ( r ) ⊥ then v is tangent to each of the surfaces Σ ij at r . It follows that ( d r φ ) v is tangent to each of the surfaces Σ mij , and so ( d r φ ) v ∈ { } × R n − .To prove the second claim, note that by Corollary 4.12, the boundary normals n ∂ij meet at 120 ○ angles, and so the cone ˜ Y = ⋃ i < j { λ n ∂ij ( r ) ∶ λ ≤ } is isometric to Y . Since ∂ij is tangent to Σ ij , ( d r φ ) n ∂ij is tangent to Σ mij , and so P E ( d r φ ) n ∂ij is tangent to Y ij .It follows that (when restricted to W ( r ) ) P E ○ d r φ maps ˜ Y to Y , and so by the firstparagraph, it is conformal on W ( r ) .A convenient consequence of Lemma 6.3 is that, after composing with a linear functionif necessary, we may always assume that each local diffeomorphism as in Theorem 4.8 (ii)is an isometry at p : Corollary 6.4.
For every p ∈ Σ there is a C ∞ diffeomorphism φ defined on a neighborhood N p ∋ p such that φ ( p ) = , φ ( N p ∩ Σ ) ⊂ Y × R n − , and d p φ is an isometry.Proof. Suppose that p ∈ Σ ijk and let W ijk = span { n ∂ij ( p ) , n ∂jk ( p ) , n ∂ki ( p )} . By Corol-lary 4.13, there exists a C ∞ diffeomorphism φ satisfying φ ( p ) = φ ( N p ∩ Σ ) ⊂ Y × R n − .By Lemma 6.3, P E ○ d p φ is conformal on W ijk . Let { v , . . . , v n } be an orthonormal basisfor R n such that { v , v } is an orthonormal basis for W ijk . Since P E ○ d p φ is conformal,there exists an orthonormal basis { w , w } for E and α > P E ( d p φ ) v i = αw i for i = ,
2; let us complete { w , w } to an orthonormal basis { w , . . . , w n } for E × R n − .Now consider the linear operator d p φ expressed as an invertible matrix M in the bases { v , . . . , v n } and { w , . . . , w n } ; that is, M ℓm = ⟨ w ℓ , ( d p φ ) v m ⟩ . We will write M in blocks as ( A BC D ) , where A is 2 × D is ( n − ) × ( n − ) . Our choice of w and w ensures that A = α Id. By Lemma 6.3, ( d p φ ) v m ∈ { } × R n − for every m ≥
3, and it follows that B = M − = ( A − − D − CA − D − ) = ( α Id 0 − α D − C D − ) . Let f ∶ E × R n − → E × R n − be the linear function that sends w m to ∑ ℓ M − ℓm w ℓ ; that is, f is the linear function that is represented in the basis { w , . . . , w n } by the matrix M − . Theform of M − above implies that f ( Y × R n − ) ⊂ Y × R n − , because if ( x, y ) ∈ Y × R n − thenthe E -component of f ( x, y ) is α − x ∈ Y . Hence, ˜ φ = f ○ φ is a local C ∞ diffeomorphismdefined on N p satisfying ˜ φ ( p ) = φ ( N p ∩ Σ ) ⊂ Y × R n − . Moreover, d p ˜ φ is an isometrybecause ( d p ˜ φ ) v m = ( d f )( d p φ ) v m = w m by the definition of f .By an analogous argument, we may also choose the local diffeomorphisms of Theo-rem 4.8 (iii) to be isometries at p : Corollary 6.5.
For every p ∈ Σ there is a C ,α diffeomorphism φ defined on a neighbor-hood N p ∋ p such that φ ( p ) = , φ ( N p ∩ Σ ) ⊂ T × R n − , and d p φ is an isometry.Proof. The proof is essentially identical to the proof of Corollary 6.4. The first main pointis that (like Y ) any linear transformation that preserves T must be conformal. The secondis that, by Lemma 6.3 and continuity up to Σ , at every p ∈ Σ and for every distinct i, j, k ∈ { , , , } , n ij , n jk and n ki are co-planar and meet at 120 ○ angles, and it followsthat at any r ∈ Σ , the cone˜ T = ⋃ { i,j,k,ℓ }={ , , , } { x ∈ R n ∶ ⟨ x, n ij ⟩ = , ⟨ x, n ik ⟩ ≤ , ⟨ x, n jk ⟩ ≤ } is isometric to T × R n − . The rest of the proof remains unchanged. Denote for brevity Σ ≥ = Σ ∪ Σ . We begin with the exact construction of the inward fieldson R n ∖ Σ ≥ . emma 6.6. There is a family of C ∞ vector-fields Z , . . . , Z q defined on R n ∖ Σ ≥ suchthat for every k , for every i ≠ j , and for every p ∈ Σ ij , ∣ Z k ( p )∣ ≤ √ / and ⟨ Z k ( p ) , n ij ( p )⟩ = δ kj − δ ki . Proof.
By applying a partition of unity, it suffices to prove the claim locally: we will showthat for every p ∈ R n ∖ Σ ≥ , we may define vector fields Z , . . . , Z q on a neighborhood of p to satisfy the required properties on that neighborhood.For p outside of (the closed) Σ, this is easy: we may choose a neighborhood of p whichis disjoint from Σ, and define Z i ≡ ⋯ ≡ Z q ≡ p ∈ Σ ij , we may (by Theorem 4.1) choose a neighborhood of p that does not intersectany other Σ kℓ ; on that neighborhood, define Z i to be a smooth extension of n ji , define Z j to be a smooth extension of n ij , and define all other Z k ’s to be identically zero.Finally, we will describe the construction on the triple-point set Σ : suppose (in orderto simplify notation) that p ∈ Σ and set E = E ( ) . We will choose a neighborhood N p and a local C ∞ diffeomorphism φ according to Corollary 6.4. By relabeling the coordinates,we may assume that φ sends N p ∩ Ω i into { x ∈ E ∶ x i > max { x j , x k }} × R n − , and it followsthat φ sends N p ∩ Σ ij into Σ mij ∶ = Y ij × R n − , for all distinct i, j, k ∈ { , , } (recall that Y ij ∶ = { x ∈ E ∶ x i = x j > x k } ). Let w i ∶ = √ ( e j + e k − e i ) ∈ E × R n − , and note thatthe vector-fields { − √ w i } satisfy the assertion on the model cone Σ m ∶ = Y × R n − : w i istangent to Σ mjk , and it has a constant normal component on Σ mij and Σ mik . The basic ideais to define Z i as the pull-back of − w i via φ , renormalized so as to satisfy the condition on ⟨ Z i , n ℓm ⟩ . Let us describe the construction a little more carefully, in order to verify thatthe renormalization preserves smoothness.Consider the metric ⟨ ⋅ , ⋅ ⟩ φ on φ ( N p ) obtained by pushing forward the Euclidean metricunder φ . According to Lemma 6.3, at every point r ∈ N p ∩ Σ , there is some c r > u, v ∈ W ( r ) , ⟨ P E ( d r φ ) u, P E ( d r φ ) v ⟩ = c r ⟨ u, v ⟩ . The same formula holdseven if only one of u and v belongs to W ( r ) , since Lemma 6.3 implies that P E ( d r φ )( u − P W ( r ) u ) = u ∈ R n . If, in addition, either ( d r φ ) u or ( d r φ ) v belongs to E × { } ,then ⟨( d r φ ) u, ( d r φ ) v ⟩ = ⟨ P E ( d r φ ) u, P E ( d r φ ) v ⟩ = c r ⟨ u, v ⟩ . After changing variables, itfollows that whenever at least one of u and v belongs to E × { } and at least one of ( d r φ ) − u and ( d r φ ) − v belongs to W ( r ) then ⟨ u, v ⟩ = c r ⟨( d r φ ) − u, ( d r φ ) − v ⟩ = c r ⟨ u, v ⟩ φ .Let ˜ n ij be a vector-field on span Σ mij = { x ∈ E ( ) ∶ x i = x j } × R n − obtained by smoothlyextending ( dφ ) n ij (for example, such a vector-field can be constructed by starting with e j − e i and applying the Gram-Schmidt process with respect to the inner product ⟨ ⋅ , ⋅ ⟩ φ ).Since, for any r ∈ Σ ∩ N p , w k ∈ E × { } and ( d r φ ) − ˜ n ij ∈ W ( r ) , the previous paragraphimplies that ⟨ w k , ˜ n ij ( φ ( r ))⟩ = c r ⟨( d r φ ) − ( w k ) , n ij ( r )⟩ =
0, because w k is tangent to Σ mij andso ( d r φ ) − ( w k ) is tangent to Σ ij at r . On the other hand, ⟨ P E ˜ n ij , P E ˜ n ik ⟩ = c r ⟨ n ij , n ik ⟩ = c r / ∣ P E ˜ n ij ∣ = c r at the point φ ( r ) . Combining the previous observations, it followsthat P E ˜ n ij ( φ ( r )) = ± √ c r / ( e j − e i ) for every i ≠ j ∈ { , , } . In fact, the assumptionthat φ maps N p ∩ Ω i into { x ∈ E ∶ x i > max { x j , x k }} × R n − implies that P E ˜ n ij ( φ ( r )) = √ c r / ( e j − e i ) .Now define the function f i on span Σ mij ∪ span Σ mik by f i ( x ) = ⎧⎪⎪⎨⎪⎪⎩⟨ w i , ˜ n ij ( x )⟩ φ for x ∈ span Σ mij ⟨ w i , ˜ n ik ( x )⟩ φ for x ∈ span Σ mik . This definition is consistent for x ∈ span Σ mij ∩ span Σ mik = { } × R n − : if r = φ − ( x ) then w i ∈ E × { } and ( d r φ ) − ˜ n ij = n ij ∈ W ( r ) , and so c r ⟨ w i , ˜ n ij ⟩ φ = ⟨ w i , ˜ n ij ⟩ = √ c r / ⟨ w i , e j − e i ⟩ = √ c r / j replaced by k ). Hence, f i may be extended to a C ∞ unction on all of R n , for example by setting f i ( x + y ) = f i ( x )+ f i ( y ) whenever x ∈ span Σ mij , y ∈ span Σ mik and x − y ∈ span Σ mij ∩ span Σ mik , i.e. x and y agree in the last n − f i does not vanish at the origin, and so by continuity, after possibly shrinking N p if necessary, f i does not vanish on the entire φ ( N p ) for all i = , , i = , ,
3, define Z i = − ( dφ ) − ( w i / f i ) . By the definition of f i , we have ⟨ Z i , n ij ⟩ = − ⟨ w i / f i , ˜ n ij ⟩ φ = − ij ,and similarly on Σ ik . On the other hand, w i is tangential (with respect to ⟨ ⋅ , ⋅ ⟩ and thereforealso with respect to ⟨ ⋅ , ⋅ ⟩ φ ) to Σ mjk , and so ⟨ Z i , n jk ⟩ ≡ jk .To prove the claim on the boundedness of Z i , recall that ⟨ ⋅ , ⋅ ⟩ φ agrees with the Euclideanmetric at the origin (as d p φ is an isometry). It follows that ∣ Z i ( p )∣ = √ /
3. By continuity,we may shrink the neighborhood N p in order to ensure that ∣ Z i ∣ ≤ √ / R n ∖ Σ . Note that we cannotsimply imitate our construction from Lemma 6.6, because in a neighborhood of p ∈ Σ , Σis only C ,α -diffeomorphic to the model cone T × R n − . In particular, if we were to imitatethat construction then we would end up with merely C ,α vector-fields. This would notbe sufficient for our purposes, for example because such non-Lipschitz vector-fields do notnecessarily admit flows. To avoid this issue, we will drop the requirement that the normalcomponents be exactly constant, and settle for a pointwise approximation. This in factmakes the construction a fair amount simpler. Lemma 6.7.
For every ǫ > , there is a family of C ∞ vector fields Y , . . . , Y q defined on R n ∖ Σ such that for every k , for every i ≠ j and every p ∈ Σ ij , ∣ Y k ( p )∣ ≤ √ / and ∣⟨ n ij ( p ) , Y k ⟩ − ( δ kj − δ ki )∣ ≤ ǫ. Proof.
By applying a partition of unity, it suffices to prove the claim locally: for every p ∈ R n ∖ Σ , we will locally define Y , . . . , Y q with the required properties. As long as p ∈ R n ∖ Σ ≥ , we may set Y i to be identical to Z i from Lemma 6.6 in a neighborhood of p ;it remains to describe the construction in a neighborhood of p ∈ Σ .Suppose (to simplify notation) that p ∈ Σ . Choose a neighborhood N p ∋ p and alocal C ,α diffeomorphism φ according to Corollary 6.5. By relabeling the coordinates, wemay assume that φ sends N p ∩ Ω i into { x ∈ E ( ) ∶ x i > max j ≠ i x j } × R n − . Since d p φ is anisometry, it follows that ( d p φ ) n ij = √ ( e j − e i ) for every distinct i, j ∈ { , , , } .For { i, j, k, ℓ } = { , , , } , let w i = √ ( e j + e k + e ℓ − e i ) ∈ E ( ) × R n − . Then each w i satisfies ⟨ n ij ( p ) , ( d p φ ) − w i ⟩ = √ /
3, and ⟨ n jk ( p ) , ( d p φ ) − w i ⟩ = i / ∈ { j, k } . For each i = , . . . ,
4, define Y i in a neighborhood of p to be the constant vector-field − √ / ( d p φ ) − w i .By continuity, there is a neighborhood of p on which ⟨ Y i , n ij ⟩ ∈ [ − − ǫ, − + ǫ ] and ⟨ Y i , n jk ⟩ ∈ [ − ǫ, ǫ ] whenever i / ∈ { j, k } . Moreover, the fact d p φ is an isometry implies that ∣ Y i ∣ = √ / Y , . . . , Y q to be zero on this neighborhood.We can now combine the exact inward fields of Lemma 6.6 with the approximate inwardfields of Lemma 6.7, using the cutoff functions of Subsection 5.1. Proof of Proposition 6.1.
Fix ǫ >
0, and choose η as in Lemma 5.5, with ǫ = ǫ . Let K be the support of η , and let K ∶ = { η = } ⊂ K ; by Lemma 5.5, K and K are compact,disjoint from Σ , and satisfy µ n − ( Σ ∖ K ) ≤ µ n − ( Σ ∖ K ) ≤ ǫ . Let Z , . . . , Z q be vectorfields as in Lemma 6.6.Let C η be the constant C U of Lemma 5.6 applied to U = R n ∖ K . Then let Y , . . . , Y q bevector-fields as in Lemma 6.7 with parameter ǫ = √ ǫ / C η . For some δ > depending on K , { Y k } and Σ), choose ξ as in Lemma 5.6, with parameter δ = min ( ǫ , δ ) .Define X k = η ⋅ ( ξZ k + ( − ξ ) Y k ) . Note that this defines a C ∞ vector-field on R n , because supp ( ηξ ) is contained in the domainof Z k , namely R n ∖ Σ ≥ , and supp η is contained in the domain of Y k , namely R n ∖ Σ ; assupp X k ⊂ supp η = K , this establishes requirement (1).Next, since ∇ t Z n k = , observe that ∇ t X n k = ( ξZ k + ( − ξ ) Y k ) n ∇ t η + η ⋅ ( Z k − Y k ) n ∇ t ξ + η ( − ξ ) ∇ t Y n k . To control the first term above, note that ∣( ξZ k + ( − ξ ) Y k ) n ∣ ≤ √ /
2. For the second term,note that ∣ η ⋅ ( Z k − Y k ) n ∣ ≤ √ ǫ / C η . For the third term (which vanishes outside the set { ξ < } ), we split ∇ t Y n k into its normal and tangential components as ( ∇ t Y k ) n + ∇ Y t k n , andobserve that ∣ ∇ Y t k n ∣ ≤ ∣ Y k ∣∥ II ∥ ≤ √ / ∥ II ∥ . Hence, ∣ ∇ t X n k ∣ ≤ ∣ ∇ η ∣ + ǫ C η ∣ ∇ ξ ∣ + { ξ < } η ( ∥ ∇ t Y k ∥ + ∥ II ∥ ) . Integrating, we use Lemma 5.5 to control ∣ ∇ η ∣ and Lemma 5.6 to control ∣ ∇ ξ ∣ : ∫ Σ ∣ ∇ t X n k ∣ dµ n − ≤ ǫ + ∫ Σ { ξ < } η (∥ ∇ t Y k ∥ + ∥ II ∥ ) dµ n − . Now, ∇ Y k is uniformly bounded on the compact support of η , and hence η ∥ ∇ t Y k ∥ isuniformly bounded and therefore integrable, as µ n − ( Σ ) < ∞ . In addition, η ∥ II ∥ isintegrable by Proposition 4.14 (1), as the cutoff function η is compactly supported awayfrom Σ . Hence, we may choose δ > ǫ . By appropriately modifying ǫ by a constant factor,this proves requirement (2).For requirement (3), note that X n ij k = Z n ij k = δ kj − δ ki whenever η = ξ =
1. Since K = { η = } , we deduce that ⋃ i < j { p ∈ Σ ij ∩ K ∶ ∃ k X n ij k ≠ δ kj − δ ki } ⊂ { p ∈ Σ ∩ K ∶ ξ < } . Hence, requirement (3) follows by the union-bound from Lemma 5.6 and the fact that µ n − ( Σ ∖ K ) ≤ ǫ . Requirements (4) and (5) follow from Lemmas 6.6 and 6.7, and thefact that both of these properties are preserved by pointwise convex combinations (the α in requirement (4) will be η ( p ) ). In this subsection we record some useful estimates which follow from the properties ofapproximate inward fields. We first recall the definition of the q × q symmetric matrix L A ,associated to A = { A ij } ≤ i < j ≤ q : L A ∶ = ∑ ≤ i < j ≤ q A ij ( e i − e j )( e i − e j ) T . (6.1) Lemma 6.8.
Let Ω be a stationary regular cluster, let A ij = µ n − ( Σ ij ) , and let ( X , . . . , X q ) be a collection of ( ǫ , ǫ ) -approximate inward fields. Given a ∈ R q , set X ∶ = ∑ qk = a k X k .Then:(1) ∫ Σ ∣ ∇ t X n ∣ dµ n − ≤ q ∣ a ∣ ǫ . ∫ Σ ( X n ) dµ n − ≥ a T L A a − ∣ a ∣ ( ǫ + ǫ ) .(3) For all i , ∣( δ X V ( Ω ) + L A a ) i ∣ ≤ max ( , ǫ )( ǫ + ǫ )√ q ∣ a ∣ .Proof. The first assertion follows since by property (2) of approximate inward fields andCauchy–Schwarz: ∫ Σ ∣ ∇ t X n ∣ dµ n − ≤ ∫ Σ ∣ a ∣ q ∑ k = ∣ ∇ t X n k ∣ dµ n − . The second assertion follows since, denoting: q ij ∶ = µ n − ( p ∈ Σ ij ; ∃ k = , . . . , q X n k ≠ δ kj − δ ki ) , we have ∑ i < j q ij ≤ ǫ + ǫ by property (3). Therefore: ∫ Σ ij ( X n ) dµ n − ≥ ( a j − a i ) ( µ n − ( Σ ij ) − q ij ) , and after summation over i < j , we obtain: ∑ i < j ∫ Σ ij ( X n ) dµ n − ≥ ∑ i < j A ij ( a j − a i ) − ( ǫ + ǫ ) max i < j ( a j − a i ) ≥ a T L A a − ( ǫ + ǫ ) ∣ a ∣ . To see the third assertion, first note that: ∫ Σ ij ∣ X n k − ( δ kj − δ ki )∣ dµ n − ≤ q ij max ( , ǫ ) , since by property (4): sup p ∈ Σ ij ∣ X n k ( p ) − ( δ jk − δ ik )∣ ≤ max ( , ǫ ) . It follows by (4.3) that: ∣( δ X V ( Ω ) + L A a ) i ∣ ≤ RRRRRRRRRRR ∑ j ≠ i ∫ Σ ij X n dγ n − − ∑ j ≠ i ( a j − a i ) A ij RRRRRRRRRRR ≤ ∑ j ≠ i ∫ Σ ij ∣ ∑ k a k X n k − ( a j − a i )∣ dγ n − ≤ ∑ j ≠ i ∫ Σ ij ∑ k ∣ a k ∣ ∣ X n k − ( δ jk − δ ik )∣ dγ n − ≤ max ( , ǫ ) ∑ j ≠ i q ij ∑ k ∣ a k ∣ ≤ max ( , ǫ )( ǫ + ǫ )√ q ∣ a ∣ . Lemma 6.9.
Let ( X , . . . , X q ) be a collection of ( ǫ , ǫ ) -approximate inward vector-fieldswith ǫ < . Then for all distinct i, j, k , and at every point of Σ ijk ,1. There exists α ∈ [ , ] so that: ∥ P W ijk ( q ∑ ℓ = X ℓ X Tℓ − α Id ) P W ijk ∥ ≤ C q αǫ , where P W ijk denote the orthogonal projection onto the two-dimensional subspace W ijk = span ( n ij , n jk , n ki ) , ∥ ⋅ ∥ denotes the Hilbert-Schmidt norm, and C q is a constant de-pending solely on q . . For any linear operator Z ∶ R n → R n so that P W ijk Z = ZP W ijk = Z and tr ( Z ) = : ∣ tr (( q ∑ ℓ = X ℓ X Tℓ ) Z )∣ ≤ C q ∥ Z ∥ ǫ . Proof.
By continuity, property (4) extends from Σ ij to Σ ijk . Given p ∈ Σ ijk , the constant α = α p ∈ [ , ] from property (4) is a common scaling factor to all { X ℓ } qℓ = , and so it isenough to prove the claim for α =
1. Since ∣ ⟨ X i , n ji ⟩ − ∣ ≤ ǫ , ∣ ⟨ X i , n ki ⟩ − ∣ ≤ ǫ and ∣ ⟨ X i , n jk ⟩ ∣ ≤ ǫ , an easy calculation verifies that: ∣ P W ijk X i − √ n ∂jk ∣ ≤ √ ǫ . Denoting V i = √ n ∂jk and E i = P W ijk X i − V i , note that: ∥ P W ijk X i X Ti P W ijk − V i V Ti ∥ ≤ ∥ E i V Ti ∥ + ∥ E i E Ti ∥ = ∣ E i ∣ ∣ V i ∣ + ∣ E i ∣ ≤ ǫ + ǫ . On the other hand, if ℓ ∉ { i, j, k } then similarly: ∥ P W ijk X ℓ X Tℓ P W ijk ∥ = ∣ P W ijk X ℓ ∣ ≤ ǫ . Since: V i V Ti + V j V Tj + V k V Tk = P W ijk , it follows that: ∥ P W ijk ( q ∑ ℓ = X ℓ X Tℓ − ) P W ijk ∥ ≤ ǫ + qǫ , establishing the first assertion.For the second assertion, denote T = ∑ qℓ = X ℓ X Tℓ and W = W ijk , and write:tr ( T Z ) = tr ( P W T P W Z ) + tr (( T − P W T P W ) Z ) = tr ( P W ( T − α Id ) P W Z ) , using both properties of Z . It remains to apply the Cauchy–Schwarz inequality and theestimate from the first assertion. From this point on, unless otherwise stated, we will switch from working with the generalmeasure µ to the standard Gaussian measure γ . In this section, we identify several crucialproperties of stable regular clusters for the Gaussian measure, when the number of cells isat most n +
1. In fact, we can say a bit more, as described below.
Definition 7.1 (Flat Cluster) . A regular cluster Ω is called flat and is said to have flatcells if for every i ≠ j , II ij = ij . Theorem 7.2 (Stable Regular Clusters) . Let Ω denote a stable regular q -cluster withrespect to γ , and assume that q ≤ n + . Then:(i) Ω has flat, connected and convex cells.(ii) n ij is constant on every non-empty Σ ij , and Σ ij is contained in a single hyperplaneperpendicular to n ij . In addition dim span { n ij } i < j ≤ q − . iii) All cells are convex polyhedra with at most q − facets, given by: Ω i = ⋂ j ≠ i ∶ A ij > { x ∈ R n ∶ ⟨ n ij , x ⟩ < λ j − λ i } , (7.1) where, recall, λ ∈ E ∗ is given by Lemma 4.2 (ii). A further refinement of this theorem will be given in Theorem 10.1. The main task inthe proof of Theorem 7.2 is to prove that the cells are flat. This task will be broken upinto two cases, depending on whether the cluster is full-dimensional or not.
Definition 7.3 (Dimension-Deficient and Full-Dimensional Clusters) . A regular cluster Ωis called dimension-deficient if there is some θ ∈ S n − which is perpendicular to all normals { n ij ( x )} for all x ∈ Σ ij and all i ≠ j . Otherwise, it is called full-dimensional .In both cases, we will assume that the cluster is not flat, and produce a vector-field todemonstrate that it is not stable. When the cluster is dimension-deficient (and withoutany restrictions on q !), we will build such a vector field by taking a carefully chosen linearcombination of inward fields, multiplied by a “height”-dependent linear function in a di-rection of dimension-deficiency. When the cluster is full-dimensional and q ≤ n +
1, we willtake again a carefully chosen linear combination of inward fields together with a constantfield. Vector-fields of the latter form were previously considered in [39] but in the contextof a single set with smooth boundary instead of a cluster (of course, in a single set setting,having moreover a smooth boundary, there is no need to construct the inward field as it issimply given by the interior unit-normal, and its variations are well-known and classical).
In this subsection we temporary revert to treating a general measure µ , as the Gaussianmeasure’s properties will not be used. Lemma 7.4.
A cluster Ω (with respect to µ ) is dimension-deficient if and only if thereexists θ ∈ S n − , a cluster ˜Ω in θ ⊥ (with respect to ˜ µ , the marginal of µ on θ ⊥ ), so that Ω = ˜Ω × R up to null-sets.Proof. The “if” direction is obvious as ∂ ∗ ( ˜Ω i × R ) = ( ∂ ∗ Ω i ) × R and modification of a clusterby null-sets does not change its interfaces. For the “only if” direction, assume without lossof generality that the cluster Ω is dimension-deficient in the direction e n ∈ S n − . Observethat by the Gauss–Green–De Giorgi theorem (3.3) and (3.5): ∫ Ω i ∂ x n ϕ d H n = ∫ ∂ ∗ Ω i ϕ ⟨ e n , n Ω i ⟩ d H n − = ∑ j ≠ i ∫ Σ ij ϕ ⟨ e n , n ij ⟩ d H n − = , for all ϕ ∈ C c ( R n ) . Applying this to − ϕ ǫ ( x − ⋅ ) where ϕ ǫ is a compactly-supported approx-imation of identity, and denoting u ǫ ∶ = Ω i ∗ ϕ ǫ , it follows that: ∂ x n u ǫ ( x ) = ∫ Ω i ∂ x n ϕ ǫ ( x − y ) d H n ( y ) = ∀ x ∈ R n . Since u ǫ ∈ C ∞ ( R n ) , it follows that u ǫ ( x ) = ˜ u ǫ ( ˜ x ) for some ˜ u ǫ ∈ C ∞ ( R n − ) , where ˜ x = ( x , . . . , x n − ) and x = ( ˜ x, x n ) . But since u ǫ → Ω i in L loc ( R n ) as ǫ →
0, it follows byFubini’s theorem that for a.e. ˜ x ∈ R n − , for a.e. x n ∈ R , 1 Ω i ( ˜ x, x n ) = ˜Ω i ( ˜ x ) for someBorel set ˜Ω i in R n − . In other words, Ω i coincides with ˜Ω i × R up to a null-set. Since { Ω i } are disjoint, by modifying { ˜Ω i } on null-sets, we can ensure that they are also disjoint.Since ∂ ∗ ( ˜Ω i × R ) = ( ∂ ∗ Ω i ) × R , it follows that P ˜ µ ( ˜Ω i ) = P µ ( Ω i ) < ∞ and ˜ µ ( R n − ∖ ∪ i ˜Ω i ) = µ ( R n ∖ ∪ i Ω i ) =
0, verifying that ˜Ω is indeed a cluster. he key lemma we will require pertains to the following quantity associated to a regularcluster Ω (with respect to µ ) and a tame vector-field X : R ( X ) ∶ = ∑ i < j [ ∫ Σ ij (∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ ) dµ n − − ∫ ∂ Σ ij X n X n ∂ II ∂,∂ dµ n − ] . Lemma 7.5.
Let Ω be a stationary regular cluster (with respect to µ ) which is non-flat.Then there exist ǫ , ǫ > , a collection { X , . . . , X q } of ( ǫ , ǫ ) -approximate inward fields,and a linear combination X = ∑ qi = a i X i , so that R ( X ) < .Proof. By assumption there is some non-flat Σ ij , and we may assume without loss ofgenerality that it is Σ . By continuity and relative openness of Σ in Σ, there exists abounded neighborhood N p of p ∈ Σ so that II ≠ ∩ N p = Σ ∩ N p , and we denote δ ∶ = ∫ Σ ∩ N p ∥ II ∥ dµ n − > . Fix ǫ ∶ = δ /( q ) , and let K be the compact set from Proposition 6.1, which is disjointfrom Σ and satisfies µ n − ( Σ ∖ K ) ≤ ǫ . We can always assume that K contains N p . Then: ∞ > ∫ Σ ∩ K ∥ II ∥ dµ n − ≥ δ > , (7.2)where the finiteness is ensured by Proposition 4.14 (1). By Proposition 6.1, for every ǫ ∈ ( , ) , there exists a collection { X , . . . , X q } of ( ǫ , ǫ ) -approximate inward vector-fields for Ω which are supported inside K . Thanks to the finiteness in (7.2), by taking ǫ > ∫ Σ ∩ K (( X n ) + ( X n ) )∥ II ∥ dµ n − ≥ δ. (7.3)As { X i } are supported in K , we will henceforth drop the intersection with K in all subse-quent integrals.For a ∈ R q , define X a ∶ = ∑ qℓ = a ℓ X ℓ , and introduce the following quadratic forms: R ( a ) ∶ = ∑ i < j ∫ Σ ij ( X n a ) ∥ II ∥ dµ n − ,R ( a ) ∶ = ∑ i < j ∫ ∂ Σ ij X n a X n ∂ a II ∂,∂ dµ n − . Note that R and R are well-defined, as II ij is square-integrable on Σ ij ∩ K and II ij∂,∂ isintegrable on ∂ Σ ij ∩ K by Proposition 4.14. Note that:tr ( R ) ≥ ∫ Σ (( X n ) + ( X n ) )∥ II ∥ dµ n − ≥ δ by (7.3). In addition, observe that given p ∈ ∂ Σ ij and setting Z = n ij n T∂ij , Lemma 6.9implies: ∣ q ∑ ℓ = X n ij ℓ X n ∂ij ℓ ∣ = ∣ tr (( q ∑ ℓ = X ℓ X Tℓ ) Z )∣ ≤ C q ǫ . Consequently, the integrability of II ∂,∂ on Σ ∩ K implies that tr ( R ) ≥ − C K C q ǫ . Hence,by choosing ǫ > ( R + R ) ≥ δ − C K C q ǫ ≥ δ . t follows that we may find a unit-vector a ∈ R q so that R ( a ) + R ( a ) ≥ δ q . (7.4)Let us fix this a = a ǫ ,ǫ and set X = X ǫ ,ǫ ∶ = X a .It remains to apply Lemma 6.8 (1), which together with ∣ a ∣ = R ( X ǫ ,ǫ ) = ∑ i < j ∫ Σ ij ∣ ∇ t X n ∣ dµ n − − R ( a ) − R ( a ) ≤ qǫ − δ q . Recalling that ǫ was chosen to be δ q , we see that R ( X ǫ ,ǫ ) < ǫ > γ solely. Proposition 7.6.
If a stable regular cluster (with respect to γ ) is dimension-deficient thenit is flat.Proof. Suppose that Ω is stationary, regular, dimension-deficient, and non-flat; we willprove that it is unstable. Without loss of generality, we will assume that Ω is dimension-deficient in the last coordinate. By Lemma 7.4, there exists a cluster ˜Ω in R n − such thatΩ = ˜Ω × R up to null-sets. Since ∂ ( ˜Ω × R ) = ∂ ˜Ω × R and since Ω is stationary and regularwith respect to γ n , it is immediate to verify that ˜Ω is equally stationary and regular withrespect to γ n − . Since Ω is assumed non-flat, the same clearly holds for ˜Ω. We will use atilde to denote everything related to ˜Ω: ˜Σ ij are the interfaces of ˜Ω, ˜ n ij are the normals tothose interfaces, and so on.By Lemma 7.5 applied to ˜Ω, there exist ǫ , ǫ >
0, a collection { ˜ X , . . . , ˜ X q } of ( ǫ , ǫ ) -approximate inward fields for ˜Ω, and a linear combination ˜ X = ˜ X ǫ ,ǫ ∶ = ∑ qi = a i ˜ X i , so that˜ R ( ˜ X ) <
0, where:˜ R ( ˜ X ) ∶ = ∑ i < j [ ∫ ˜Σ ij (∣ ∇ t ˜ X ˜ n ∣ − ( ˜ X n ) ∥ ˜II ∥ ) dγ n − − ∫ ∂ ˜Σ ij ˜ X ˜ n ˜ X ˜ n ∂ ˜II ∂,∂ dγ n − ] . Now take f = f ǫ ∈ C ∞ c ( R ) so that ∫ f dγ = ∫ f dγ = ∫ ( f ′ ( x )) dγ ≤ + ǫ ;note that the Gaussian Poincar´e inequality (e.g. [4]) ensures that ∫ ( f ′ ( x )) dγ ≥ f ( x ) = x , so an appropriatesmooth cut-off of this function will do the job. We define a tame vector-field X = X ǫ on R n by X ( x ) = f ( x n ) ˜ X ( ˜ x ) , where x = ( ˜ x, x n ) . By (5.1), as ∇ n , n W = Q ( X ) = R ( X ) − ∑ i < j ∫ Σ ij ( X n ) dγ n − , (7.5)where: R ( X ) ∶ = ∑ i < j [ ∫ Σ ij (∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ ) dγ n − − ∫ ∂ Σ ij X n X n ∂ II ∂,∂ dγ n − ] . Note that for x = ( ˜ x, x n ) ∈ Σ ij , we have ˜ x ∈ ˜Σ ij , X n ( x ) = f ( x n ) ˜ X ˜ n ( ˜ x ) , X n ∂ ( x ) = f ( x n ) ˜ X ˜ n ∂ ( ˜ x ) , ∥ II ( x )∥ = ∥ ˜II ( ˜ x )∥ , and for x ∈ ∂ Σ ij , we have ˜ x ∈ ∂ ˜Σ ij and II ∂,∂ ( x ) = ˜II ∂,∂ ( ˜ x ) . onsequently, thanks to the product structure of Ω and γ n − = γ n − ⊗ γ , Fubini’s theoremand the fact that ∫ R f dγ = ∑ i < j ∫ Σ ij ( X n ) ∥ II ∥ dγ n − = ∑ i < j ∫ ˜Σ ij ( ˜ X ˜ n ) ∥ ˜II ∥ dγ n − , ∑ i < j ∫ ∂ Σ ij X n X n ∂ II ∂,∂ dγ n − = ∑ i < j ∫ ∂ ˜Σ ij ˜ X ˜ n ˜ X ˜ n ∂ ˜II ∂,∂ dγ n − , ∑ i < j ∫ Σ ij ( X n ) dγ n − = ∑ i < j ∫ ˜Σ ij ( ˜ X n ) dγ n − . As for the integral involving ∣ ∇ t X n ∣ , observe that for x ∈ Σ ij : ∇ t X n ( x ) = e n f ′ ( x n ) ˜ X ˜ n ( ˜ x ) + f ( x n )( ˜ ∇ t ˜ X ˜ n ( ˜ x ) , ) , and since the two terms on the right are orthogonal, ∣ ∇ t X n ( x )∣ = ( f ′ ( x n )) ( ˜ X ˜ n ( ˜ x )) + f ( x n )∣ ˜ ∇ t ˜ X ˜ n ( ˜ x )∣ . By Fubini’s theorem and the fact that ∫ R f dγ = ∫ R ( f ′ ) dγ ≤ + ǫ , ∫ Σ ij ∣ ∇ t X n ∣ dγ n − ≤ ∫ ˜Σ ij (( + ǫ )( ˜ X ˜ n ) + ∣ ˜ ∇ t ˜ X ˜ n ∣ ) dγ n − . Plugging all of these relations into (7.5), we conclude that: Q ( X ǫ ) ≤ ∑ i < j ∫ ˜Σ ij ǫ ( ˜ X ˜ n ) dγ n − + ˜ R ( ˜ X ) . But by Proposition 6.1, ˜ X n = ∑ qi = a i ˜ X n i is uniformly bounded on ˜Σ , and since γ n − ( ˜Σ ) = γ n − ( Σ ) < ∞ , we see that the contribution of the first term on the right can be made arbi-trarily small. Recalling that ˜ R ( ˜ X ) <
0, it follows that Q ( X ǫ ) < ǫ > δ X ǫ V = ∫ f dγ =
0. By definition, thisverifies that the cluster Ω is unstable.
Proposition 7.7. If q ≤ n + then every full-dimensional, stable regular cluster (withrespect to γ ) is flat and necessarily q = n + . For the proof, let us first expand a bit on Definition 7.3.
Definition 7.8 (Effective Dimension and Dimension of Deficiency) . Given a regular clusterΩ, consider the linear span N of { n ij ( x )} for all x ∈ Σ ij and all i ≠ j . The dimension e of N is called the effective dimension of Ω and Ω is said to be effectively e -dimensional. Thedimension of N ⊥ is called the dimension of deficiency of Ω, and Ω is said to be dimension-deficient in the directions of N ⊥ . If N = R n , recall that Ω is called full-dimensional.We will make use of the following linear operator associated to a q -cluster Ω (note thatby [40, Corollary 4.4], any Borel set is volume regular with respect to γ , so the definitionis indeed well justified without requiring regularity): M ∶ R n → E ( q − ) , M w ∶ = δ w V ( Ω ) . We first observe a general relation between the kernel of M and the directions of dimension-deficiency of the cluster Ω. emma 7.9. For any stable regular q -cluster Ω , Ker M coincides with Ω ’s directions ofdimension-deficiency. In particular, if d denotes Ω ’s dimension of deficiency and e denotesits effective dimension, then:(i) d = dim Ker M and e = dim Im M .(ii) Ω is full-dimensional iff M is injective.(iii) Ω is effectively ( q − ) -dimensional iff M is surjective.(iv) The effective dimension of Ω is at most q − .(v) If q < n + then e < n and d > , so Ω must be dimension-deficient.Proof. Clearly, if w is a direction of dimension-deficiency then δ w V = ∇ w, n W = w n for the Gaussian measure,we have: Q ( w ) = − ∑ i < j ∫ Σ ij ( w n ) dγ n − ≤ . Consequently, stability implies that if δ w V = Q ( w ) ≥
0, which means that w isperpendicular to all normals n ij , and is thus a direction of dimension-deficiency. Since M ∶ R n → E ( q − ) , we have dim Im M = n − dim Ker M = n − d = e , and e ≤ dim E ( q − ) = q − M has trivial kernel and q ≥ n + q ≤ n + q = n + M must be surjective. Thisreduces the proof of Proposition 7.7 to the following slightly more general claim: Proposition 7.10.
A stable regular cluster (with respect to γ ) for which M is surjectiveis flat.Proof. Let Ω be a stationary, regular and non-flat cluster, for which M is surjective; wewill show that it is unstable.By Lemma 7.5, there exist ǫ , ǫ > X = ∑ qi = a i X i of ( ǫ , ǫ ) -approximate inward fields { X , . . . , X q } , so that R ( X ) <
0, where recall: R ( X ) ∶ = ∑ i < j [ ∫ Σ ij (∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ ) dγ n − − ∫ ∂ Σ ij X n X n ∂ II ∂,∂ dγ n − ] . Now choose w ∈ R n such that δ w V ( Ω ) = M w = − δ X V ( Ω ) , which is always possiblebecause M is surjective, and denote Y = X + w . By Theorem 5.3 and the fact that ∇ n , n W = ∇ w t , n W = Q ( Y ) = R ( X ) − ∑ i < j ∫ Σ ij ( Y n ) dγ n − ≤ R ( X ) < . But since δ Y V ( Ω ) = δ X V ( Ω ) + δ w V ( Ω ) =
0, it follows by definition that Ω is unstable.
By combining Proposition 7.6 and Proposition 7.7, we know that all stable regular clustersare flat. To show that their cells must be connected, we will show that if there is adisconnected cell of a stationary, regular, flat cluster Ω then Ω is unstable.The following useful lemma was already proved in [40, Corollary 4.6]; for completeness,we provide a somewhat more direct proof. Recall the definition of L A from (6.1). emma 7.11. Let Ω be a q -cluster with γ ( Ω ) ∈ int ∆ ( q − ) and let A ij = γ n − ( Σ ij ) . Considerthe undirected graph G with vertices { , . . . , q } and an edge between i and j if A ij > . Then G is connected and L A is positive-definite as an operator on E ( q − ) (in particular, it hasfull-rank q − ).Proof. The graph G is connected because if S ⊂ { , . . . , q } were a non-trivial connected-component then U = ⋃ i ∈ S Ω i would satisfy γ ( U ) ∈ ( , ) . On the other hand, we would have P γ ( U ) = ∑ i ∈ S,j ∉ S A ij = L A is positive semi-definite as v T L A v = ∑ i ≠ j A ij ( v i − v j ) with A ij ≥
0. More-over, the fact that G is connected implies that v T L A v = v ∈ R q if and only if thecoordinates of v are all identical, which means that v = v ∈ E ( q − ) . It follows that L A is strictly positive-definite on E ( q − ) .Let Ω be a stationary regular flat cluster, and recall our convention from Theorem 4.1(ii). If Ω has a disconnected cell, we may suppose without loss of generality that Ω isdisconnected. We can then define a ( q + ) -cluster ˜Ω by splitting Ω into two non-trivialpieces ˜Ω and ˜Ω q + , each consisting of unions of connected components of Ω , while leavingthe other cells unaltered, i.e. ˜Ω i = Ω i for i = , . . . , q . Note that ∂ Ω = ∂ ˜Ω ∪ ∂ ˜Ω q + and that ∂ ∗ Ω = ∂ ∗ ˜Ω ∪ ∂ ∗ ˜Ω q + . Consequently, ˜Ω is still a stationary and regular cluster. It alsofollows that δ kX V ( Ω ) = δ kX V ( ˜Ω ) + δ kX V ( ˜Ω ) q + and δ kX A ( Ω ) = δ kX A ( ˜Ω ) for all k ≥ C ∞ c vector-fields X . In addition, denoting ˜ H ij,γ = H ˜Σ ij,γ , since ˜ H j,γ = ˜ H ( q + ) j,γ = H j,γ forall j = , . . . , q , it follows that ˜ H ij,γ = ˜ λ i − ˜ λ j for all i ≠ j = , . . . , q + λ = ˜ λ q + = λ and ˜ λ j = λ j for all j = , . . . , q . In particular, it follows that Q ˜Ω ( X ) = Q Ω ( X ) .We will construct a C ∞ c vector-field X with δ X V ( ˜Ω ) = e q + − e and Q ˜Ω ( X ) < . (7.6)Applying the same vector field to Ω, it will follow by the previous comments that δ X V ( Ω ) = Q Ω ( X ) <
0, demonstrating that Ω is unstable.To construct X , take ǫ > { X ǫi } i = ,...,q + be an ( ǫ, ǫ ) -approximate family ofinward fields for ˜Ω. Given a ∈ R q + , set X ǫa = ∑ q + i = a i X ǫi , and let V ǫ ∶ E ( q ) → E ( q ) bethe linear map defined by V ǫ a = δ X ǫa V ( ˜Ω ) . Observe that by Lemma 6.8 (3), V ǫ → − L A as ǫ → A = A ( ˜Ω ) and L A was defined in (6.1). ByLemma 7.11 L A is non-degenerate, and hence V − ǫ exists and is bounded uniformly in ǫ for all ǫ > a ǫ = V − ǫ ( e q + − e ) , denote X ǫ = X ǫa ǫ , and note that δ X ǫ V ( ˜Ω ) = V ǫ a ǫ = e q + − e by construction. By Theorem 5.2 and the flatness of Ω andhence of ˜Ω, we have: Q ˜Ω ( X ǫ ) = ∑ i < j ∫ ˜Σ ij (∣ ∇ t X n ǫ ∣ − ( X n ǫ ) ) dγ n − . Applying Lemma 6.8 (1) and (2), it follows that: Q ˜Ω ( X ǫ ) ≤ − ( a ǫ ) T L A a ǫ + ( q + ) ǫ ∣ a ǫ ∣ . Since a ǫ → − L − A ( e q + − e ) as ǫ → ∣ a ǫ ∣ is bounded away from 0 and ∞ for small-enough ǫ >
0. Recalling that L A is strictly positive-definite, we conclude that Q ( X ǫ ) < ǫ >
0. This completes the construction of a vector-field satisfying (7.6),and thus the proof of Theorem 7.2. .5 Stable regular clusters have convex cells To proceed with the proof of Theorem 7.2, we will show that the cells of a stationary,regular, flat cluster Ω with connected open cells must necessarily be convex.We will require the following proposition, which may be of independent interest. Aswe could not find a reference for this in the literature (in the case that A ≠ ∅ below), weprovide a proof. Proposition 7.12 (From Almost Local to Global Convexity) . Let Ω be an open connectedsubset of R n , and let A ⊂ ∂ Ω be a Borel set with H n − ( A ) = . Assume that for every p ∈ ∂ Ω ∖ A there exists an open neighborhood N p of p so that Ω ∩ N p is convex. Then Ω isconvex. It is easy to see that the claim is false when H n − is replaced by H n − k with k < Lemma 7.13 (From Local to Global Convexity) . Let Ω be an open connected subset of R n , so that for any p ∈ ∂ Ω , there exists an open neighborhood N p of p so that Ω ∩ N p isconvex. Then Ω is convex.Proof of Proposition 7.12. Given distinct x , z ∈ Ω, we will show that: ∀ ǫ > ∃ x ∈ B ( x , ǫ ) ∃ z ∈ B ( z , ǫ ) [ x, z ] ⊂ Ω . (7.7)This will be enough to conclude the claim, since this will imply that [ x , z ] lies inside Ω,which will imply that Ω is convex, which in turn will imply that its interior Ω is convex.Recall that connectedeness and path-connectedeness are equivalent topological proper-ties on open subsets of R n . As Ω is path-connected (being open), there exists a closed pathfrom x to z which lies in Ω. By openness of Ω and compactness of the path, a standard ar-gument implies that we may assume that the path is piecewise linear, i.e. consists of a finitenumber of intervals {[ y i , y i + ]} i = ,...,N − , with y = x and y N = z , so that [ y i , y i + ] ⊂ Ω.We will show, by induction on N , that this implies (7.7). The base case N = N =
2. To establish the case of N +
1, by openness of Ω andcompactness of intervals, there exists δ > y ′ N ∈ B ( y N , δ ) , [ y ′ N , y N + ] ⊂ Ω;by applying the induction hypothesis with min ( ǫ / , δ ) to the piecewise linear path from y to y N , and the case N = [ y ′ , y ′ N ] ∪ [ y ′ N , y N + ] , we find x ∈ B ( x , ǫ ) and z ∈ B ( z , ǫ / ) with [ x, z ] ⊂ Ω, as required.To establish the case N =
2, let y ∈ Ω be such that [ x , y ] ⊂ Ω and [ y , z ] ⊂ Ω.Consider the affine subspace E x,y,z spanned by x, y, z . If E x ,y ,z is one-dimensional, ittrivially follows that [ x , z ] ⊂ Ω and (7.7) is established, so let us assume that E x ,y ,z is two-dimensional. By openness of Ω and compactness of the intervals, there exists δ > x ∈ B ( x , δ ) , y ∈ B ( y , δ ) and z ∈ B ( z , δ ) , [ x, y ] ⊂ Ω, [ y, z ] ⊂ Ω and E x,y,z is two-dimensional. We now claim that for all ǫ ∈ ( , δ ) , there exist x ∈ B ( x , ǫ ) , y ∈ B ( y , ǫ ) and z ∈ B ( z , ǫ ) so that E = E x,y,z is disjoint from A . Once this is established,we will consider the set Ω ∶ = Ω ∩ E , which is relatively open in E ; we subsequentlyconsider the relative topology of sets in their corresponding affine hulls. We do not a-priori know that Ω is a (path-)connected set, but as [ x, y ] and [ y, z ] are both in Ω ,we do know that x, y, z all lie in the same (path-)connected component Ω ⊂ Ω . Since ∂ Ω ⊂ ∂ Ω ⊂ ∂ Ω ∩ E ⊂ ∂ Ω ∖ A , we know that for any p ∈ ∂ Ω there exists an openneighborhood N p of p so that Ω ∩ N p = Ω ∩ N p = ( Ω ∩ N p ) ∩ E is convex. Applying Lemma7.13 to the open connected Ω , it follows that Ω is convex, and hence [ x, z ] ⊂ Ω ⊂ Ω,thereby establishing (7.7). o see the existence of a subspace E = E x,y,z which is disjoint from A , let σ n denote thenatural Haar measure on E n , the homogeneous space of all two-dimensional affine subspacesin R n , normalized so that σ { E ∈ E n ∶ E ∩ B ( , ) ≠ ∅ } = H n − ( B n − ) , where B n − denotesthe Euclidean unit-ball in R n − . Introduce the following intergral-geometric Favard outermeasure: F n − ( U ) ∶ = ∫ E n H ( U ∩ E ) dσ n ( E ) . (7.8)It is known that F n − is a Borel measure, and by a generalized Crofton formula proved byFederer (see [25, Theorem 3.2.26]), that F n − and H n − coincide on H n − -rectifiable sets.While this may not be the case on general Borel sets, we always have F n − ( U ) ≤ H n − ( U ) for all Borel sets U (see [25, Theorem 2.10.15 and Section 2.10.6]). Given ǫ >
0, considerthe following variant: F n − ǫ ( U ) ∶ = ∫ E n ( ǫ ) H ( U ∩ E ) dσ n ( E ) , where: E n ( ǫ ) ∶ = { E ∈ E n ∶ E ∩ B ( x , ǫ ) ≠ ∅ , E ∩ B ( y , ǫ ) ≠ ∅ , E ∩ B ( z , ǫ ) ≠ ∅ } . Clearly F n − ǫ ( U ) ≤ F n − ( U ) for all U .Applying this to our set A , we deduce that F n − ( A ) = F n − ǫ ( A ) =
0, and we deduce thatfor σ n -a.e. E ∈ E n ( ǫ ) , H ( A ∩ E ) =
0, i.e. E is disjoint from A . Applying this to all ǫ ∈ ( , δ ) and using that σ n ( E n ( ǫ )) >
0, the existence of the desired E , disjoint from A andintersecting B ( x , ǫ ) , B ( y , ǫ ) and B ( z , ǫ ) , is deduced, concluding the proof.Now let Ω be a stationary, regular, flat cluster with connected open cells { Ω i } . Notethat by Theorems 4.1, 4.8 and flatness of the interfaces, the local convexity condition ofProposition 7.12 holds on ∂ Ω i ∖ Σ : indeed, around p ∈ ∂ Ω i ∩ Σ , p ∈ ∂ Ω i ∩ Σ and p ∈ ∂ Ω i ∩ Σ , Ω i looks locally like the convex flat cell of a model simplicial cluster aroundits interface, triple points and quadruple points, respectively. As H n − ( ∂ Ω i ∩ Σ ) =
0, wededuce by Proposition 7.12 that Ω i are all convex. Every open convex set Ω ⊂ R n is the intersection of its open supporting halfspaces over allboundary points ∂ Ω (see e.g. [54]). Clearly, by continuity (as the boundary of a convex setis a Lipschitz manifold), it is enough to intersect the supporting halfspaces of any subset A ⊂ ∂ Ω which is dense in ∂ Ω. Recalling our convention from Theorem 4.1 (ii) and (3.6),it follows that Ω i is the intersection of its open supporting halfspaces over the points ofits interfaces ∪ j ≠ i Σ ij . As the interfaces are smooth, the unique open supporting halfspaceto Ω i at p ∈ Σ ij is of the form { x ∈ R n ∶ ⟨ n ij ( p ) , x ⟩ < C ij ( p )} . On the other hand, by theflatness of the interfaces, we know that for all x ∈ Σ ij : H ij,γ = H Σ ij ( x ) − ∇ n ij W ( x ) = − ⟨ n ij , x ⟩ , and hence C ij ≡ − H ij,γ is constant on Σ ij .From this, we deduce that n ij must be constant over (any non-empty) Σ ij . Otherwise,if there exist p k ∈ Σ ij , k = ,
2, with n ij ( p ) ≠ n ij ( p ) , it would follow (as n ji = − n ij and H ji,γ = − H ij,γ ) that:Ω i ⊂ ⋂ k = , { x ∈ R n ∶ ⟨ n ij ( p k ) , x ⟩ < − H ij,γ } , Ω j ⊂ ⋂ k = , { x ∈ R n ∶ ⟨ n ij ( p k ) , x ⟩ > − H ij,γ } . his would either imply that one of the cells Ω i or Ω j is empty (if n ij ( p ) = − n ij ( p ) ), orthat ∂ Ω i ∩ ∂ Ω j is contained in the n − ⋂ k = , { x ∈ R n ∶ ⟨ n ij ( p k ) , x ⟩ = − H ij,γ } (otherwise); in either case, it would follow that A ij = γ n − ( Σ ij ) =
0, i.e. that Σ ij isempty, a contradiction. As n ij and H ij,γ are constant on Σ ij , Σ ij is contained in a singlehyperplane. By Lemma 7.9, dim span { n ij } i < j ≤ q − i are convex polyhedra, obtained as the intersection of a finitenumber (at most q −
1) open supporting halfspaces to its non-empty interfaces Σ ij . Theexplicit formula in (7.1) is obtained after recalling that H ij,γ = λ i − λ j , where λ ∈ E ∗ isgiven by Lemma 4.2 (ii). This concludes the proof of Theorem 7.2. All of the work we have done so far culminates in the following key proposition, which isthe main new tool we require for proving Theorem 1.1.
Proposition 8.1.
Let Ω be a minimizing cluster with respect to the Gaussian measure γ with γ ( Ω ) = v ∈ int ∆ , and let A = A ( Ω ) = { A ij } denote the collection of its interfacemeasures. Then for every y ∈ E and every ǫ > , there exists a C ∞ c vector-field X such that ∣ δ X V − y ∣ ≤ ǫ and Q ( X ) ≤ − y T L − A y + ǫ . Remark 8.2.
In fact, it is clear from the proof that Ω need only be stable and regular forthe statement to hold.
Proof of Proposition 8.1.
By the results of Section 4, Ω stationary, stable and regular.Given y ∈ E , since L A is full-rank by Lemma 7.11, there exists a ∈ E ( q − ) so that L A a = − y .Fix ǫ ∈ ( , ) and construct a family { X , . . . , X q } of ( ǫ, ǫ ) -approximate inward fields byProposition 6.1. Setting X = ∑ k a k X k , it follows by Lemma 6.8 (3) that for all i : ∣( δ X V ( Ω ) + L A a ) i ∣ ≤ ǫ √ q ∣ a ∣ . As L A a = − y , we conclude that ∣ δ X V − y ∣ ≤ ǫq ∣ a ∣ .Since Ω is stable and regular, Theorem 7.2 asserts that it is in addition flat. Hence,Theorem 5.2 implies that: Q ( X ) = ∑ i < j ∫ Σ ij (∣ ∇ t X n ∣ − ( X n ) ) dγ n − . Applying Lemma 6.8 (1) and (2), it follows that: Q ( X ) ≤ − a T L A a + ( q + ) ǫ ∣ a ∣ . Finally, since a T L A a = y T L − A y , the assertion is proved (after adjusting ǫ ).Having Proposition 8.1 at hand, the conclusion of the proof of Theorem 1.1 is essentiallyidentical to the one of the double-bubble case ( q =
3) from our previous work [40]. Forcompleteness, we provide the details. We will require the following standard fact (see,e.g. [40, Lemma 7.1]):
Lemma 8.3. I is lower semi-continuous on ∆ .Proof of Theorem 1.1. Our task is to prove that I ( q − ) = I ( q − ) m on ∆ ( q − ) for 2 ≤ q ≤ n + n , we will prove this by induction on q . Our base case is q =
2, which is theclassical (single-bubble) Gaussian isoperimetric inequality. By induction, we will assumethat I ( q − ) = I ( q − ) m on ∆ ( q − ) . Since (by Lemma 2.3) I ( q − ) m agrees with I ( q − ) m on ∂ ∆ ( q − ) , nd since I ( q − ) agrees with I ( q − ) on ∂ ∆ ( q − ) by definition, it follows by the inductionhypothesis that I ( q − ) = I ( q − ) m on ∂ ∆ ( q − ) . For brevity, we will henceforth write I , I m and∆ for I ( q − ) , I ( q − ) m and ∆ ( q − ) , respectively.By the construction of Section 2, I ≤ I m , so we will show the converse inequality. Since I m is continuous on ∆ by Lemma 2.3 and I is lower semi-continuous on ∆ by Lemma 8.3, I − I m must attain a global minimum at some v ∈ ∆. Assume in the contrapositive that I ( v ) − I m ( v ) <
0; by the comments in the previous paragraph, necessarily v ∈ int ∆.Let Ω be a minimizing cluster with γ ( Ω ) = v , as ensured by Theorem 4.1. Since v isa minimum of I − I m , for any smooth flow F t along an admissible vector-field X , we have I m ( γ ( F t ( Ω ))) ≤ I ( γ ( F t ( Ω )) − ( I − I m )( v ) ≤ P γ ( F t ( Ω )) − ( I − I m )( v ) , (8.1)where the second inequality follows from the definition of I . The two sides are equal when t =
0, and twice differentiable in t . By comparing first and second derivatives at t = ⟨ ∇ I m ( v ) , δ X V ⟩ = δ X A (8.2) ∇ δ X V,δ X V I m ( v ) + ⟨ ∇ I m ( v ) , δ X V ⟩ ≤ δ X A. (8.3)On the other hand, recall that δ X A = ⟨ λ, δ X V ⟩ by Lemma 4.4, where λ is the uniqueelement of E ∗ such that λ i − λ j = H ij,γ ( Ω ) . Comparing with (8.2), and using that (byProposition 8.1) for y = ∇ I m ( v ) − λ and any ǫ > C ∞ c vector-field X with ∣ δ X V − y ∣ ≤ ǫ , it follows that ∇ I m ( v ) = λ . Plugging this into (8.3), we obtain: ∇ δ X V,δ X V I m ( v ) ≤ δ X A − ⟨ λ, δ X V ⟩ = Q ( X ) . Now by Proposition 8.1, for any y ∈ E and ǫ > C ∞ c vector-field X so that ∣ δ X V − y ∣ ≤ ǫ and ( δ X V ) T ∇ I m ( v ) δ X V ≤ Q ( X ) ≤ − y T L − A y + ǫ (where recall A = A ( Ω ) = { A ij } is the collection of interface measures). Taking ǫ →
0, itfollows that ∇ I m ( v ) ≤ − L − A in the positive semi-definite sense, and since L A is positive-definite, we deduce that −∇ I m ( v ) − ≤ L A . It follows by Proposition 2.4 that:2 I m ( v ) = − tr [( ∇ I m ( v )) − ] ≤ tr ( L A ) = ∑ i < j A ij = I ( v ) , in contradiction to the assumption that I ( v ) < I m ( v ) . Having proved the multi-bubble inequality (Theorem 1.1), the uniqueness of minimizers(Theorem 1.2) follows in essentially the same way as in our previous work on the double-bubble case ( q = ≤ q ≤ n + v ∈ int ∆ ( q − ) ),must have been equalities. In particular, this leads to the following conclusion, which wasalready established in [40, Section 8]. Lemma 9.1.
Let Ω be a minimizing q -cluster in R n with respect to γ with γ ( Ω ) = v ∈ int ∆ ( q − ) and ≤ q ≤ n + . Let Ω m be a model q -cluster in R n with γ ( Ω m ) = v . Let A ij and A mij denote the Gaussian-weighted areas of the interfaces of Ω and Ω m , respectively.Then: i) ∇ I ( v ) = − L − A ;(ii) A ij = A mij for all i ≠ j ; and(iii) the first and second variations of Ω satisfy the following inequality for every admissiblevector-field X : ( δ X V ) T L − A ( δ X V ) ≥ − Q ( X ) . (9.1)For the rest of this section, we will fix a minimizing q -cluster Ω with γ ( Ω ) = v ∈ int ∆ ( q − ) and 2 ≤ q ≤ n +
1. Note that A mij > i ≠ j , and so Lemma 9.1 implies that A ij > i ≠ j as well – this is the crucial property of a minimizing cluster which we willrepeatedly use (see the next section for a more general result involving this property). Itfollows by Theorem 7.2 that n ij is constant on (the non-empty) Σ ij for all i ≠ j .Note that for the constant vector field w , we have ( δ w V ) i = ∑ j ≠ i ∫ Σ ij ⟨ w, n ij ⟩ dγ n − = ∑ j ≠ i A ij ⟨ n ij , w ⟩ . Since n ij = − n ji , we may express this as δ w V = M w , M ∶ = ∑ i < j A ij ( e i − e j ) n Tij . As for the second variation, by (5.15): − Q ( w ) = ∑ i < j ∫ Σ ij ⟨ n , w ⟩ dγ n − = ∑ i < j A ij ⟨ n ij , w ⟩ . (9.2)Comparing with (9.1), it follows that as quadratic forms on w ∈ R n : M T L − A M ≥ − Q = N , N ∶ = ∑ i < j A ij n ij n Tij . (9.3)On the other hand, we recall the following matrix version of the Cauchy-Schwarz in-equality (see e.g. [40, Lemma 6.4]): Lemma 9.2. If X is a random vector in R n and Y is a random vector in R m such that E ∣ X ∣ < ∞ , E ∣ Y ∣ < ∞ , and E Y Y T is non-singular, then ( E XY T )( E Y Y T ) − ( E Y X T ) ≤ E XX T as quadratic forms on R n , with equality if and only if X = BY almost surely, for a deter-ministic n × m matrix B . Applying this to random vectors X ∈ R n and Y ∈ E ( q − ) so that with probability A ij / ∑ k < ℓ A kℓ , X = n ij and Y = e i − e j , observe that: E XX T = ∑ k < ℓ A kℓ N , E Y X T = ∑ k < ℓ A kℓ M , EY Y T = ∑ k < ℓ A kℓ L A , and so Lemma 9.2 implies that M T L − A M ≤ N . Together with (9.3), it follows that thelatter inequality is in fact an equality, and so there a linear map B ∶ E ( q − ) → R n sothat n ij = B ( e j − e i ) for all i ≠ j (using that A ij > ∑ ( ℓ,m )∈ C ( i,j,k ) n ℓm = B ∑ ( ℓ,m )∈ C ( i,j,k ) ( e m − e ℓ ) = i, j, k , and it follows that n ij , n jk , and n ki areunit vectors with 120 ○ angles between them.Moreover, since E ij = ( e j − e i )( e j − e i ) T span the space of all symmetric matrices on E ( q − ) (by checking dimensions), and tr ( B T BE ij ) = ∣ n ij ∣ = i < j (as A ij > B T B = Id, i.e. √ B is an isometry from E ( q − ) onto Im B = span { n ij } . pplying Theorem 7.2 and using that A ij > i ≠ j one last time, we know thatafter modifying Ω by null-sets according to our convention from Theorem 4.1 (ii), we have:Ω i = { x ∈ R n ∶ ∀ j ≠ i ⟨ n ij , x ⟩ < λ j − λ i } = { x ∈ R n ∶ ∀ j ≠ i ⟨ e j − e i , B T x ⟩ < λ j − λ i } = { x ∈ R n ∶ ∀ j ≠ i ⟨ e j − e i , B T x − λ ⟩ < } . In other words, Ω is the pull-back via B T ∶ R n → E ( q − ) of the canonical model q -clusteron E ( q − ) centered at λ . Since √ B is an isometry, it follows that { Ω i } are the Voronoicells of q equidistant points in R n , i.e. that Ω is a simplicial q -cluster. This concludes theproof of Theorem 1.2.
10 Concluding Remarks
It is an interesting problem to obtain a complete characterization of stable regular q -clustersfor the Gaussian measure when q ≤ n +
1. We add here an additional necessary propertyof stable regular clusters on top of the information given in Theorem 7.2. As usual, wedenote by A ij = γ n − ( Σ ij ) the measure of the interfaces of Ω. Theorem 10.1.
Let Ω denote a stable regular q -cluster in R n with respect to γ , and assumethat q ≤ n + . Then Ω is the pull-back via a linear map R n ↦ E ( q − ) of a canonical modelsimplicial q -cluster in E ( q − ) . Specifically:1. There exists a linear operator B ∶ E ( q − ) → R n so that n ij = B ( e j − e i ) (10.1) for all i ≠ j such that A ij > ; and2. Ω is the pull-back via B T of the canonical model simplicial q -cluster in E = E ( q − ) centered at λ : Ω i = ⋂ j ≠ i { x ∈ R n ∶ ⟨ e j − e i , B T x − λ ⟩ < } , (10.2) where λ ∈ E ∗ is given by Lemma 4.2 (ii). Note that this is indeed a strengthening of Theorem 7.2; the additional information that(10.1) holds is a strong way to encapsulate the fact that Ω is stationary, since it impliesthat n ij + n jk + n ki = i, j, k so that A ij , A jk , A ki > ijk is non-empty). Before providing a proof of Theorem 10.1, let us remark thatit immediately yields an alternative proof of the uniqueness of minimizers from the previoussection, and in fact implies the following strengthening: Corollary 10.2.
Let Ω denote a stable regular q -cluster in R n with respect to γ , andassume that q ≤ n + . Assume that A ij > for all i ≠ j , where A ij = γ n − ( Σ ij ) denote asusual the interface measures. Then up to null-sets, Ω is a simplicial cluster. Indeed, for any minimizing Ω with γ ( Ω ) ∈ int ∆ ( q − ) , Lemma 9.1 implies that A ij = A mij > i ≠ j , which together with Corollary 10.2 yields Theorem 1.2 on the uniqueness ofminimizers. To see why Corollary 10.2 follows from Theorem 10.1, simply note that theonly other crucial property which was used in the proof of Theorem 1.2 in the previoussection, besides the property that A ij > i ≠ j , was that n ij = B ( e j − e i ) for all i ≠ j ;once this is known, it follows that √ B is an isometry onto its image, and so Theorem { Ω i } are the Voronoi cells of q equidistant points in R n , i.e. that Ω is asimplicial q -cluster.The proof of Theorem 10.1 is based on an interesting topological observation. Considerthe following two-dimensional simplicial complex S associated to a regular stable clusterΩ with respect to γ : its vertices are given by {( i ) ∶ Ω i ≠ ∅ } , its edges are given by {( i, j ) ∶ Σ ij ≠ ∅ } , and its triangles are given by {( i, j, k ) ∶ Σ ijk ≠ ∅ } . This is indeed asimplicial complex since the face of any simplex in S is clearly also in S . By a trivialadaptation of the proof of Lemma 7.11, it is immediate to see that the 1-skeleton of S , i.e.the graph obtained by considering only its edges, is necessarily connected. In other words,the zeroth (simplicial) homology of S is trivial. To this we add: Theorem 10.3.
The first (simplicial) homology of S is trivial. It is easy to check that Theorem 10.3 is false for a general finite convex tessellation of R n .The crucial property we will use in the proof is the combinatorial incidence structure ofcells along triple-points Σ , and the fact that H n − ( Σ ∪ Σ ) = Sketch of Proof of Theorem 10.3.
Let C denote a directed cycle in the 1-skeleton of S . Ourgoal will be to show that it is the boundary of a 2-chain ∑ Th = T h , where T h are orientedtriangles in S . Consider a closed piecewise linear directed path P in R n which emulates C ,meaning that it crosses from Ω i to Ω j transversely through Σ ij in the order specified by C (and without intersecting Σ ∪ Σ ). Note that it is always possible to construct such apath thanks to the connectivity of the cells (and moreover, the convexity of the cells makesthe construction especially simple). Assume that P = ∪ N − i = [ y i , y i + ] with y N ∶ = y , andfix a point o in one of the (non-empty open) cells which is not co-linear with any of thesesegments. Consider the convex interpolation P t = ( − t ) ⋅ P + t ⋅ o , so that [ , ] ∋ t ↦ P t is a contraction of P onto o . We claim that there exists a perturbation of the points Y ∶ = { y i } i = ,...,N − ∪ { o } so that:(i) o remains in its original cell, P still emulates C , and o is not co-linear with any of thesegments [ y i , y i + ] .(ii) P t does not intersect Σ ∪ Σ for all times t ∈ [ , ] .(iii) P t does not intersect Σ except for a finite set of times t ∈ ( , ) .(iv) For all times t ∈ [ , ] , P t crosses Σ transversely.The argument is similar to the one detailed in the proof of Proposition 7.12, and we employthe same notation introduced there. Consider open balls of radius ǫ > Y ; by selecting ǫ small-enough, we may always ensure that (i) holds for any perturbationinside these balls. Let K denote a closed ball containing the convex hull of the abovesmall balls around Y . Consider a perturbation of the points of Y selected uniformly andindependently inside each small ball. For each i = , . . . , N −
1, consider the two-dimensionalaffine subspace F i spanned by the perturbed points { y i , y i + , o } ; it is easy to see that thedistribution of F i is absolutely continuous with respect to σ n , the Haar measure on thehomogeneous space of two-dimensional affine subspaces of R n . Recall the definition (7.8)of the integral-geometric Favard measure F n − , for which we know by Theorem 4.8 that F n − ( Σ ∪ Σ ) = H n − ( Σ ∪ Σ ) = F n − ( K ∩ Σ ) = H n − ( K ∩ Σ ) < ∞ (notethat the latter statement employs for simplicity the local finiteness asserted in Theorem4.8 (ii), but as promised in Remark 4.11, this can be avoided by performing the aboveperturbation in two separate steps). It follows that with probability 1, a random two-dimensional subspace selected according to σ n will be disjoint from Σ ∪ Σ and intersect K ∩ Σ a finite number of times. By absolute continuity, the same holds for all of our andomly selected F i with probability 1. In particular, this ensures (ii) and (iii). Finally,note that the linear segments of P t remain parallel to those of P along the contraction to o ,and since Σ has a finite number of normal directions n ij , with probability 1 the segmentsof P (and hence of P t ) will cross Σ transversely, thereby ensuring (iv).The strategy is now clear: for every t ∈ [ , ) , consider the directed cycle C t in the1-skeleton of S which the path P t traverses; it is well-defined since P t crosses from one cellto the next transversely through Σ , and is of finite length (the length is bounded e.g. by N ( q − ) since every segment can cross at most q − C t remains constant between consecutive times when P t intersects Σ , and at such times thereare two possibilities: either a directed edge ( i, j ) in C t will transform into two consecutivedirected edges ( i, k ) , ( k, j ) , or vice versa, two consecutive directed edges ( i, k ) , ( k, j ) willcollapse into a single directed edge ( i, j ) (depending on how P t traverses Σ ijk as t varies).By (iii), this can happen only a finite number of times, and for times t close enough to 1, C t must be empty since the path P t is already close enough to o so as to remain inside a singlecell. This procedure produces a description of our original cycle C = C as the boundary ofthe 2-chain ∑ Th = T h , where T h are the oriented triangles ( i, j, k ) or ( i, k, j ) correspondingto the above two possibilities for how P t traversed Σ ijk . This concludes the proof. Proof of Theorem 10.1.
We first claim that for any directed cycle C in the 1-skeleton of S : ∑ ( i,j )∈ C n ij = . (10.3)Indeed, by Theorem 10.3, any such directed cycle is the boundary of a 2-chain ∑ Th = T h ,where T h are oriented triangles in S . Note that n ij + n jk + n ki = T = ( i, j, k ) in S , since this holds by Corollary 4.12 at any point x in the non-empty Σ ijk .Summing this up over all oriented triangles { T h } h = ,...,T , and using that n mℓ = − n ℓm , (10.3)immediately follows.Clearly, for each coordinate a = , . . . , n , (10.3) implies the existence of b a ∈ E ( q − ) sothat ⟨ n ij , e a ⟩ = ⟨ b a , e j − e i ⟩ for all edges ( i, j ) ∈ S ; this may be seen as the triviality of thefirst (simplicial) cohomology. In fact, b a is necessarily unique if there are no empty cells, byconnectivity of the 1-skeleton on the set of all vertices { , . . . , q } . Defining B = ∑ na = e a b Ta ,this establishes (10.1).It follows by Theorem 7.2 that:Ω i = ⋂ j ≠ i ∶ A ij > { x ∈ R n ∶ ⟨ e j − e i , B T x − λ ⟩ < } . Note the subtle difference with (10.2), where the intersection is taken over all j ≠ i (norequirement that A ij >
0) – let us denote the latter variant by ˜Ω i . We claim that Ω i = ˜Ω i for all i . Indeed, ˜Ω is a genuine cluster, being the pull-back of a canonical model clusterin E ( q − ) centered at λ , and so its cells cover R n (up to null-sets). On the other hand, Ωis itself a cluster by assumption, and clearly ˜Ω i ⊂ Ω i for all i . Since all cells are open, itfollows that necessarily Ω = ˜Ω, concluding the proof.An interesting question, which we leave for another investigation, is whether Theorem10.1 admits a converse:Is every q -cluster which is the pull-back via a linear map B T ∶ R n → E ( q − ) of a canonical model q -cluster in E ( q − ) , so that ∣ B ( e i − e j )∣ = i ≠ j such that A ij >
0, necessarily stable when q ≤ n + q -clusters when q ≤ n +
1. To appreciate the difficulty in determining the stability of a given cluster, note hat we do not even know how to directly show that the model simplicial clusters are stable,without invoking our main theorem to show that they are in fact minimizing (and hencein particular stable). Let us summarize the characterizing properties of minimizing clusters among all stableregular clusters which we have obtained so far (as usual, when q ≤ n + Theorem 10.4.
Let Ω denote a stable regular q -cluster in R n with respect to γ with γ ( Ω ) ∈ int ∆ ( q − ) , and assume that q ≤ n + . Let A ij = γ n − ( Σ ij ) denote the measures of itsinterfaces, and let d denote its dimension of deficiency. Then the following are equivalent:(i) Ω is a minimizing cluster.(ii) Ω is (up to null-sets) a model simplicial cluster.(iii) A ij > for all i ≠ j .(iv) Ω is effectively ( q − ) -dimensional (recall Definition 7.8). Indeed, (i) implies (iii) by Lemma 9.1, (iii) implies (ii) by Corollary 10.2, and (ii) implies(i) by Theorem 1.1. To this list of equivalences, we add (iv); recall that by Lemma 7.9, theeffective dimension of any stable regular q -cluster is at most q −
1. We remark that for theproof that (iv) implies (iii) and hence (ii), we do not need to invoke our Partial DifferentialInequality argument from Section 8 (as none of the required implications invoke Theorem1.1 or Lemma 9.1). In other words, if for a given v ∈ int ∆ ( q − ) , a minimizing cluster wereknown to be effectively ( q − ) -dimensional, we could directly deduce that it must be amodel simplicial cluster, thereby simultaneously verifying Theorems 1.1 and 1.2. Whilethis would result in a conceptual simplification of the overall argument for establishing ourmain results, unfortunately we do not know how to a-priori exclude the possibility that agiven minimizing cluster is effectively lower-dimensional.We first establish an almost obvious lemma regarding the convex polyhedral cells of Ω. Lemma 10.5.
With the same assumptions as in Theorem 10.4, assume that Σ ij is non-empty for some i ≠ j . If F ij denotes the (closed) facet of Ω i which contains Σ ij , thennecessarily F ij = Σ ij . In particular, F ij = F ji .Proof. Recall that the interfaces Σ kℓ are relatively open in Σ = Σ by Theorem 4.1, andhence Σ ij ∖ Σ ij ⊂ Σ ∪ Σ ∪ Σ . Assume in the contrapositive that F ij ≠ Σ ij . Since F ij is the closure of its relative interior (by convexity), it follows that H n − ( F ij ∖ Σ ij ) > H n − ( Σ ij ) > F ij that H n − ( relint F ij ∩ ( Σ ij ∖ Σ ij )) > F ij with a Gaussian density and employing the Gaussiansingle-bubble isoperimetric inequality inside a convex set, as in Theorem 10.7 below). SinceΣ ij ∖ Σ ij ⊂ Σ ∪ Σ ∪ Σ and H n − ( Σ ∪ Σ ) = p ∈ relint F ij ∩ Σ . But this contradicts Theorem 4.8 and Corollary 4.12, because p ∈ relint F ij implies that Ω i has only one outer normal at p , and so it cannot have twoouter normals with a 120 ○ angle between them. Proof of equivalence with (iv) in Theorem 10.4.
Clearly, (ii) implies (iii) and (iv), so let usassume that (iv) holds and establish (iii). By Lemma 7.4 and the product structure of theGaussian measure, we may assume that Ω is not dimension-deficient (if it were, we couldpass to the subspace perpendicular to the directions of dimension-deficiency). That is, Ωis full-dimensional and so q = n + s a first step, we observe that there is no cell Ω i for which Ω i contains a line. Assumein the contrapositive that Ω i contains a line parallel to the one-dimensional subspace E .Since Ω i is convex and open, it follows that Ω i can be written as ˜Ω i × E . Now let j ≠ i be such that A ij >
0; we will show that Ω j also contains a line parallel to E . Since theundirected graph in which ( i, j ) is an edge if A ij > E , which will imply that every cell Ω j can be written as ˜Ω j × E , contradicting the assumption that Ω is full-dimensional.To see that Ω j contains a line parallel to E , let F ij be the facet of Ω i that contains Σ ij and let F ji be the facet of Ω j that contains Σ ij . Since Ω i is a cylinder in the direction E ,it follows that F ij contains a line parallel E , and so by Lemma 10.5 so does F ji and henceΩ j .Now fix i ∈ { , . . . , q } . Since Ω i does not contain a line, it follows that Ω i has a faceof dimension zero (if not, a face of minimal dimension would necessarily contain a line).So let { p } ⊂ Ω i be a face of dimension zero. Then p is contained in at least n = q − i . Since Ω i has at most q − i has exactly q − H n − -measure, it follows by Lemma 10.5 that A ij > j . Since i was arbitrary, this proves (iii). Moreover, it actually follows that p ∈ Ω i ∩ ( ∩ j ≠ i Σ ij ) ⊂ ∩ j Ω j , and hence our two-dimensional simplicial complex S is complete(i.e. contains all vertices, edges and triangles), and so trivially has vanishing first homology,without requiring to pass through the proof of Theorem 10.3 to deduce Corollary 10.2. We have shown in Theorem 7.2 that whenever 2 ≤ q ≤ n +
1, a stable regular q -clusterin R n must be flat (thus having convex polyhedral cells). In fact, by Proposition 7.6, norestriction on q is necessary for the latter statement to hold if the cluster is dimension-deficient. One could wonder whether this is also the case for full-dimensional clusters. Inother words:Is every stable regular cluster necessarily flat?A positive answer would resolve a conjecture of Corneli-et-al [20, Conjecture 2.1], whichasserts that all minimizing clusters in R are flat.One possible way to tackle the above question is to show that stability tensorizes,namely that if Ω is a stable cluster in R n then so is Ω × R in R n + , since then we couldinvoke Proposition 7.6 for the dimension-deficient cluster Ω × R . Unfortunately, we do nothave a clear feeling of how reasonable this might be. Finally, we mention that Theorem 1.1 may be immediately extended to probability mea-sures having strongly convex potentials.
Definition 10.6.
A probability measure µ on R n is said to have a K -strongly convexpotential, K >
0, if µ = exp ( − W ( x )) dx with W ∈ C ∞ and Hess W ≥ K ⋅ Id.
Theorem 10.7.
Let µ be a probability measure having K -strongly convex potential. Denoteby I ( q − ) µ ∶ ∆ ( q − ) → R + its associated ( q − ) -bubble isoperimetric profile, given by: I ( q − ) µ ( v ) ∶ = inf { P µ ( Ω ) ∶ Ω is a q -cluster with µ ( Ω ) = v } . Then: I ( q − ) µ ≥ √ KI ( q − ) m on ∆ ( q − ) . he proof is an immediate consequence of Caffarelli’s Contraction Theorem [17]. Asthe argument is identical to the one for the case q = A Calculation of First and Second Variations
Appendix A is dedicated to establishing Lemmas 5.4 and 5.11. In the first two subsectionsbelow, let U denote a Borel set of locally finite perimeter so that ∂ ∗ U = Σ ∪ Ξ, whereΣ is a smooth ( n − ) -dimensional manifold co-oriented by the outer unit-normal n and H n − ( Ξ ) =
0. Let X be an admissible vector-field, and let F t denote the associated flowgenerated by X via (3.7).For a fixed point p ∈ Σ, let t , . . . , t n − be normal coordinates for Σ. Then t , . . . , t n − , n is an orthonormal basis for R n at p ; we write X i for ⟨ t i , X ⟩ and X n for ⟨ n , X ⟩ . Moreover,we will write X t for the tangential component of X : X t = X i t i . We freely employ Einsteinsummation convention, summing over repeated matching upper and lower indices.Recall that the second fundamental form is the symmetric bilinear form defined on T Σby II ( Y, Z ) = ⟨ Y, ∇ Z n ⟩ . We write { II ij } n − i,j = for its coordinates II ij = ⟨ t i , ∇ j n ⟩ = − ⟨ n , ∇ j t i ⟩ ,where ∇ j means ∇ t j . The mean-curvature H Σ is defined as tr II. Given a measure µ = e − W dx with W ∈ C ∞ ( R n ) , the weighted mean curvature H Σ ,µ is defined as H Σ − ∇ n W .The surface unweighted and weighted divergences are defined, respectively, bydiv Σ X = div X − ⟨ n , ∇ n X ⟩ , div Σ ,µ X = div Σ X − ∇ X W. A.1 Second variation of perimeter
Assuming X is compactly supported or U is perimeter regular, Lemma 3.3 implies that δ X A ( U ) = ∫ ∂ ∗ U d ( dt ) ∣ t = ( J Φ t e − W ○ F t ) d H n − = ∫ Σ ⎡⎢⎢⎢⎢⎣ d J Φ t ( dt ) ∣ t = − ∇ X W dJ Φ t dt ∣ t = + d e − W ○ F t ( dt ) ∣ t = e + W ⎤⎥⎥⎥⎥⎦ dµ n − , (A.1)where, recall from Subsection 3.3, J Φ t denotes the (surface) Jacobian of F t ∣ Σ . An imme-diate calculation yields: d e − W ○ F t dt ∣ t = e + W = − ⟨ ∇ X X, ∇ W ⟩ + ( ∇ X W ) − ∇ X,X W. (A.2)An explicit computation of the first and second derivatives of J Φ t in t (see e.g. [56, formula(2.16)]) reveals that: dJ Φ t dt ∣ t = = div Σ Xd J Φ t ( dt ) ∣ t = = div Σ ∇ X X + ( div Σ X ) + n − ∑ i = ⟨ n , ∇ i X ⟩ − n − ∑ i,j = ⟨ t i , ∇ j X ⟩⟨ t j , ∇ i X ⟩ . lugging the above into (A.1) yields δ X A ( U ) = ∫ Σ [ div Σ ∇ X X + ( div Σ X ) + n − ∑ i = ⟨ n , ∇ i X ⟩ − n − ∑ i,j = ⟨ t i , ∇ j X ⟩⟨ t j , ∇ i X ⟩ − ∇ X W div Σ X − ⟨ ∇ X X, ∇ W ⟩ + ( ∇ X W ) − ∇ X,X W ] dµ n − . (A.3) Lemma A.1. div Σ ( ∇ X t X ) + ( div Σ X ) + n − ∑ i = ⟨ n , ∇ i X ⟩ − n − ∑ i,j = ⟨ t j , ∇ i X ⟩⟨ t i , ∇ j X ⟩ = div Σ ( X div Σ X ) + ∣( ∇ X n ) t ∣ − ( X n ) ∥ II ∥ − div Σ ( X n ∇ X t n ) + X n ∇ X t H Σ . Proof.
In coordinates ∇ X t ( div Σ X ) = n − ∑ i = X j ∇ j ⟨ t i , ∇ i X ⟩ = n − ∑ i = X j ⟨ t i , ∇ j ∇ i X ⟩ + X j ⟨ ∇ j t i , ∇ i X ⟩ = n − ∑ i = X j ⟨ t i , ∇ j ∇ i X ⟩ − X j II ij ⟨ n , ∇ i X ⟩ . The last term can be manipulated by writing X j II ij = ⟨ ∇ i n , X ⟩ = ∇ i X n − ⟨ n , ∇ i X ⟩ , and hence ∇ X t ( div Σ X ) = n − ∑ i = X j ⟨ t i , ∇ j ∇ i X ⟩ + ⟨ n , ∇ i X ⟩ − ( ∇ i X n ) + X j II ij ∇ i X n . (A.4)On the other hand,div Σ ( ∇ X t X ) = n − ∑ i = ⟨ t i , ∇ i ( X j ∇ j X )⟩ = n − ∑ i = X j ⟨ t i , ∇ i ∇ j X ⟩ + ∇ i X j ⟨ t i , ∇ j X ⟩ = n − ∑ i,j = X j ⟨ t i , ∇ i ∇ j X ⟩ + ⟨ t j , ∇ i X ⟩⟨ t i , ∇ j X ⟩ − II ij X n ⟨ t i , ∇ j X ⟩ . Comparing this with (A.4), we obtain (using that ∇ i ∇ j X = ∇ j ∇ i X since ∇ is the flatconnection on R n and [ t i , t j ] = Σ ( ∇ X t X ) + n − ∑ i = ⟨ n , ∇ i X ⟩ − n − ∑ i,j = ⟨ t j , ∇ i X ⟩⟨ t i , ∇ j X ⟩ = ∇ X t ( div Σ X ) + n − ∑ i = ( ∇ i X n ) − n − ∑ i,j = ( X j II ij ∇ i X n + II ij X n ⟨ t i , ∇ j X ⟩) . (A.5) e now concentrate on the last two terms above. First, note that ⟨ t i , ∇ j X ⟩ = ∇ j ⟨ t i , X ⟩ − ⟨ ∇ j t i , X ⟩ = ∇ j X i + X n II ij . In conjunction with the Codazzi equation ∑ n − i = ∇ i II ij = ∑ n − i = ∇ j II ii , we obtain: n − ∑ i,j = ( X j II ij ∇ i X n + II ij X n ⟨ t i , ∇ j X ⟩) = ( X n ) ∥ II ∥ + n − ∑ i,j = II ij ( X j ∇ i X n + X n ∇ j X i ) = ( X n ) ∥ II ∥ + n − ∑ i,j = II ij ∇ i ( X n X j ) = ( X n ) ∥ II ∥ + n − ∑ i,j = ∇ i ( II ij X n X j ) − X n X j ∇ i II ij = ( X n ) ∥ II ∥ + n − ∑ i = ∇ i ⟨ X n ∇ X t n , t i ⟩ − X n ∇ X t II ii = ( X n ) ∥ II ∥ + div Σ ( X n ∇ X t n ) − X n ∇ X t H Σ , where in the last equality we used that ∇ X t n is tangential, and ∇ i ⟨ Y, t i ⟩ = ⟨ ∇ i Y, t i ⟩ fortangential fields Y . Plugging this final equation into (A.5) yieldsdiv Σ ( ∇ X t X ) + n − ∑ i = ⟨ n , ∇ i X ⟩ − n − ∑ i,j = ⟨ t j , ∇ i X ⟩⟨ t i , ∇ j X ⟩ = ∇ X t ( div Σ X ) + ∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ − div Σ ( X n ∇ X t n ) + X n ∇ X t H Σ . Finally, add ( div Σ X ) to both sides and note thatdiv Σ ( X div Σ X ) = ∇ X t ( div Σ X ) + ( div Σ X ) . Going back to (A.3) and plugging in Lemma A.1, we obtain δ X A ( U ) = ∫ Σ [ div Σ ( X n ∇ n X ) + div Σ ( X div Σ X ) − div Σ ( X n ∇ X t n ) + ∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ + X n ∇ X t H Σ − ⟨ ∇ X X, ∇ W ⟩ − ∇ X W div Σ X + ( ∇ X W ) − ∇ X,X W ] dµ n − . Now, expanding the definition of weighted divergence, we observe thatdiv Σ ,µ ( X div Σ ,µ X ) (A.6) = div Σ ( X div Σ X ) − ∇ X W div Σ X + ( ∇ X W ) − ∇ X t ∇ X W = div Σ ( X div Σ X ) − ∇ X W div Σ X + ( ∇ X W ) − ∇ X t ,X W − ⟨ ∇ X t X, ∇ W ⟩ . Hence, δ X A ( U ) = ∫ Σ [ div Σ ( X n ∇ n X ) + div Σ ,µ ( X div Σ ,µ X ) − div Σ ( X n ∇ X t n ) + ∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ + X n ∇ X t H Σ − X n ⟨ ∇ n X, ∇ W ⟩ − X n ∇ X t , n W − ( X n ) ∇ n , n W ] dµ n − . ow, the first term of the first line combines with the first term of the third line to makea weighted divergence, and X n ∇ X t , n W = X n ∇ X t ∇ n W − X n ⟨ ∇ W, ∇ X t n ⟩ . The second ofthese terms combines to make div Σ ( X n ∇ X t n ) a weighted divergence, while the first onecombines with X n ∇ X t H Σ to make the mean-curvature weighted. To summarize: Lemma A.2.
For U and X as above, δ X A ( U ) = ∫ Σ [ div Σ ,µ ( X n ∇ n X ) + div Σ ,µ ( X div Σ ,µ X ) − div Σ ,µ ( X n ∇ X t n ) + ∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ + X n ∇ X t H Σ ,γ − ( X n ) ∇ n , n W ] dγ n − . A.2 Second variation of volume
Assuming X is compactly supported or U is volume and perimeter regular, Lemma 3.3implies that: δ X V ( U ) = ∫ U d ( dt ) ∣ t = ( JF t e − W ○ F t ) dx = ∫ U ⎡⎢⎢⎢⎢⎣ d JF t ( dt ) ∣ t = − ∇ X W dJF t dt ∣ t = + d e − W ○ F t ( dt ) ∣ t = e + W ⎤⎥⎥⎥⎥⎦ dµ, where, recall from Subsection 3.3, JF t denotes the Jacobian of F t . Recalling (A.2), andusing (see e.g. [56, formula (2.13) and the derivation after (2.35)]): dJF t dt ∣ t = = div X , d JF t ( dt ) ∣ t = = div ( X div X ) , we obtain: δ X V ( U ) = ∫ U [ div ( X div X ) − ∇ X W div X − ⟨ ∇ X X, ∇ W ⟩ + ( ∇ X W ) − ∇ X,X W ] dµ = ∫ U div µ ( X div µ X ) dµ, where the last equality follows by expanding out the weighted divergences as in (A.6). Wenow claim that we may integrate-by-parts and proceed as follows: = ∫ ∂ ∗ U X n div µ X dµ n − . Indeed, if X is compactly supported, this holds by the Gauss–Green–De Giorgi theorem(3.3). For admissible X and volume and perimeter regular U , this follows by employing inaddition a truncation argument, exactly as the one employed in the proof of [40, Lemma5.2]. To summarize: Lemma A.3.
For U and X as above: δ X V ( U ) = ∫ Σ X n div µ X dµ n − . .3 Index form of a stationary cluster Let us apply the above calculations to the cells { Ω i } of a stationary cluster Ω, assuming thateither X is compactly supported, or that X is admissible and { Ω i } are volume and perimeterregular. Stationarity implies there is some λ ∈ E ∗ such that H ij,µ = λ i − λ j for all i ≠ j .Define Ψ ij = ∫ Σ ij X n div µ X dµ n − and note that Ψ ji = − Ψ ij and that δ X V ( Ω ) i = ∑ j ≠ i Ψ ij by Lemma A.3 and (3.5). It follows that: ⟨ λ, δ X V ( Ω )⟩ = ∑ i λ i ∑ j ≠ i Ψ ij = ∑ i ≠ j ( λ i − λ j ) Ψ ij = ∑ i < j H ij,µ ∫ Σ ij X n div µ X dµ n − . (A.7)As for δ X A ( Ω ) , stationarity implies that the term ∇ X t H Σ ij ,µ in Lemma A.2 vanishes. Weare left with: δ X A ( Ω ) − ⟨ λ, δ X V ( Ω )⟩ = ∑ i < j ∫ Σ ij [ div Σ ,µ ( X n ∇ n X ) + div Σ ,µ ( X div Σ ,µ X ) − div Σ ,µ ( X n ∇ X t n ) + ∣ ∇ t X n ∣ − ( X n ) ∥ II ∥ − ( X n ) ∇ n , n W − H ij,µ X n div µ X ] dµ n − , which is the formula claimed in Lemma 5.4. A.4 Polarization with respect to constant vector-fields
Finally, assume that Ω is a stationary cluster whose cells are volume and perimeter regular.Let X be an admissible vector-field, and set Y = X + w , where w ∈ R n is a constant vector-field. For brevity, denote δ Z A = δ Z A ( Ω ) and δ Z V = δ Z V ( Ω ) for Z = Y, X, w . Since ∇ w = Σ w =
0, we may plug Y into (A.3) to obtain δ Y A = ∑ i < j ∫ Σ ij [ div Σ ∇ Y X + ( div Σ X ) + n − ∑ i = ⟨ n , ∇ i X ⟩ − n − ∑ i,j = ⟨ t i , ∇ j X ⟩⟨ t j , ∇ i X ⟩ − ⟨ ∇ Y X, ∇ W ⟩ − ∇ Y W div Σ X + ( ∇ Y W ) − ∇ Y,Y W ] dµ n − . Expanding Y = X + w in all the remaining places, we obtain: δ Y A = δ X A + δ w A + ∑ i < j ∫ Σ ij [ div Σ ∇ w X − ⟨ ∇ w X, ∇ W ⟩ − ∇ w W div Σ X + ∇ X W ∇ w W − ∇ w,X W ] dµ n − = δ X A + δ w A + ∑ i < j ∫ Σ ij [ div Σ ,γ ( ∇ w X ) − Σ ,µ ( ∇ w W ⋅ X ) − X n ∇ w, n W ] dµ n − . or the second variation of volume, (A.7) implies that ⟨ λ, δ Y V ⟩ = ∑ i < j H ij,γ ∫ Σ ij Y n div µ Y dµ n − = ∑ i < j H ij,γ ∫ Σ ij ( X n + w n )( div µ X − ∇ w W ) dµ n − = ⟨ λ, δ X V + δ w V ⟩ + ∑ i < j H ij,µ ∫ Σ ij ( w n div µ X − X n ∇ w W ) dµ n − . The polarization formula for Q ( X + w ) = δ Y A − ⟨ λ, δ Y V ⟩ asserted in Lemma 5.11 thenfollows by subtracting the above expressions. B Curvature Blow-Up Near Quadruple Points
Appendix B is dedicated to providing a proof of Proposition 4.14. The latter is an easyconsequence of the following statement:
Proposition B.1.
Let Ω be a stationary regular q -cluster, and let i, j, k, l ∈ { , . . . , q } be distinct. For any q ∈ Σ ijkl and open neighborhood N q of q , there exists an open sub-neighborhood ˜ N q ⊂ N q of q so that:(i) ∃ C q > ∀ p ∈ Σ ijk ∩ ˜ N q ∥ II ij ( p )∥ ≤ C q d ( p, Σ ijkl ∩ N q ) − α . (B.1) (ii) ∃ C q > ∀ w ∈ Σ ij ∩ ˜ N q ∥ II ij ( w )∥ ≤ C q d ( w, Σ ijkl ∩ N q ) − α . (B.2)Of course statement (B.2) is more general than (B.1), as Σ ij is C ∞ smooth all the wayup to Σ ijk , but it will be more convenient to first establish (B.1) and then obtain (B.2).Let us start by showing that Proposition B.1 implies Proposition 4.14. Proof that Proposition B.1 implies Proposition 4.14.
As explained in the comments follow-ing the formulation of Proposition 4.14, it is enough by compactness to verify the assertedintegrability in a small-enough open neighborhood of a quadruple point in Σ . By The-orem 4.8, given q ∈ Σ ijkl , there exists a small enough neighborhood N q of q where Σ isan embedded C ,α -diffeomorphic image of T × R n − ; namely, there exists a neighborhood U q of the origin in E ( ) × R n − and a C ,α -diffeomorphism ϕ ∶ U q → N q so that ϕ ( ) = q , ϕ (( T × R n − ) ∩ U q ) = Σ ∩ N q and so that ∥ ϕ ∥ C ,α ( U q ) and ∥ ϕ − ∥ C ,α ( N q ) are bounded (seee.g. [13] for an inverse function theorem for C ,α functions).Denote Σ m ∶ = { x ∈ E ( ) ∶ x = x > x , x } × R n − , Σ m ∶ = { x ∈ E ( ) ∶ x = x = x > x } × R n − and similarly for Σ m and Σ m . We may suppose without loss of generality that ϕ maps Σ m ∩ U q onto Σ ij ∩ N q , Σ m ∩ U q onto Σ jk ∩ N q , and Σ m ∩ U q onto Σ ki ∩ N q (andhence Σ m ∩ U q onto Σ ijk ∩ N q ).We now apply Proposition B.1. By part (i), there exists a sub-neighborhood ˜ N q ⊂ N q of q satisfying (B.1), and we denote ˜ U q = ϕ − ( ˜ N q ) . To see that ∫ Σ ijk ∩ ˜ N q ∥ II ij ∥ dµ n − < ∞ ,observe that ∫ Σ ijk ∩ ˜ N q ∥ II ij ( p )∥ dµ n − ( p ) ≤ M q ∫ Σ m ∩ ˜ U q ∥ II ij ∥ ( ϕ ( z )) J ( z ) d H n − ( z ) ≤ M q J q ∫ Σ m ∩ ˜ U q C q d ( ϕ ( z ) , Σ ijkl ∩ N q ) − α d H n − ( z ) , here M q is an upper bound on the density of µ on ˜ N q , J ( z ) is the Jacobian at z of ϕ restricted as a map from Σ m ∩ ˜ U q to Σ ijk ∩ ˜ N q and J q is an upper bound on the latterJacobian on ˜ U q (as it only depends on first-order derivatives of ϕ ). Since ∣ ϕ ( z ) − ϕ ( z )∣ and ∣ z − z ∣ are equivalent on U q up to a factor D q , we deduce that: ∫ Σ ijk ∩ ˜ N q ∥ II ij ∥ ( p ) dµ n − ( p ) ≤ M q J q D − αq C q ∫ Σ m ∩ ˜ U q d ( z, ({ } × R n − ) ∩ U q ) − α d H n − ( z ) . But the latter integral is immediately seen to be finite after applying Fubini’s theorem,since Σ m = ( , , , − ) R + × R n − and the function r ↦ / r − α is integrable at 0 ∈ R + .The proof that ∫ Σ ij ∩ ˜ N q ∥ II ij ∥ dµ n − < ∞ follows similarly from Proposition B.1 part(ii). Indeed, after pulling back via ϕ to the model cluster, we obtain: ∫ Σ ij ∩ ˜ N q ∥ II ij ( w )∥ dµ n − ( w ) ≤ M q J q D ( − α ) q C q ∫ Σ m ∩ ˜ U q d ( z, ({ } × R n − ) ∩ U q ) ( − α ) d H n − ( z ) . To see that the integral on the right-hand-side is finite, note that:Σ m = (( , , , − ) R + + ( , , − , ) R + ) × R n − , and so an application of Fubini’s theorem and integration in polar coordinates on the twodimensional sector above, which incurs an additional r factor from the Jacobian, boilsthings down to the integrability of the function r ↦ r / r ( − α ) at 0 ∈ R + . B.1 Schauder estimates
For the proof of Proposition B.1 we will require the following weighted Schauder estimate forsystems of linear elliptic and coercive PDEs. As explained in [32], this is a generalization of[1, Theorem 9.1] for systems, modeled on the generalization of [1, Theorem 7.2] to systemsin [2, Theorem 9.2]. We will only formulate the version which we will require here.Let ˚Ω R , denote an open hemisphere of radius R ∈ ( , ] around the origin in R n , letΣ R denote the open ( n − ) -dimensional disc which constitutes its flat boundary, and setΩ R ∶ = ˚Ω R ∪ Σ R . We will use the following notation for a function f defined on Ω R or Σ R ,a non-negative integer l and α ∈ ( , ) : [ f ] l ∶ = max ∣ µ ∣= l ∥ D µ f ∥ ∞ , [ f ] l + α ∶ = max ∣ µ ∣= l ∥ ∣ D µ f ( x ) − D µ f ( y )∣∣ x − y ∣ α ∥ ∞ , ∣ f ∣ l ∶ = l ∑ i = [ f ] i , ∣ f ∣ l + α ∶ = ∣ f ∣ l + [ f ] l + α , where D µ = ∂ µ x . . . ∂ µ n x n denotes partial differentiation with respect to the multi-index µ ,and ∣ µ ∣ denotes its total degree µ + . . . + µ n . We will also require the following weightedversions of the above norms and semi-norms. Let d S ( x ) denote the distance of a point x ∈ Ω R from the spherical part of ∂ Ω R . Given a homogeneity excess parameter p ≥ ̃[ f ] p,l ∶ = max ∣ µ ∣= l ∥ d p + lS D µ f ∥ ∞ , ̃[ f ] p,l + α ∶ = max ∣ µ ∣= l sup { d p + l + αS ( x ) ∣ D µ f ( x ) − D µ f ( y )∣∣ x − y ∣ α ; 4 ∣ x − y ∣ < min ( d S ( x ) , d S ( y ))} , ̃∣ f ∣ p,l ∶ = l ∑ i = ̃[ f ] p,i , ̃∣ f ∣ p,l + α ∶ = ̃∣ f ∣ p,l + ̃[ f ] p,l + α . inally, set: ̃[ f ] a ∶ = ̃[ f ] ,a , ̃∣ f ∣ a ∶ = ̃∣ f ∣ ,a . Theorem B.2 (Agmon–Douglis–Nirenberg) . Let α ∈ ( , ) . Fix integers N, M ≥ , let i, j = , . . . , N , k = , . . . , M , and denote the following linear differential operators on x ∈ Ω R and y ∈ Σ R , respectively: L ij ∶ = ∑ ∣ β ∣∈{ , } , ∣ µ ∣∈{ , } D β a ijβµ ( x ) D µ , B kj ∶ = ∑ ∣ µ ∣∈{ , } b kjµ ( y ) D µ , where a ijβµ ∈ C ,α ( Ω R ) and b kjµ ∈ C ,α ( Σ R ) satisfy ∣ a ijβµ ∣ α , ∣ b kjµ ∣ α ≤ K for some parame-ter K > . Here and below differentiation is understood in the distributional sense. Given F iβ ∈ C ,α ( Ω R ) and Φ k ∈ C ,α ( Σ R ) , denote also: F i ∶ = ∑ ∣ β ∣∈{ , } D β F iβ . Assume that { f j } ⊂ C ,α ( Ω R ) satisfy the following system of equations in the distributionalsense: ∀ i ∑ j L ij f j = F i on Ω R , ∀ k ∑ j B kj f j = Φ k on Σ R . Finally, assume that the above system is uniformly elliptic and coercive on Ω R . Then: ∀ j ̃∣ f j ∣ + α ≤ C ⎛⎝ ∑ i ∑ ∣ β ∣∈{ , } ̃∣ F iβ ∣ − ∣ β ∣ ,α + ∑ k ̃∣ Φ k ∣ ,α + ∑ j ̃∣ f j ∣ ⎞⎠ . where the constant C depends only on n, N, M, K, α and the ellipticity and coercitivityconstants. We refer to [2] for the definitions of ellipticity and coercitivity (the latter is a cer-tain compatibility requirement for the boundary conditions, a generalization of an obliquederivative condition). We will use Theorem B.2 in the following form:
Corollary B.3.
With the same notation and assumptions of Theorem B.2, assume inaddition that F iβ ≡ and Φ k ≡ for all i, k, β . Then: ∀ j ̃[ f j ] ≤ C ⎛⎝ ∑ i,j ̃∣ a ij ∣ ,α ∣ f j ( )∣ + ∑ k,j ̃∣ b kj ∣ ,α ∣ f j ( )∣ + ∑ j ̃∣ f j − f j ( )∣ ⎞⎠ . Proof.
Immediate after applying Theorem B.2 to g j = f j − f j ( ) which satisfies: ∀ i ∑ j L ij g j = − ∑ j a ij ( x ) f j ( ) on Ω R , ∀ k ∑ j B kj g j = − ∑ j b kj ( y ) f j ( ) on Σ R , and noting that ̃∣ g j ∣ + α ≥ ̃[ g j ] = ̃[ f j ] .We are now ready to prove Proposition B.1. When referring to the topology of a set U (and in particular, its topological boundary ∂U ), we always employ the relative topologyin U ’s affine hull. .2 Blow-up on Σ ijk To establish (B.1), first note that by replacing the neighborhood N q by a smaller one ifnecessary, we may always assume that Σ ∩ N q is an embedded local C ,α -diffeomorphicimage of T × R n − as in the proof that Proposition B.1 implies Proposition 4.14, since thisonly makes the estimate (B.1) stronger. We continue with the notation introduced in thelatter proof and proceed as in [32]. Let H denote the tangent plane to Σ ki at q (which iswell-defined by continuity and the fact that ϕ is a C -diffeomorphism). By again shrinking N q if necessary, we may represent Σ ij , Σ jk and Σ ki in N q as graphs of C ,α functions,denoted u , u , u respectively, over their orthogonal projections onto H , denoted Ω , Ω and Ω respectively. Indeed, this is immediate by the implicit function theorem for Σ ki ,but also follows for Σ ij and Σ jk as they form 120 ○ degree angles with Σ ki on Σ ijk , and thisextends by C regularity of ϕ to q ∈ Σ ijkl ; hence Σ ij and Σ jk form 60 ○ degree angles with H at q , and the angle remains bounded away from 90 ○ by C regularity at a small-enoughneighborhood of q .Denote by Γ the orthogonal projection P H of Σ ijk ∩ N q onto H . Note that Γ is thecommon boundary of Ω , Ω and Ω in P H N q , and that Ω and Ω lie on one side ofΓ, whereas Ω lies on the other side. Also denote Γ ∶ = P H ( Σ ijkl ∩ N q ) ⊂ Γ. As weshall essentially reproduce below, it was shown in [32] that in fact u i are C ∞ ( Ω i ∖ Γ ) . Forconvenience, we choose a coordinate system so that H is identified with R n − , q is identifiedwith the origin, T q Γ is identified with T { x = } with ∂ x pointing into Ω and Ω at theorigin, and T q Σ ijkl is identified with T { x = x = } .To elucidate the picture, let us describe how this works for a model 4-cluster in R : allthe Σ ij ’s are two-dimensional sectors of angle cos − ( − / ) ≃ ○ , and after projecting theseonto the plane tangent to (and spanned by) Σ ki , which we identify with R , clearly Ω isthe same sector of angle cos − ( − / ) , but Ω = Ω are sectors of angle cos − ( − /√ ) ≃ ○ ,with common boundary Γ = R + e and with Γ = { } .Now assume, by relabeling indices if necessary, that u ≥ u on Ω ∩ Ω . Each heightfunction u i , i = , ,
3, satisfies the following constant weighted mean-curvature equation: n − ∑ a = ∂ x a ⎛⎜⎝ ∂ x a u i √ + ∣ ∇ u i ∣ ⎞⎟⎠ − ⟨ ∇ W ( x, u i ( x )) , ( ∇ u i ( x ) , − )√ + ∣ ∇ u i ( x )∣ ⟩ ≡ C i in int Ω i . (B.3)The boundary conditions are obtained from the fact that Σ ij , Σ jk and Σ ki meet at 120 ○ degrees on Σ ijk according to Corollary 4.12 (and this extends by C regularity to Σ ijkl ),and hence: ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩⟨ ∇ u , ∇ u ⟩ + − cos ( ○ )√ + ∣ ∇ u ∣ √ + ∣ ∇ u ∣ = , ⟨ ∇ u , ∇ u ⟩ + − cos ( ○ )√ + ∣ ∇ u ∣ √ + ∣ ∇ u ∣ = ,u = u = u , on Γ . (B.4)Of course, this still leaves a vertical degree of freedom (adding a common constant to all u i ’s), which we may remove by setting u ( ) =
0, but we do not require this here.Denoting x ′ = ( x , . . . , x n − ) , we now apply the following hodograph transform T map-ping x ∈ Ω ∩ Ω to y , where: ⎧⎪⎪⎨⎪⎪⎩ w ( x ) ∶ = u ( x ) − u ( x ) ,y = T ( x ) ∶ = ( w ( x ) , x ′ ) . The inverse T − of this transform is given by: x = T − ( y ) = ( ψ ( y ) , y ′ ) , here ψ is determined by the inverse function theorem as solving x = ψ ( y ) . This isalways well-defined, perhaps after replacing N q by a smaller neighborhood, since ∂ x w ( ) = ( ○ ) >
0, and C regularity ensures that ∂ x w will remain positive on the entire Ω ∩ Ω if N q is chosen sufficiently small. In particular w ( x ) > x ∈ ( Ω ∩ Ω ) ∖ Γ, and ∂ y ψ > T ( Ω ∩ Ω ) .Denote V ∶ = T ( Ω ∩ Ω ) , Ξ ∶ = T ( Γ ) and Ξ ∶ = T ( Γ ) , and observe that Ξ ⊂ { y ∈ R n − ∶ y = } , V ∖ Ξ ⊂ { y ∈ R n − ∶ y > } and T ( ) =
0. Also note that V is “convex in the first co-ordinate”, i.e. if ( t , y ′ ) , ( t , y ′ ) ∈ V then (( − λ ) t + λt , y ′ ) ∈ V for all λ ∈ [ , ] , bycontinuity of w . Finally note that ψ ∈ C ,α ( V ) ∩ C ∞ ( V ∖ Ξ ) by the inverse functiontheorem.We now apply the reflection method of [32], and for y ∈ V , denote: S ( y ) ∶ = ( ψ ( y ) − Cy , y ′ ) , C ∶ = sup y ∈ V ∣ ∇ ψ ( y )∣ + . Clearly S is a diffeomorphism of class C ,α from V to S ( V ) and of class C ∞ from V ∖ Ξ to S ( V ∖ Ξ ) , and hence by the inverse function theorem, so is its inverse on thecorresponding domains. Obviously S maps Ξ onto Γ and Ξ onto Γ , but also note thatit maps V ∖ Ξ to the complement of Ω ∩ Ω , since otherwise we would have z ∈ V ∖ Ξand y ∈ V with ( ψ ( z ) − Cz , z ′ ) = ( ψ ( y ) , y ′ ) , which is impossible since t ↦ ψ ( t, y ′ ) isincreasing whereas t ↦ ψ ( t, y ′ ) − Ct is decreasing on { t ≥ ∶ ( t, y ′ ) ∈ V } . Finally, we set V ∶ = S − ( Ω ) ⊂ V , which by the preceding comments must contain an open neighborhoodof K in { y ∈ R n − ∶ y ≥ } for any compact subset K of relint ( Ξ ) ; we will return to thegeometry of V later. We can now define for y ∈ V :Φ + ( y ) ∶ = u ( x ) , x = T − ( y ) , Φ − ( y ) ∶ = u ( x ) , x = S ( y ) , and note that: u ( x ) = w ( x ) + u ( x ) = y + Φ + ( y ) , y = T ( x ) . Also note that since u i ∈ C ,α ( Ω i ) ∩ C ∞ ( Ω i ∖ Γ ) then Φ + , Φ − ∈ C ,α ( V ) ∩ C ∞ ( V ∖ Ξ ) .As verified in [32], the hodograph and reflection maps transform our quasilinear systemof PDEs in divergence form (B.3), for functions u i defined on different domains Ω i , intoa system of quasilinear PDEs, still in divergence form, for Φ + , Φ − , Ψ, defined on the samedomain V , which is of the following form: ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ M ( D Ψ , D Φ + , D Ψ , D Φ + , Φ + ) = M ( D Ψ , D Φ + , D Ψ , D Φ + , Φ + ) = M ( D Ψ , D Φ − , D Ψ , D Φ − , Φ − ) = V . (B.5)Our boundary conditions (B.4) on Γ are transformed into boundary conditions on Ξ of thefollowing form: ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ N ( D Ψ , D Φ + , D Φ − ) = N ( D Ψ , D Φ + ) = N ( Φ + , Φ − ) = Φ + − Φ − = . (B.6)Unfortunately, these boundary conditions are quadratic in the first-order leading terms,which prevents us from directly employing C ,α Schauder estimates for elliptic linear sys-tems [2] after treating the first and zeroth order terms as coefficients, which we do knoware in C ,α ( V ) .Consequently, as in [32], we linearize (B.5) and (B.6) by considering their first order per-turbation in a given direction θ ∈ R n − , treating the lower order terms as C ,α coefficients. s verified in [32], this results in a linear system of PDEs with boundary conditions whichis elliptic and coercive at y = C ,α ( V ) coefficients, by replacing N q by a smaller neighborhoodif necessary, we may always assume that the system is uniformly elliptic and coercive onthe entire V .It is worth pointing out that the linearized system remains in divergence form with C ,α coefficients – to see why, recall that the system before linearization consisted of quasi-linearterms in divergence form: ∑ ∣ β ∣∈{ , } , ∣ µ ∣∈{ , } D β c iβµ ( Df , Df , Df , f , f , f ) D µ f i , with c iβµ being smooth functions of their arguments and { f , f , f } = { Φ + , Φ − , Ψ } ⊂ C ,α ( V ) ∩ C ∞ ( V ∖ Ξ ) . Taking partial derivative ∂ θ in the direction of θ ∈ S n − , we obtain: ∑ ∣ β ∣∈{ , } , ∣ µ ∣∈{ , } D β [ c iβµ ( Df , Df , Df , f , f , f )] D µ ( ∂ θ f i ) + ∑ j = ∑ ∣ β ∣∈{ , } , ∣ δ ∣= D β ⎡⎢⎢⎢⎢⎣ ∑ ∣ µ ∣∈{ , } D µ f i ⋅ D δj c iβµ ( Df , Df , Df , f , f , f )⎤⎥⎥⎥⎥⎦ D δ ( ∂ θ f j ) + ∑ j = ∑ ∣ β ∣∈{ , } D β ⎡⎢⎢⎢⎢⎣ ∑ ∣ µ ∣∈{ , } D µ f i ⋅ ∂ j + c iβµ ( Df , Df , Df , f , f , f )⎤⎥⎥⎥⎥⎦ ∂ θ f j , where D δj denotes partial differentiation of c iβµ according to the multi-index δ of its j ’thargument. Since Df i ∈ C ,α ( V ) , we see that the system for { ∂ θ f i } i = , , is still in di-vergence form with C ,α ( V ) coefficients (delineated by rectangular brackets above). Afterlinearization, the quadratic boundary conditions become linear in D µ ∂ θ f i with C ,α ( Ξ ) coefficients. Note that these coefficients (of both the system and the boundary conditions)are bounded in C ,α norm on the entire V and Σ, respectively, since all of the estimatesdepend only on our initial C ,α estimates for ϕ and ϕ − . We are thus in a position to applythe Schauder estimates of Corollary B.3.Let p ∈ Σ ijk ∩ N q . Then x p ∶ = P H p ∈ Γ ∖ Γ and y p ∶ = T ( x p ) ∈ Ξ ∖ Ξ . Let Ω R ⊂ R n − denote a maximal hemisphere centered at y p of radius R = R p so that its flat partΣ R ⊂ { y = } and Ω R ⊂ V ∖ Ξ . By the preceding discussion and since Ξ is a relativelyclosed set in P H N q , necessarily R >
0. Applying Corollary B.3 to our linearized system for { ∂ θ f , ∂ θ f , ∂ θ f } = { ∂ θ Φ + , ∂ θ Φ − , ∂ θ ψ } ⊂ C ∞ ( Ω R ) , and recalling the weighing used in thenorms appearing in the resulting estimate, we deduce for all j = , , R ∣ D∂ θ f j ( y p )∣ ≤ C q ⎛⎝ ∑ i,j R + α ∥ a ij ∥ C ,α ( Ω R ) ∣ ∂ θ f j ( y p )∣ + ∑ k,j R + α ∥ b kj ∥ C ,α ( Σ R ) ∣ ∂ θ f j ( y p )∣ + ∑ j ∥ ∂ θ f j − ∂ θ f j ( y p )∥ C ( Ω R ) ⎞⎠ , where a ij and b ij are the zeroth order coefficients of the linearized system and boundaryconditions, and C q depends on quantities which are uniform for all p ∈ Σ ijk ∩ N q ∖ { q } .Since ∥ a ij ∥ C ,α ( V ) , ∥ b kj ∥ C ,α ( Ξ ) < ∞ , ∥ ∂ θ f j ∥ C ( V ) < ∞ and since we may assume R ≤ N q small enough, it follows that for another constant C ′ q which is independentof p we have: R ∣ D∂ θ f j ( y p )∣ ≤ C ′ q ⎛⎝ R + α + ∑ j ∥ ∂ θ f j − ∂ θ f j ( y p )∥ C ( Ω R ) ⎞⎠ . (B.7) sing the fact that f j ∈ C ,α ( V ) , it follows that ∥ ∂ θ f j − ∂ θ f j ( y p )∥ C ( Ω R ) ≤ c q R α where c q is independent of p , and we conclude that: ∀ j = , , ∥ D f j ( y p )∥ ≤ C ′′ q R − αp , for some constant C ′′ q independent of p . As a side note, we remark that Kinderlehrer–Nirenberg–Spruck deduce in [32] that f j ∈ C ,α all the way up to the triple points, byconsidering the difference-quotients of f j (instead of working with D θ f j as we did above,since we already know the functions are smooth by [32]), and using the resulting C ,α Schauder estimate on the difference-quotient and the Arzel`a–Ascoli theorem to pass to thelimit, yielding C ,α regularity.We now claim that: ∀ j = , , ∥ D u j ( x p )∥ ≤ C ( ) q R − αp , (B.8)where C ( ) q is independent of p . Indeed, by the chain-rule, if we denote ¯ T = T − : D ab T c ( x p ) = − D b T e ( x p ) D ae ¯ T d ( y p ) D d T c ( x p ) . Recalling that ¯ T ( y ) = ( ψ ( y ) , y ′ ) , we know that ∥ D ¯ T ( y p )∥ ≤ A q / R − αp ; and recalling that T ( x ) = ( u ( x ) − u ( x ) , x ′ ) , since u i ∈ C ,α ( Ω ∩ Ω ) , we see that ∥ DT ( x p )∥ ≤ B q , where A q , B q are constants independent of p . Consequently, ∥ D T ( x p )∥ ≤ C q / R − αp . Taking twoderivatives of u ( x p ) = Φ + ( T ( x p )) and applying the chain-rule, the latter estimates on ∥ DT ( x p )∥ and ∥ D T ( x p )∥ in conjunction with ∥ D Φ + ( y p )∥ ≤ A q / R − αp and ∥ D Φ + ( y p )∥ ≤ B q , readily establishes (B.8) for u . As u ( x p ) = w ( x p ) + u ( x p ) and w ( x ) = T ( x ) , theprevious estimates imply (B.8) for u as well; the case of u follows similarly.As II ij ( p ) is computed quasilinearly from D u ( x p ) (leading order) and Du ( x p ) , itimmediately follows that: ∀ p ∈ Σ ijk ∩ N q ∥ II ij ( p )∥ ≤ C ( ) q R − αp . It remains to establish that: ∀ p ∈ Σ ijk ∩ ˜ N q R p ≥ c q d ( p, Σ ijkl ∩ N q ) , (B.9)for some constant c q > p and some sub-neighborhood ˜ N q ⊂ N q of q . Wewill use ˜ N q = ϕ ( ˜ U q ) , where: ˜ U q ∶ = { z ∈ U q ∶ d ( z, ) < d ( z, ∂U q )} . Indeed, observe that R p = min ( d ( y p , Ξ ) , d ( y p , ∂V ∖ Ξ )) , and so if we show that: ∀ y p ∈ Ξ ∖ Ξ ∩ T ( P H ( ˜ N q )) d ( y p , ∂V ∖ Ξ )) ≥ c ′ q d ( y p , Ξ ) , (B.10)then (B.9) will follow, since d ( y p , Ξ ) is equivalent to d ( p, Σ ijkl ∩ N q ) up to a factor of D q (as T ○ P H is a C ,α diffeomorphism on Σ ijk ∩ N q ). To show (B.10), observe that this isindeed the case on our model cluster: ∀ z ∈ Σ m d ( z, ∂ Σ m ∖ Σ m ) ≥ d ( z, Σ m ) , (in fact, with equality above, since Σ m and Σ m , the two boundary components of Σ m ,form an obtuse angle of cos − ( / ) ≃ ○ ). Consequently, it is easy to see that: ∀ z ∈ Σ m ∩ ˜ U q d ( z, ∂ ( Σ m ∩ U q ) ∖ Σ m ) ≥ d ( z, Σ m ∩ U q ) . he same holds with Σ m replaced by Σ m and Σ m . But since P H ○ ϕ is a C ,α diffeomorphismfrom Σ m ∩ U q to Ω , from Σ m ∩ U q to Ω , and from Σ m ∩ U q to Ω , we have: ∀ x ∈ Γ ∖ Γ ∩ P H ( ˜ N q ) ∀ i = , , d ( x, ∂ Ω i ∖ Γ ) ≥ c ′′′ q d ( x, Γ ) . Finally, since V = T ( Ω ∩ Ω ) ∩ S − ( Ω ) , and T and S − are C ,α diffeomorphisms on theircorresponding domains, (B.10) follows with an appropriate constant c ′ q . This concludes theproof of (B.1). B.3 Blow-up on Σ ij Let us now sketch the argument for establishing (B.2). Let N q be a (possibly readjusted)neighborhood of q as in the proof of (B.1) described above. We will use ˜ N q = ˜ N q ( c q ) ∩ ˜ N q ,where: ˜ N q ( c ) ∶ = ϕ ({ z ∈ U q ∶ d ( z, ) < cd ( z, ∂U q )}) , and: ˜ N q ∶ = { w ∈ N q ∶ d ( w, q ) < d ( w, ∂N q )} . The constant c q ≤ w ∈ ˜ N q ( c q ) and p ∈ ∂ Σ realizes thedistance d ( w, ∂ Σ ) , then necessarily p ∈ ˜ N q ( ) , which was exactly the requirement on p which we used in the proof of (B.1). Indeed, it is always possible to choose such a constant c q since ϕ is a C ,α diffeomorphism from U q to N q , and thus distances are preserved up toconstants.We first establish (B.2) for points w ∈ Σ ij ∩ ˜ N q so that: d ( w, Σ ijkl ∩ N q ) > A q d ( w, ∂ ( Σ ij ∩ N q )) , (B.11)for some constant A q > p ∈ ∂ ( Σ ij ∩ N q ) realize the distance onthe right-hand-side above. Then necessarily p ∈ ∂ Σ ij ∩ N q , since d ( w, ∂N q ) > d ( w, q ) ≥ d ( w, ∂ Σ ij ) by our assumption that w ∈ ˜ N q . Hence p ∈ Σ ijk ∩ N q or p ∈ Σ ijl ∩ N q , and weassume without loss of generality it is the former case (otherwise exchange the index k withthe index l in all of our previous arguments). Clearly p ∉ Σ ijkl since otherwise we wouldhave d ( w, Σ ijkl ∩ N q ) = d ( w, p ) , in violation of (B.11) and the fact that A q >
1. In addition,since w ∈ ˜ N q ( c q ) , we are guaranteed that p ∈ ˜ N q ( ) .Now let y p = T ( P H p ) ∈ Ξ ∖ Ξ ∩ T ( P H ( ˜ N q ( ))) . By (B.10), we know that there existsa constant c ′ q ∈ ( , ] so that Ω R yp ( y p ) ⊂ V with R y p ≥ c ′ q d ( y p , Ξ ) , where recall Ω R ( y ) denotes the hemisphere of radius R centered at y with flat part in { y = } . Since T and S − are C ,α diffeomorphisms from Ω ∩ Ω and Ω ∩ S ( V ) into V , respectively, there existsa constant c ′′ q > R x p ∶ = c ′′ q d ( x p , Γ ) , we have B R xp ( x p ) ⊂ ( Ω ∩ Ω ) ∪ Ω ,where B R ( x ) is a ball of radius R around x in H .By (B.11), we know that: d ( w, p ) < A q d ( w, Σ ijkl ∩ N q ) . (B.12)Since P H is a C ,α diffeomorphism from Σ ij ∩ N q to Ω , there exists a constant C q > d ( x w , x p ) < C q A q d ( x w , Γ ) . By the triangle inequality, it follows that: d ( x w , x p ) ≤ C q A q − d ( x p , Γ ) . (B.13)Therefore, choosing A q large enough, we can ensure that C q A q − ≤ c ′′ q , and conclude that x w ∈ B R xp ( x p ) ; in particular, x w ∈ Ω ∩ Ω ∩ S ( V ) and y w ∶ = T ( x w ) ∈ V . n addition, since T is a C ,α diffeomorphism from Ω ∩ Ω into V , (B.13) implies thatby choosing A q large enough, we can also ensure that d ( y w , y p ) ≤ ( c ′ q / ) d ( y p , Ξ ) , whichrecall is at most R y p /
2. Hence the distance of y w to the spherical part of the boundary ofΩ R yp ( y p ) is at least R y p /
2. Applying the Schauder estimate for systems as in (B.7) andtesting it at y w , we conclude that: ∥ D Φ + ( y w )∥ ≤ C ( ) q R − αy p . Arguing as in (B.8), we deduce that: ∥ D u ( x w )∥ ≤ C ( ) q R − αy p , which implies: ∥ II ij ( w )∥ ≤ C ( ) q R − αy p . (B.14)Recall that R y p ≥ c q d ( p, Σ ijkl ∩ N q ) by (B.9). On the other hand, (B.12) implies that: d ( p, Σ ijkl ∩ N q ) ≥ d ( w, Σ ijkl ∩ N q ) − d ( w, p ) ≥ A q − A q d ( w, Σ ijkl ∩ N q ) . Hence, we conclude that R y p ≥ c q A q − A q d ( w, Σ ijkl ∩ N q ) , which together with (B.14) concludesthe proof of (B.2) under the assumption that (B.11).The case when (B.11) does not hold is much simpler, and there is no need for systemsof PDEs nor boundary conditions. Set x w = P H w and let B R w be a ball centered at x w ofmaximum radius R w > ⊂ H . Since u is C ,α uniformly on the entire Ω , a standardtext-book C ,α Schauder estimate for the quasilinear elliptic equation (B.3) satisfied by u in B R w immediately verifies that: ∥ D u ( x w )∥ ≤ C ′ q R − αw , which implies as before that: ∥ II ij ( w )∥ ≤ C ′′ q R − αw . Since w violates (B.11), it remains to establish that R w ≥ c q d ( w, ∂ ( Σ ij ∩ N q )) . But thisis immediate, since P H is a C ,α diffeomorphism from Σ ij ∩ N q to Ω . This concludes theproof of (B.2) and thus of Proposition B.1. References [1] S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions ofelliptic partial differential equations satisfying general boundary conditions. I.
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