Mean curvature self-shrinkers of high genus: Non-compact examples
MMEAN CURVATURE SELF-SHRINKERS OF HIGH GENUS:NON-COMPACT EXAMPLES
N. KAPOULEAS, S. J. KLEENE, AND N. M. MØLLER
Abstract.
We give the first rigorous construction of complete, em-bedded self-shrinking hypersurfaces under mean curvature flow, sinceAngenent’s torus in 1989. The surfaces exist for any sufficiently largeprescribed genus g , and are non-compact with one end. Each has 4 g + 4symmetries and comes from desingularizing the intersection of the planeand sphere through a great circle, a configuration with very high sym-metry.Each is at infinity asymptotic to the cone in R over a 2 π/ ( g + 1)-periodic graph on an equator of the unit sphere S ⊆ R , with the shapeof a periodically ”wobbling sheet”. This is a dramatic instability phe-nomenon, with changes of asymptotics that break much more symmetrythan seen in minimal surface constructions.The core of the proof is a detailed understanding of the linearizedproblem in a setting with severely unbounded geometry, leading to spe-cial PDEs of Ornstein-Uhlenbeck type with fast growth on coefficientsof the gradient terms. This involves identifying new, adequate weightedH¨older spaces of asymptotically conical functions in which the operatorsinvert, via a Liouville-type result with precise asymptotics. Introduction
In studying the flow of a hypersurface by mean curvature in Euclidean n -space as well as in general ambient Riemannian n -manifolds ( M n , g ), n ≥ R n ,viz. solitons moving by an ambient conformal Killing field, and of these theself-shrinkers are the most important. Taking center stage when identifiedby Huisken in 1988 (and the compact H ≥ n = 3 only a few complete, embedded self-shrinking surfaces in R are to this date rigorously known: Flat planes,round cylinders, round spheres and a (not round-profile) torus of revolutiondiscovered by Angenent in [An] (this list exhausts the rotationally symmetric The results and methods in this paper have been presented at conferences in Princeton(February 2011), and at seminars at the Max Planck Institute in Golm (March 2011) andat MIT (May 2011). The first and second authors were supported partially by NSF grantsDMS1105371 and DMS1004646 , respectively. a r X i v : . [ m a t h . DG ] N ov xamples, although the uniqueness of the torus is still open; see [KM]).Note also that several results involving self-shrinkers in some generality haveappeared, most prominently a smooth compactness theorem (for closed,fixed genus surfaces [CM1]) and a theory of generic singularities of Colding-Minicozzi, including classification of all H ≥ R . Sincethe local considerations involved in the constructions would work in somegenerality (see [Ka05] and [Ka11]), it has long been expected that such con-structions could work for self-similar surfaces under mean curvature flow,and indeed there are constructions for the self-translating case in the inter-esting work by X.H. Nguyen (see [Ng1]-[Ng2]).The existence of self-shrinkers with the topology we consider in this paperwas conjectured by Tom Ilmanen in 1995 (from numerics, using Brakke’ssurface evolver; see [Il95]), while their asymptotic geometry was not clear atthat point.Our main theorem is the following: Theorem 1.1.
For every large enough integer g there exists a complete,embedded, orientable, smooth surface Σ g ⊆ R , with the properties: (i) Σ g is a mean curvature self-shrinker of genus g . (ii) Σ g is invariant under the dihedral symmetry group with g + 4 ele-ments. (iii) Σ g has one non-compact end, and separates R into two connectedcomponents. (iv) The end is outside some Euclidean ball a graph over a plane, asymp-totic to the cone on a non-zero vertical smooth (4g+4)-symmetricgraph over a great circle in S (hence the visual appearance of a”wobbling sheet”). (v) Inside any fixed ambient ball B R (0) ⊆ R , the sequence { Σ g } con-verges in Hausdorff sense to the union S ∪ P , where P is a planethrough the origin in R . In fact, the bounds (1.1) d H (cid:2) Σ g ∩ B R (0) , ( S ∪ P ) ∩ B R (0) (cid:3) ≤ C Rg , on the Hausdorff distance d H hold for some constant C > . Theconvergence is furthermore locally smooth away from the intersectioncircle. igure 1. Tom Ilmanen’s conjectural shrinker of genus 8with 9 Scherk handles.
Corollary 1.2.
Euclidean flat cylinders over Σ g are shrinkers. So, in anyfixed dimension n ≥ we obtain self-shrinking hypersurfaces Σ ng = Σ g × R n − ⊆ R n +1 , with arbitrary large first Betti number b (Σ g × R n − ) = b (Σ g ) = 2 g. The general approach of this article is the same as that of [Ka97], whichfollows the general methodology developed in [Ka95]. Our construction isanalogous to a specific instance of the main theorem in [Ka97], the case ofa catenoid intersecting a plane through its waist, which is simpler than thegeneral case because of the extra symmetry. On the other hand, we mustcontend with major analytic difficulties arising from the unbounded natureof the self-shrinker equation, which do not arise in minimal and constantmean curvature constructions.To look further into the analytical difficulties faced here, it is instructiveto use the mentioned characterization of self-shrinkers (which shrink towardsthe origin, with scaling factor (cid:112) − t )): Minimal surfaces S ⊆ R w.r.t.the conformal metric g ij = e −| x | / δ ij , where | x | is the distance to the originand δ ij is the Euclidean standard metric. All previous desingularization onstructions – and indeed much of geometric analysis – rely on some kindof reasonably bounded geometry such as for example geodesic completeness,curvature bounds, or even stronger assumptions such as asymptotic flatness.We must however here face that the metric is geodesically incomplete (non-extendible: the distance to infinity is finite) and the Ricci curvature of aplane through the origin in the unit normal direction, respectively the Gausscurvature of the induced metric on such a plane, are (see Appendix C):Ric( (cid:126)ν, (cid:126)ν ) = e | x | / (1 − | x | / → −∞ , for | x | → ∞ ,K ( R ,g ) = 12 e | x | / → + ∞ , for | x | → ∞ . It should hence come as no surprise that the analysis we need to performcould not follow from any very general principle, and in fact this paper alsogives the first successful example of a construction for such an unboundedgeometry. Our new (anisotropically) weighted H¨older spaces and accom-panying Liouville-type result and global Schauder-type estimates for theexterior linear problem of Ornstein-Uhlenbeck type, that are pivotal to thecompletion of the construction, arise from homogeneity properties of the lin-earized operator, which in turn lend their origin to the parabolic self-similarnature: It is the sum of homogeneous operators, with a homogeneity zeroterm which annihilates cones. We consider the problem of solving the equa-tion for homogeneous functions and find good (sharp) choices for weightedH¨older norms, and then proceed for general functions with those very samespaces.Note that the global Schauder estimates have no obvious extensions togeneral Laplace-type operators under the same growth rates on the coeffi-cients, and there are counterexamples by Priola for a very similar equation(see [Pr]).It is fruitful to compare our construction with that of desingularizing, inthe H ≡ / | x | g +1 as | x | → ∞ . In our construction no suchimprovement shows up, the self-shrinkers constructed have regardless of g the (likely sharp) asymptotics:(1.2) σ ( θ ) | x | + O ( | x | − ) , | x | → ∞ . Another difference from the previously known constructions for minimalsurfaces is that the surfaces we construct must be entropy unstable (sinceby [CM2] the only stables ones are of the form S n − k × R k , k = 0 , . . . , n ),and this is another way of viewing some of the complications that arisehere. However, it is from the desingularization viewpoint not presence ofthe instability per se that is the problem, it is the severe way in which t happens, witnessed by Equation (1.2): Imposing ever so much dihedralsymmetry never renders it negligible.Finally, we will mention that X.H. Nguyen via nonlinear parabolic meth-ods has studied a related, truncated nonlinear exterior problem for the self-shrinker equation and obtained existence results (see [Ng3]-[Ng4]). Also, L.Wang has announced interesting existence and uniqueness result for exteriorgraphs with prescribed cones at infinity (see [Wa]), which provides separateevidence of the dramatic change of asymptotics of the non-compact ends,i.e. that our examples are not asymptotic to planes.After this work was completed, we learned of a preprint by X.H. Nguyen[Ng5] which announces results very similar to ours.2. Overview of the paper
The basic philosophy of the desingularization procedure is as follows: Con-sider the initial configuration of a plane intersecting a sphere through a greatcircle. For each τ with τ − = k ∈ N a positive integer, define a one param-eter family of surfaces M [ τ, θ ] that serve as approximate solutions to theself-shrinker equation. The surfaces M [ τ, θ ] are invariant under the action ofthe dihedral group with 4 k elements, and under various normalizations con-verge either to the initial configuration or to Scherk’s singly-periodic surfaceΣ as the parameters τ and θ tend to zero.On each of these surfaces, we consider graphs of small functions u , andproduce via an incarnation of Newton’s method, here Schauder’s fixed pointtheorem, a pair ( θ ∗ , u ∗ ) such that the graph over M [ τ, θ ∗ ] by u ∗ solves theself-shrinker equation exactly. Naturally, to apply the Schauder fixed pointtheorem, one needs to first understand the linearized equation on these sur-faces, and to do this one needs to understand the linearized equation on thelimits under both normalizations; that is to say on the initial configurationand on Scherk’s surface. That is, we need to solve the equation L u = E on the initial surface M [ τ, θ ] with reasonable estimates, where L is the lin-earized operator for the self-shrinker equation (note that the study of thisoperator played an important role in [CM1]-[CM3]) and the function E isthe initial error in the self-shrinker equation on M [ τ, θ ].On the pieces of the initial configuration (that is, the surfaces with bound-ary determined by the intersection circle), we prove that the linearized equa-tion is always solvable with Dirichlet boundary conditions (and here we are,on the outer plane, forced to allow a dramatic change of asymptotics toinclude conical functions that are oscillatory in the angular variable). Nearthe intersection circle, the linearized equation turns out to be a perturbationof the stability operator on Scherk’s surface.The linearized equation on Scherk’s surface is not solvable, with appro-priate bounds on the norm of the inverse, in any bounded function space,in general, due to the persistence of a one-dimensional kernel spanned bya translational Killing field. But as long as the inhomogeneous term E is orthogonal” to this kernel, we can solve the equation in a weighted H¨olderspace with exponential decay. The decay then allows a solution to be patchedup globally to a solution on the entire initial surface. The role of the param-eter θ in the surfaces M [ τ, θ ] is then to arrange for the initial error term E to be orthogonal to the kernel. As θ changes, two of the pieces of M [ τ, θ ]move within a family of perturbed cap-shaped self-shrinkers near the roundspherical caps. Note therefore that the role of the chosen θ ∗ in this problemis of a more technical nature (unlike for example the case of catenoidal endsfor the H ≡ M [ τ, θ ] are introduced, and their basicproperties – smoothness in parameters, symmetries – are established.Section 6 gives necessary estimates for the mean curvature of the desin-gularizing surfaces Σ[ τ, θ ] and its variation under the θ parameter.In Section 7, the linearized operator L on the curled up Scherk belt Σ[ τ, θ ]is studied. We prove that the operator is invertible as a map between H¨olderspaces with decay, modulo a one-dimensional cokernel, and we show that thiscokernel can indeed be geometrically generated by varying the θ parameter.In Section 8, we study the exterior Ornstein-Uhlenbeck problem and iden-tify the correct weighted H¨older cone spaces which have all desired proper-ties (such as a compact inclusion hierarchy), and in which we invert thelinearized operator.In Section 9, the patching up of solutions of the linear problem on thevarious pieces of the initial surfaces M [ τ, θ ] to a global solution is under-taken.In Section 10, we verify the important fact that the nonlinear part ofthe problem closes up in the norms from Section 8, that is we prove thequadratic improvement required for Newton’s method to be applicable.Finally, in Section 11 we then complete the argument by setting up andcarrying out the Schauder fixed point procedure. The Appendix at the endrecords various computations which were needed throughout.3. Notation and conventions
Throughout R will denote Euclidean 3-space, (cid:126)X will denote a point in R , ( x, y, z ) the Cartesian coordinates of the point, and { (cid:126)e x , (cid:126)e y , (cid:126)e z } the as-sociated standard basis, so that (cid:126)X = ( x, y, z ) = x(cid:126)e x + y(cid:126)e y + z(cid:126)e z . We denoteby P xy , P yz , and P xz the xy -, yz -, and xz -coordinate planes respectively.We adopt the convention in this article that for a surface S , all associ-ated geometric objects and quantities will bear “ S ” as a subscript, with the xception of Scherk’s singly-periodic surface Σ and the surfaces Σ[ τ, θ ] de-fined in Section 5 . Objects associated with Σ will at times simply bear thesubscript “0”. In most cases, the surfaces Σ[ τ, θ ] will appear with the τ and θ arguments suppressed - so, for example, as simply Σ - and their associatedquantities will be identified without subscript. The reader should take careto distinguish subscripts from superscripts, as “0” will appear throughoutthe article as superscript as well.We denote by (cid:126)ν S the Gauss map of an oriented surface S . Given a func-tion f : S → R on a surface S , we use the shorthand { S : f ≤ } todenote the set { p ∈ S : f ( p ) ≤ } ⊂ S , and likewise for “ ≥ ”. Note thatunder appropriate assumptions on f , { S : f ≤ } is a smooth surface withsmooth (possibly empty) boundary, and we view { S : f ≤ } as inheritingall geometric quantities from S – i.e. first and second fundamental forms– via the inclusion mapping. Also, for a function f , we denote by S f thenormal graph of f over S . Note that when f and S are class C k,α and f issufficiently small, then S f is a C k − ,α surface naturally parametrized by S .Geometric objects defined on any of the surfaces Σ given in Section 5 maybe viewed as objects on Σ via the map Z : Σ → Σ.We denote by H + the upper half plane { ( s, z ) : s > } and by C itsquotient (a cylinder) under the action z (cid:55)→ z + 2 π . Throughout this article,we fix a smooth, non-decreasing function ψ : R → R which vanishes on( −∞ , /
3) and has ψ ≡ / , ∞ ). Also, we let ψ [ a, b ] : R → [0 ,
1] be ψ [ a, b ]( s ) := ψ (cid:0) s − ab − a (cid:1) , so that ψ [ a, b ] transitions from 0 at a to 1 at b .We will for the compact pieces in our construction work in the usualweighted H¨older spaces C k,α ( S, g S , f ) on Riemannian surfaces ( S, g S ), de-fined by finiteness of the corresponding norms(3.1) (cid:13)(cid:13)(cid:13) u : C k,α ( S, g S , f ) (cid:13)(cid:13)(cid:13) := sup x ∈ S f ( x ) (cid:13)(cid:13)(cid:13) u : C k,α ( S ∩ B ( x ) , g S ) (cid:13)(cid:13)(cid:13) , with weight function f : S → R , where g S is the metric for which the usual C k,α -norm is taken and B ( x ) the geodesic ball of radius 1 centered at x .When the metric is understood, we sometimes drop it from the notationwriting C k,α ( S, f ) = C k,α ( S, g S , f ).4. The self-shrinker equation
Recall that the PDE to be satisfied for a smooth oriented surface S ⊆ R to be a self-shrinker (shrinking towards the origin with singular time T = 1)is(4.1) H S ( (cid:126)X ) − (cid:126)X · (cid:126)ν S ( (cid:126)X ) = 0 , for each (cid:126)X ∈ S , where by convention H S = (cid:80) n κ i is the sum of the signedprincipal curvatures w.r.t. the chosen normal (cid:126)ν S (i.e. H = 2 for the spherewith outward pointing (cid:126)ν ). Such surfaces shrink by homothety towards the rigin under flow by the (orientation-independent) mean curvature vector (cid:126)H = − H(cid:126)ν , by the factor (cid:112) − t ). In particular, we have normalizedEquation (4.1) so that T = 1 is the singular time.The surface ˜ S obtained by dilating a self-shrinker S about the origin bya factor of τ − satisfies the corresponding rescaled equation(4.2) H ˜ S ( (cid:126)X ) − τ (cid:126)X · (cid:126)ν ˜ S ( (cid:126)X ) = 0 . For a smooth normal variation (cid:126)X t determined by a function u via X t = X + tu(cid:126)ν ˜ S , where (cid:126)X parametrizes ˜ S , the pointwise linear change in (minus) thequantity on the left hands side in (4.2) is given by the stability operator (seethe Appendix, and also [CM1]-[CM2] for more properties of this operator)(4.3) L ˜ S u = ∆ ˜ S u + | A ˜ S | u − τ (cid:16) (cid:126)X · ∇ ˜ S u − u (cid:17) . Because at times we want to treat Equation (4.2) as a perturbation of themean curvature equation, we isolate the part of the linear change due tovarying the mean curvature of S and set(4.4) L S = ∆ S + | A S | . Note that Equation (4.1) and its dilated version (4.2) are invariant underthe orthogonal group O(3).5.
The Initial Surfaces
In this section we describe in detail the construction of the initial surfaces M [ τ, θ ], depending on parameters τ and θ which we assume satisfy0 < τ ≤ δ τ , | θ | ≤ δ θ , throughout for appropriate constants that will later be chosen. The surfacesare approximate solutions to Equation (4.1), and by means of a fixed pointargument we will for each small enough τ produce a function on them (forappropriately chosen θ ) whose graph satisfies Equation (4.1) exactly. Thebasic ingredients are the singly periodic Scherk’s singly-periodic surface Σ and a family of half surfaces K [ θ ] that are rotationally symmetric (aboutthe y -axis) perturbations of the round hemisphere of radius 2. The crucialproperties of the half-surfaces K [ θ ] are that they satisfy Equation (4.1) ex-actly, intersect the plane P = P xz at the angle π/ − θ and when θ vanishesagree with the hemisphere S (2) ∩ { y ≥ } .Let C [ θ ] denote the configuration consisting of the plane P together with K [ θ ] and a copy of K [ θ ] reflected through P and let c [ θ ] denote their circleof intersection. For each τ with τ − an integer, the surfaces M [ τ, θ ] out-side of a neighborhood of c [ θ ] of uniformly fixed radius will agree with C [ θ ].Inside this neighborhood they will consist, loosely speaking, of τ − funda-mental domains of Σ , rescaled by a factor of τ that have been “curled”and appropriately smoothed out to replace the singular intersection circle inthe configuration. The analysis is simplified by identifying the symmetriespreserved by this procedure and then imposing these from the beginning. efinition 5.1. Let G τ be the subgroup generated by ω τ , ξ τ ∈ O (3) , where: (1) ω τ is the rotation about the y -axis by a positive angle πτ followed bythe reflection y (cid:55)→ − y . (2) ξ τ , is the reflection through a plane P τ , which is { z = 0 } rotated anangle of ( π/ τ around the y -axis.Denote also by σ τ = ω τ the rotation about the y -axis by a positive angle πτ . We will construct the surfaces M [ τ, θ ] so that they are invariant under G τ , with σ τ orientation preserving and ω τ orientation reversing. We assumeimplicitly that τ − is a positive integer. These symmetries will be reflectedin the analysis by working with functions on M [ τ, θ ] that are invariant under σ τ and ξ and anti-invariant under ω τ . As the parameter τ →
0, the surfaces M [ τ, θ ] converge, under an appropriate renormalization, to a surface Σ[ θ ],singly periodic in the direction of the z -axis and invariant under the actionof a group G , as follows: Definition 5.2.
Let G be the group generated by the Euclidean isometries ω and ξ , where: (1) ω is the translation z (cid:55)→ z + π followed by the reflection y (cid:55)→ − y . (2) ξ is the reflection through the plane { z = π/ } .Denote also by σ = ω the translation z (cid:55)→ z + 2 π . The geometrically correct notion of symmetric functions is as in the nextdefinition, the point being to ensure that normal graphs (using the fixedunit normal giving the orientation) over the symmetric surface inherit thesymmetries.
Definition 5.3.
Let S be an oriented surface invariant under G τ (resp. G ). By the G τ -equivariant (resp. G -invariant) functions we will mean all f : S → R such that β ∗ f = (cid:104) (cid:126)ν S , β(cid:126)ν S (cid:105) f, ∀ β ∈ G τ (resp. G ) . Now, recall Scherk’s minimal surface Σ (cf. [Ka97] p. 101–106) withangle π between the asymptotic planes:(5.1) Σ = { ( x, y, z ) ⊆ R | sinh x sinh y − sin z = 0 } . In addition to G , the isometries of Σ include reflection in the planes { x = y } and { x = − y } . The regions Σ ∩ {± x > } and Σ ∩ {± y > } aregraphs over P xz and P yz respectively, and the symmetries of Σ give that itis globally determined by the graph of a single function(5.2) f : H + → R . where H + = { ( s, z ) | s > } . That is, in the half space I = { ( x, y, z ) | x > } )we have Σ ∩ I = { ( x, f ( x, z ) , z ) } with function f ( s, z ) satisfying the estimate(5.3) (cid:107) f : C ( { H + : s ≥ } , e − s ) (cid:107) ≤ C. simple rephrasing of this estimate is as follows: Let Proj { x · y=0 } : R →{ x · y = 0 } = P xz ∪ P yz denote the nearest point projection to this closedset. Then Proj { x · y=0 } is well defined away from the planes { x = ± y } and itsrestriction to Σ satisfies the estimate (cid:107) Proj { x · y=0 }− − Id (cid:107) : C ( { H + : s ≥ } , e − s ) (cid:107) ≤ C . On Σ we define the function s by(5.4) s (( x, y, z )) = max {| x | , | y |} . Note that since Σ is minimal, Σ / (cid:104) σ (cid:105) is conformal under the Gaussmap (cid:126)ν Σ with conformal factor | A Σ | to the punctured sphere { S : x ≥ } \ { ( ± , , , (0 , ± , } .Let ω ∗ and ξ ∗ denote the Euclidean isometries given by ( x, y, z ) (cid:55)→ ( − x, y, − z )and ( x, y, z ) (cid:55)→ ( x, y, − z ), respectively. By computing the gradient of thefunction defining Σ we obtain the intertwining relations (cid:126)ν Σ ◦ ω ( (cid:126)X ) = ω ∗ ◦ (cid:126)ν Σ ( (cid:126)X ) , (5.5) (cid:126)ν Σ ◦ ξ ( (cid:126)X ) = ξ ∗ ◦ (cid:126)ν Σ ( (cid:126)X ) . Thus, functions on Σ that are invariant under ξ and anti-invariant under ω (i.e. G -equivariant) push forward under the Gauss map to functions thatare invariant under ξ ∗ and anti-invariant with respect to the inversion ω ∗ .Since the Gauss map will be the fundamental tool in understanding thelinear operator L Σ on Σ we record the following lemma. Lemma 5.4.
The kernel of the operator ∆ S + 2 on the unit sphere in thespace of L -functions that are invariant under ξ ∗ and anti-invariant under ω ∗ is one-dimensional, spanned by the ambient coordinate function x . Proposition 5.5.
For | θ | ≤ δ θ with δ θ sufficiently small, there is a smoothone parameter family of surfaces K [ θ ] , with the following properties: (0) Each K [ θ ] satisfies Equation (4.1). (1) K [0] is the upper hemisphere of radius and the surfaces K [ θ ] aregiven as normal graphs over K [0] . (2) The surfaces K [ θ ] are invariant with respect to rotations about the y -axis. (3) The boundary ∂ K [ θ ] is a circle in the plane P xz of radius r [ θ ] , andthe inward pointing co-normal η θ to ∂ K [ θ ] at the x -axis satisfies η θ · (cid:126)e x = sin( θ ) . (4) There are conformal parametrizations κ [ θ ] : C (cid:55)→ K [ θ ] \ { y -axis } of the surfaces K [ θ ] , where C = H + / { z (cid:55)→ z + 2 π } is the flat cylinderof radius such that: (i) κ [ θ ]( { ( s, z ) : s = const. } )) is a circle with center on the y -axisparallel to the xz plane. (ii) κ [ θ ]( { s = 0 } ) = ∂ K [ θ ] . (iii) The conformal factor is (cid:37) κ [ θ ] ( s, z ) = x ( s, z ) + z ( s, z ) . iv) There are bounds (5.6) |∇ k κ [ θ ] | , |∇ k ˙ κ [ θ ] | ≤ C ( k ) where “ · ” denotes derivation in the θ parameter.Proof. See Appendix. (cid:3)
Definition 5.6.
We denote by K [ τ, θ ] the surface K [ θ ] dilated by the factor τ − and κ [ τ, θ ] : H + → K [ τ, θ ] the map given by κ [ τ, θ ]( s, z ) = τ − κ [ θ ]( τ s, τ z ) . Definition 5.7.
Let ψ = ψ [1 / , . Then define the maps B [ τ, θ ] : R → R and Z [ τ, θ ] : R → R by B [ τ, θ ]( x, y, z ) = r [ θ ] τ − e τx (cos τ z, , sin τ z ) + r [ θ ] y(cid:126)e y , and Z [ τ, θ ]( x, y, z ) = ψ ( y )( κ [ τ, θ ]( y, z )+ r [ θ ] x(cid:126)ν κ [ τ,θ ] ( y, z ))+(1 − ψ ( y )) B [ τ, θ ]( x, y, z ) . Proposition 5.8.
The maps Z [ τ, θ ] have the following properties: (1) They depend smoothly on the parameters τ and θ with bounds (cid:12)(cid:12)(cid:12) ∇ k Z [ τ, θ ] (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) ∇ k ˙ Z [ τ, θ ] (cid:12)(cid:12)(cid:12) ≤ Cτ k − , k > . (2) We have that Z [ θ ] := lim τ → Z [ τ, θ ] − τ − r [ θ ] (cid:126)e x = r [ θ ]( ψR θ + (1 − ψ )Id) where R θ ∈ SO (3) is the rotation determined by (cid:126)e x (cid:55)→ cos θ(cid:126)e x − sin θ(cid:126)e y (cid:126)e z (cid:55)→ (cid:126)e z (cid:126)e y (cid:55)→ cos θ(cid:126)e y + sin θ(cid:126)e x , In particular Z [0] is globally the identity transformation.Proof. Claim (1) follows directly from the estimates 5.6 recorded in Propo-sition 5.5. Part (2) can be seen by applying l’Hˆopital’s rule. (cid:3)
We now are ready to define the “desingularizing” and “initial” surfaces,and to set notation for various distinguished subsurfaces. For technicalreasons, we work with a family of cut-off Scherk surfaces that agree withthe asymptotic planes P xz and P yz outside of a cylinder around the line { x = y = 0 } and of a fixed radius proportional to τ − . The reason for thisis that the image of these cut-off surfaces under the maps τ Z [ τ, θ ], outsideof a tubular neighborhood (of fixed radius independent of τ and θ ) of thecircle c [ θ ], is thus contained in the initial configuration C [ θ ]. Proposition 5.9.
We obtain “desingularizing” surfaces Σ[ τ, θ ] as follows: For a constant δ s > to be determined later, assume τ ≤ δ s anddefine the immersion ϕ τ : Σ → R by ϕ τ ( (cid:126)X ) = ψ [3 δ s τ − , δ s τ − ] (cid:126)X + (1 − ψ [3 δ s τ − , δ s τ − ]) Proj { x · y=0 } ( (cid:126)X ) , where the cut-off function is evaluated at s = s ( (cid:126)X ) . (2) The surface Σ[ τ, θ ] is Σ[ τ, θ ] := Z [ τ, θ ] ◦ ϕ τ ( { Σ : s ≤ δ s τ − } ) , which with sufficiently small δ θ , δ τ > is well-defined, smooth andembedded for τ < δ τ and | θ | ≤ δ θ . The set T := { τ Σ[ τ, θ ] : δ s τ ≤ s ≤ δ s τ } , where the transition happens,consists of four connected components each of which is by construction asubregion of either a top/bottom spherical cap K [ θ ] or of the plane P .Considering the singular initial configuration C [ θ ], the set C [ θ ] \ T there-fore has 5 connected components. One is the central piece containing thecurve c [ θ ], but this singular component is now discarded and replaced bythe smooth desingularizing surface Σ[ τ, θ ] to obtain the initial surface: Definition 5.10.
The initial surface M [ τ, θ ] is the union of τ Σ[ τ, θ ] withthe four components of C [ θ ] \ T that do not contain the singular curve c [ θ ] . Since τ Σ[ τ, θ ] overlaps with C [ θ ] in the set T , and we have excised theset containing the singular curve c [ θ ], the surfaces M [ τ, θ ] are smooth. Theconstructed surfaces are orientable, but notice the topology is such that if weorient, say, the top sphere with outward pointing normal then the bottomsphere has inwards pointing normal. Proposition 5.11.
For δ θ , δ τ > chosen sufficiently small, the surfaces M [ τ, θ ] are smooth, embedded, oriented and invariant under the action of G τ . Remark 5.12.
Note that when τ − = k ∈ N , we have replaced a greatcircle by k Scherk handles. Hence, as computing the Euler characteristicreveals, the initial surface M [ τ, θ ] has topological genus g = k − and k symmetries. Thus we have: (5.5) τ = 1 g + 1 and | G τ | = 4 g + 4 . Definition 5.13.
We define the function s on the surfaces Σ[ τ, θ ] and M [ τ, θ ] as follows. (1) On Σ[ τ, θ ] , we take s to be the push forward by Z [ τ, θ ] · φ τ of thefunction s defined on Σ . (2) s is then extended continuously to a constant on the remainder of M [ τ, θ ] ⊃ Σ[ τ, θ ] . Remark 5.14.
The reader will note that the surfaces Σ[ τ, θ ] are by con-struction diffeomorphic to { Σ : s ≤ δ s τ − } under the map Z [ τ, θ ] ◦ ϕ τ . e will, throughout this article, identify functions, tensors, and operatorson Σ[ τ, θ ] with their pull-backs by Z [ τ, θ ] ◦ ϕ τ , and vice versa. Geometric quantities on the initial surfaces
Proposition 6.1.
Let < γ < . Then on { Σ[ τ, θ ] : s ≥ } we have: (cid:13)(cid:13)(cid:13) H Σ − τ (cid:126)X · (cid:126)ν Σ : C ( { Σ[ τ, θ ] : s ≥ } , e − γs ) (cid:13)(cid:13)(cid:13) ≤ Cτ, and (cid:13)(cid:13)(cid:13)(cid:13) ∂∂θ (cid:110) H Σ − τ (cid:126)X · (cid:126)ν Σ (cid:111) : C ( { Σ[ τ, θ ] : s ≥ } , e − γs ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ Cτ. for τ > sufficiently small.Proof. In the following we let δ = δ ij denote the flat standard metric onthe upper half plane H + . Note that the surface Σ[ τ, θ ] ∩ { y ≥ } has theparametrization ϕ : { H + : y ≥ } → R given by ϕ ( s, z ) = κ [ τ, θ ]( s, z ) + ψ [4 δ s τ − , δ s τ − ]( s ) f ( s, z ) r [ θ ] (cid:126)ν κ ( s, z ) , We will in the rest of this proof denote κ = κ [ τ, θ ]. When s ≥ δ s /τ the estimates are trivially satisfied since ϕ ≡ κ [ τ, θ ] in this region. For s belonging to the interval [3 δ s /τ, δ s /τ ], we have that H Σ − τ (cid:126)X · (cid:126)ν Σ = − ∆ K ˆ f − | A K | ˆ f + τ (cid:16) ∇ K ˆ f · (cid:126)X − ˆ f (cid:17) + Q ˆ f + τ (cid:126)X · (cid:126)Q ˆ f , where ˆ f = ψ [4 δ s τ − , δ s τ − ]( s ) f ( s, z ) and Q ˆ f and (cid:126)Q u denote terms thatare at least quadratic in ˆ f and its derivatives. In this region, we may (since γ <
1) arrange that e − s < τ e − γs by taking τ sufficiently small in terms of γ . The estimate then follows by observing that |∇ kδ ˆ f | ≤ Ce − s , k = 0 , , (cid:37) − κ ∆ δ = ∆ K , and that both (cid:37) κ and | A K | are uniformly bounded inthis region.We now treat the case s ≤ δ s /τ as follows. Since { Σ : y ≥ } is a graphover H + which is itself minimal, and since dilations preserve minimality, wehave from the variation formula (11.11) in the Appendix the relation(6.1) ∆ δ f = r θ Q f . We then estimate the error term on Σ = { Σ[ τ, θ ] : s ≤ δ s /τ } , using that itis a graph over K = K [ τ, θ ], as follows: H Σ − τ (cid:126)X · (cid:126)ν Σ = − r θ L K f + Q r θ f + τ (cid:126)X · (cid:126)Q r θ f = − r θ ∆ K f − | A K | r θ f + τ r θ (cid:16) (cid:126)X · ∇ K f − f (cid:17) + Q r θ f + τ (cid:126)X · (cid:126)Q r θ f = −| A K | r θ f + τ r θ (cid:16) (cid:126)X · ∇ K f − f (cid:17) + r θ ( Q f − (cid:37) − κ Q f ) + τ (cid:126)X · (cid:126)Q r θ f , where in the last equality we have used (6.1).Note that as a consequence of the estimates for Z [ τ, θ ] recorded in (5.8)the terms | A K | r θ f and τ r θ ( (cid:126)X · ∇ K f − f ) appearing above and their ariations by θ satisfy the desired estimates, so it remains to estimate theterms R := r θ ( Q f − (cid:37) − κ Q f ). At τ = 0 one has that R ≡
0, and since onemay verify that the map ( τ, θ ) (cid:55)→ R ( · ) is C in the parameters τ ≥ θ as a map into C ( H + , δ, e − γs ), we get the claimed estimates by one-sidedTaylor expansion. (cid:3) The linearized equation away from the end
Definition 7.1.
Set Σ[ θ ] = Z [ θ ](Σ ) and let the function ˙ H Σ[ θ ] : Σ[ θ ] → R denote the variation under θ of the mean curvature H Σ[ θ ] of Σ[ θ ] , that is forall x ∈ Σ ˙ H Σ[ θ ] ◦ Z [ θ ]( x ) := ∂∂θ (cid:104) H Σ[ θ ] ◦ Z [ θ ]( x ) (cid:105) Then the function w : Σ[ τ, θ ] → R is given by w [ τ, θ ] = ˙ H Σ[ θ ] ◦ Z [ θ ] ◦ ( Z ◦ ϕ τ ) − [ τ, θ ] . where we are viewing Z [ τ, θ ] ◦ ϕ τ as a diffeomorphism of { Σ : s ≤ δ s /τ } onto Σ[ τ, θ ] . Lemma 7.2.
The function w has the following properties: (1) w is supported on { Σ[ τ, θ ] : s ≤ } . (2) The estimate (cid:13)(cid:13)(cid:13)(cid:13) ∂∂θ (cid:110) H Σ − τ ( (cid:126)X · (cid:126)ν Σ ) (cid:111) − w : C (Σ , g, e − γs ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ Cτ, holds for all sufficiently small θ and τ . (3) When τ = 0 and θ = 0 it holds (cid:90) Σ / (cid:104) σ (cid:105) w ( (cid:126)e x · (cid:126)ν ) dµ Σ = 8 π. where w ≡ w [0 , .Proof. (1) and (2) follow directly from Definition 7.1 and Proposition 6.1.To see (3), set S c = { Σ : s ≤ c } / (cid:104) σ (cid:105) . We then we have (cid:90) S c w ( (cid:126)e x · (cid:126)ν Σ ) = (cid:90) S c u L Σ ( (cid:126)e x · (cid:126)ν Σ ) + (cid:90) ∂S c (cid:104) ( (cid:126)e x · (cid:126)ν )( ∇ u · (cid:126)η ) − u(cid:126)η · ∇ ( (cid:126)e x · (cid:126)ν ) (cid:105) where ∇ = ∇ Σ , (cid:126)η is the co-normal at the boundary of S c , and u = ∂∂θ Z [ θ ] (cid:12)(cid:12) θ =0 · (cid:126)ν Σ , so that L Σ u = w . The claim then follows by taking c to ∞ and noting that |∇ ( (cid:126)e x · (cid:126)ν Σ )( s, z ) | ≤ Ce − s , | (cid:126)η − (cid:126)e y | ≤ Ce − c , and | ( ∇ u ( s, z ) − (cid:126)e y ) | ≤ Ce − s . (cid:3) By Proposition 6.1, the quantity E = H Σ − τ ( (cid:126)X · (cid:126)ν Σ ) and its variationsunder θ lie in the weighted H¨older spaces C ,α (Σ , g, e − γs ). The symmetriesof Σ give that E is G τ -equivariant, and that its pull-back to Σ by Z is G -equivariant. For the remainder of this article, all functions defined onΣ[ τ, θ ] are assumed to be invariant under the symmetry group G τ . roposition 7.3. Given any E ∈ C ,α (Σ , g, e − γs ) , there is a constant b = b E and a function v = v E such that L Σ v = E − bw,v = 0 , on ∂ Σ , | b | , (cid:107) v : C ,α (Σ , g, e − γs ) (cid:107) ≤ C (cid:107) E : C ,α (cid:0) Σ , g, e − γs (cid:1) (cid:107) . Moreover, the pair ( v E , b E ) depends continuously on the parameters τ and θ (see Remark (5.14)) We prove first Proposition 7.3 in the limiting case τ = 0, θ = 0, andhandle the general case as a perturbation. Proposition 7.4.
Given any E ∈ C ,α (Σ , g , e − γs ) , there is a constant b = b E and a function v = v E such that (7.1) L Σ v = E − bw , and such that | b | , (cid:107) v : C ,α (Σ , g , e − γs ) (cid:107) ≤ C (cid:107) E : C ,α (Σ , g , e − γs ) (cid:107) Proof.
Let E ∈ C ,α (Σ , e − γs ) be a given G -equivariant function, and as-sume for the moment that E is supported on { Σ : s ≤ a } where a > ν Σ : Σ → S is a conformal covering which descends to a diffeomorphism from Σ /σ onto the punctured sphere S \ { ( ± , , , (0 , ± , } , with the four punc-tures corresponding to the four asymptotic ends of Σ . The function ¯ E = (cid:126)ν Σ ∗ ( E/ | A Σ | ) ∈ L ( S , g S ) is then well-defined and satisfies (cid:107) ¯ E (cid:107) L ( S ) ≤ C (cid:107) E (cid:107) C ,α (Σ ,e − γs ) where the constant C depends on a . It is easily verified that since E is G -equivariant, the function ¯ E satisfies the identities (5.5), which then givethat ¯ E is L orthogonal to the functions y and z on S from Lemma 5.4.Now, (3) in Lemma 7.2 gives that (cid:90) S ¯ wx = 8 π, where ¯ w = (cid:126)ν Σ ∗ ( w / | A Σ | ). Thus, we may find a constant b such that ¯ E − b ¯ w is L -orthogonal to x . We then get a function v : S → R satisfying(∆ S + 2) v = ¯ E − ¯ w and the identities (5.5), from which we conclude that v (1 , ,
0) = − v ( − , , v (0 , ± ,
0) = 0. Define then the G -equivariant function u : Σ → R by u = (cid:126)ν ∗ Σ ( v − v (1 , , x ) . We then get immediately that u satisfies L Σ u = E − bw . hat u has the appropriate decay, i.e. lies in the space C ,α (Σ , g , e − γs ),follows by observing that the operator L Σ is asymptotically a perturbationof the Laplace operator on the flat cylinder C , for which the decay estimateshold. To conclude the proof, note that we may reduce to the case that E is supported in { Σ : s ≤ a } as follows: Recall that each component of { Σ : s ≥ a } is given by the graph of a small function f : H + → R with f satisfying (5.3). For a sufficiently large, the operator L Σ on { Σ : s ≥ a − } is then a perturbation of the Laplace operator ∆ H + on the flat half cylinder H + . Proposition 11.5 then gives a function u (cid:48) on { Σ : s ≥ a } satisfying L Σ u (cid:48) = E,u (cid:48) = c, on ∂ { Σ : s ≥ a − } . for a constant c with | c | ≤ C (cid:107) E (cid:107) . Define the smooth cutoff function ψ = ψ [ a − , a ]. We then get that ψu (cid:48) is defined on all of Σ and satisfies L Σ ( ψu (cid:48) ) = ψE + E where E is an error term introduced by smoothing out u (cid:48) on the boundaryof { Σ : s ≥ a − } . The function F = (1 − ψ ) E − E is then supported on { Σ : s ≤ a } . This concludes the proof. (cid:3) Remark 7.5.
The reader will note that the highly symmetric nature of ourconstruction, in contrast with the general situation and in particular theconstruction in [Ka97] , allow us to obtain a solution with the appropriatedecay with a single parameter.In particular, the function v satisfying the equivalent problem on thesphere has opposite values at ( ± , , , which allows simultaneous cancella-tion of both values by a single multiple of the kernel element x . Corollary 7.6.
Given E ∈ C ,α ( { Σ : s ≤ δ s τ − } , g , e − γs ) , there is a constant b ∈ R and a function v ∈ C ,α ( { Σ : s ≤ δ s τ − } , g , e − γs ) such that L Σ v = E − bw on { Σ : s ≤ δ s τ − } v = 0 on ∂ { Σ : s ≤ δ s τ − } with the bounds (cid:107) v : C ,α ( { Σ : s ≤ δ s τ − } , g , e − γs ) (cid:107) ≤ C (cid:107) E : C ,α ( { Σ : s ≤ δ s τ − } , g , e − γs ) (cid:107) and | b | ≤ C (cid:107) E : C ,α ( { Σ : s ≤ δ s τ − } , g , e − γs ) (cid:107) . Proof.
First, we apply Proposition 7.4 to obtain a function v satisfying(7.1). Now, note that for a large constant a >
0, the operator L Σ =∆ Σ + | A Σ | on { Σ : a ≤ s ≤ δ s τ − } is a perturbation of the Laplacian ona long cylinder. This allows us (see Proposition 11.5) to solve the following irichlet problem, with ∂ a and ∂ τ denoting the boundary components of { Σ : a ≤ s ≤ δ s τ − } in the obvious way: L Σ v = 0(7.2) v = v + c on ∂ τ v = 0 on ∂ a with the bounds | c | , (cid:107) v : C ,α ( { Σ : a ≤ s ≤ δ s τ − } , g ) (cid:107) ≤ C (cid:107) v : C ,α ( ∂ τ , g ) (cid:107)≤ Ce − γδ s τ − (cid:107) E : C ,α ( { Σ : s ≤ δ s τ − } , g , e − γs ) (cid:107) . The function v = v − ψ [ a, a + 1]( v − c ) then solves L Σ v = E − bw + E on { Σ : s ≤ δ s τ − } v = 0 on ∂ { Σ : s ≤ δ s τ − } . for an error term E and has the required bounds on the norm, and by taking τ sufficiently small and using that | A Σ | < Ce − s (a consequence of (5.3))we get that (cid:107)E : C ,α ( { Σ : s ≤ δ s τ − } , g , e − γs ) (cid:107) ≤ / (cid:107) E : C ,α ( { Σ : s ≤ δ s τ − } , g , e − γs ) (cid:107) . We then iterate this process to obtain an exact solution. (cid:3)
We now prove Proposition 7.3 in full generality.
Proof.
Recall that
Z ◦ ϕ τ : { Σ : s ≤ δ s /τ } → Σ is a diffeomorphism. By re-ferring to the derivative bounds on the maps Z [ τ, θ ] recorded in Proposition5.8 it is clear that we can arrange so that (cid:107) g Σ − ( Z ◦ ϕ τ ) ∗ g Σ : C ,α (Σ , g ) (cid:107) < (cid:15) by choosing the constant δ s sufficiently small for arbitrary positive (cid:15) . Now,by choosing a sufficiently large and τ sufficiently small, we can arrange that(7.3) ( Z ◦ ϕ τ ) ∗ | A Σ | , | A Σ | < (cid:15) on { Σ : a ≤ s ≤ δ s /τ } . It follows that the operator norm of ( Z ◦ ϕ τ ) ∗ L Σ −L Σ : C ,α (Σ ) → C ,α (Σ ) can be made arbitrarily small. The propositionthen follows by formally treating ( Z ◦ ϕ τ ) ∗ L Σ as a perturbation of L Σ . (cid:3) Lemma 7.7.
For any γ ∈ (0 , there exists C = C ( γ ) such that (cid:13)(cid:13)(cid:13) H Σ − τ (cid:126)X · (cid:126)v Σ − θw : C ,α (Σ , g Σ , e − γs ) (cid:13)(cid:13)(cid:13) ≤ C ( τ + | θ | ) , where H Σ is the mean curvature of Σ .Proof. This is a consequence of the smooth dependence of the surfaces Σ onthe parameters θ, τ and the definition of w . (cid:3) . An exterior linear problem of Ornstein-Uhlenbeck type
On a flat plane P through the origin, with the induced standard Euclideanmetric, the Dirichlet problem for the linearized operator L P in (4.3) at unitscale becomes:(8.1) (cid:40) L P u = ∆ u − (cid:0) (cid:126)X · ∇ u − u (cid:1) = E,u | ∂ Ω = 0 . for u : Ω → R , where the domain Ω = R \ B R (0) is the exterior of a diskwith radius R (cid:39)
2. The Laplacian and gradient are taken with respect tothe standard Euclidean metric on the plane. The function E is implicitlyassumed to be G τ -equivariant.The operator L P is of Ornstein-Uhlenbeck type (such operators are re-lated to Brownian motion and number operators in quantum mechanics). Itis of course clear that the local theory for this equation is classic, using forexample standard Schauder estimates. On the non-compact exterior domainhowever, with such fast growth on the gradient term, there is generally noreasonable global elliptic theory available (see for example the counterexam-ples [Pr]) and it is not a priori clear even what spaces to study the problemin. There exists in fact a vast literature on Ornstein-Uhlenbeck operators forvarious restrictive assumptions on the coefficients and corresponding choicesof function spaces (see for example [CV] and [DL]), but since remarkablythere is nothing in the literature that is adequate for our construction, wemust develop our theory from scratch.Firstly, note that the connection with the stability operator as a minimalsurface in the Gaussian metric (see (11.14) in the Appendix), is via thefollowing conjugation identity,(8.2) L P u = ∆ u − ( (cid:126)X · ∇ − u = e | x | / (cid:16) ∆ − | x |
16 + 1 (cid:17) e −| x | / u, where the exponential functions act by multiplication.The operator in the parentheses in (8.2) is of course nothing but theHamilton operator for the two-dimensional quantum harmonic oscillator,plus a constant. Rescaling coordinates, it has the expression(8.3) ˆ H = ∆ − | x | + 2 . This connection to the harmonic oscillator turns out to be about as mis-leading as it is helpful, for as we will see below, it is certainly not a naturalpoint of departure for our applications, because of the involved conjugationwith the Gaussian densities.We get however from (8.2) the following elementary lemma. The notation H s ( R ) refers to the Sobolev space of functions with s derivatives in L ( R ). Lemma 8.1.
Given G τ -equivariant E ∈ e | x | / L ( R ) and assuming τ ≤ ,there is a unique G τ -equivariant u ∈ e | x | / H ( R ) such that L P u = E . urthermore, there is a uniform constant C > such that (8.4) | u ( x ) | ≤ C (cid:16) sup x ∈ R (1 + | x | ) | E ( x ) | (cid:17) (1 + | x | ) , x ∈ R , for all E ∈ C ( R ) s.t. sup x ∈ R (1+ | x | ) | E ( x ) | < ∞ (hence E ∈ e | x | / L ( R ) ).The same statements hold if we replace R by Ω = R \ B R (0) and addthe condition u | ∂ Ω = 0 .Proof. Since the L -eigenvalues of ˆ H are λ ( n ,n ) ( ˆ H ) = n + n + 1, for n i ≥
0, and the well-known L -basis for ˆ H consists of Hermite functions, the e | x | / L ( R )-kernel of L P thus corresponds to the first excited eigenmodes,ker L P = span { x , x } , which thus disappears under the assumption of G τ -equivariance (given weinsert at least 2 τ − = 2 k ≥ L − P : L ( R ) → H ( R ), which by isometry invariance of L P preservesthe imposed symmetries.If we consider the disk B √ (0) = {| x | ≤ } , then if v ∈ H ( R ) satisfiesˆ Hv ≥ v ≤ ∂B √ (0), we conclude the simple maximum principleresult that v ≤ R \ B √ (0). This is standard, but we brieflysketch the proof. Namely, let w := max(0 , v ) so that(8.5) w ∆ v ≥ ( | x | − w ≥ v ∈ H ( R ) and w ∈ H ( R ),(8.6) − (cid:90) R \ B √ (0) |∇ w | ≥ , where we used w | ∂B √ (0) = 0. Thus w = 0 which proves the claim.We now take, for numbers A, B > v ( x , x ) := e −| x | / (cid:0) u − Ax + B x | x | (cid:1) . We consider a fundamental domain θ ∈ [ − π/k, π/k ] positioned inside thesupport of the test functions and compute:ˆ Hv = e −| x | / (cid:16) E + B x | x | (cid:17) ≥ e −| x | / (cid:18) B − (cid:12)(cid:12)(cid:12)(cid:12) | x | E cos( πk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) x | x | ≥ e −| x | / (cid:18) B − x ∈ R (1 + | x | ) | E ( x ) | (cid:19) x | x | , where we have used that k ≥
3, so that cos( π/k ) ≥ . Thus we get thatpicking B := 2 sup x ∈ R (1 + | x | ) | E ( x ) | ensures ˆ Hv ≥ A large depending linearly on B and on (cid:107) u | ∂B √ (cid:107) ∞ , e arrange v ≤ ∂B √ , and hence the resultfollows by the above maximum principle combined with the estimate (cid:107) u | ∂B √ (cid:107) ∞ ≤ C (cid:107) e −| x | / u (cid:107) H ( B (0)) ≤ C (cid:107) e −| x | / E ( x ) (cid:107) L ( R ) ≤ C (cid:32)(cid:90) R e −| x | / (1 + | x | ) dx (cid:33) / sup x ∈ R (1 + | x | ) | E ( x ) | , using the Sobolev inequality on the larger disk B (0). Hence the estimate(8.4) follows. The argument in the case with Ω instead of R is similar. (cid:3) However nice such simple lemmas may appear, the truth is that the spaces e | x | / L (Ω) are not well-suited for our geometric analysis purposes, in par-ticular they do not have any good compact embedding properties, becauseof the Gaussian (and linear) growth involved. What we would like is toseparate out the conical asymptotics and obtain sharp, uniform control inadequate weighted spaces, with second order derivative bounds, in such away that we can proceed with our geometric construction. To accomplishthis, we first introduce in the next section the appropriate new cone spaces.8.1. H¨older cone spaces for the exterior problem.
In this section wedefine the weighted H¨older spaces suitable for working with homogeneousfunctions. Note that these are different from the standard spaces consideredin Equation (3.1), although they could be naturally rephrased as such witha different metric (in fact the pull-back metric under the projection fromany fixed symmetric cone) on the plane.
Definition 8.2 (Homogeneously weighted H¨older spaces) . We define theappropriate weighted spaces of H¨older functions for decay rate k ∈ N , C ,α hom (Ω , | x | − k ) = { f ∈ C ,α loc (Ω) : (cid:107) f : C ,α hom (Ω , | x | − k ) (cid:107) < ∞} , with norms (cid:107) f : C ,α hom (Ω , | x | − k ) (cid:107) := [ f ] Ω , − k − α + sup x ∈ Ω | x | k | f ( x ) | , where the weighted H¨older coefficients of decay rate − k − α are defined as: [ f ] Ω ,α, − k − α := sup x,y ∈ Ω | x | − k − α + | y | − k − α | f ( x ) − f ( y ) || x − y | α . We then let: C ,α hom (Ω , | x | − ) := { f ∈ C ,α loc (Ω) : D β f ∈ C ,α hom (Ω , | x | − ) , | β | ≤ } , where β ranges over all multiindices, with norm given by (8.7) (cid:107) f : C ,α hom (Ω , | x | − ) (cid:107) := (cid:88) | β |≤ (cid:107) D β f : C ,α hom (Ω , | x | − ) (cid:107) . efinition 8.3. The anisotropically homogeneously weighted H¨older spacesare the following: C ,α an (Ω , | x | − ) := { f ∈ C ,α hom (Ω , | x | − ) : (cid:126)X · ∇ f ∈ C ,α hom (Ω , | x | − ) } , with norms (cid:107) f : C ,α an (Ω , | x | − ) (cid:107) := (cid:107) f : C ,α hom (Ω , | x | − ) (cid:107) + (cid:107) (cid:126)X · ∇ f : C ,α hom (Ω , | x | − ) (cid:107) . The definition of the homogeneously weighted spaces are motivated partlyby the following lemma. Note also that C ,α hom (Ω , | x | k ) ⊆ e | x | / L (Ω). Lemma 8.4.
Let h ( x ) = c ( x | x | ) | x | k be homogeneous of degree k ∈ Z , where c ∈ C ,α ( S ) , then ( ∇ ) l h ∈ C ,α hom (Ω , | x | k − l ) , l = 0 , , , with (cid:107) ( ∇ h ) l (cid:107) C ,α hom (Ω , | x | k − l ) ≤ (cid:107) c (cid:107) C l,α ( S ) . (8.8) When k = 1 , then we have the property L P h ∈ C ,α hom (Ω , | x | − ) . Furthermore L P : C ,α an (Ω , | x | − ) → C ,α hom (Ω , | x | − ) is a bounded operator.Proof. The first claim for homogeneous functions h is elementary from thedefinitions, using scaling.When k = 1, L P h = ∆ h − ( (cid:126)X · ∇ − h = ∆ h is a sum of homogeneousfunctions, namely one of degree − −
2, and the secondand third result also follow. (cid:3)
Definition 8.5.
The (anisotropic homogeneous) H¨older cone space of func-tions asymptotic to graphical cones over the plane, are: CS ,α (Ω , | x | − ) := C ,α hom (Ω , | x | − ) , (8.9) CS ,α (Ω , | x | − ) := C ,α ( ∂ Ω) × C ,α an (Ω , | x | − ) , (8.10) the latter equipped with the product norm (cid:107) ( c, f ) : CS ,α (Ω , | x | − ) (cid:107) := (cid:107) c (cid:107) C ,α ( S ) + (cid:107) f : C ,α an (Ω , | x | − ) (cid:107) . Remark 8.6. (i)
The pairs ( c, f ) injectively model graphs u : Ω → R as follows, (8.11) u = u ( c,f ) ( r, θ ) := c ( θ ) r + f ( r, θ ) , in polar coordinates, and by abuse of notation we write u = ( c, f ) . (ii) An important consequence in this context, is that our linearized oper-ator in (8.1) induces a well-defined bounded map ( c, f ) (cid:55)→ L P ( u ( c,f ) ) , (8.12) L P : CS ,α (Ω , | x | − ) → CS ,α (Ω , | x | − ) = C ,α hom (Ω , | x | − ) , as opposed to second order operators generally (e.g. ∆ + 1 ). roposition 8.7. The spaces C k,α hom (Ω , | x | − ) , C k,α (cid:48) an (Ω , | x | − ) and CS ,α (Ω , | x | − ) are Banach, and the natural inclusions for < α < α (cid:48) < , C k,α (cid:48) hom (Ω , | x | − ) (cid:44) → C k,α hom (Ω , | x | − l ) , (8.13) C k,α (cid:48) an (Ω , | x | − ) (cid:44) → C k,α an (Ω , | x | − l ) , (8.14) CS ,α (cid:48) (Ω , | x | − ) (cid:44) → CS ,α (Ω , | x | − l ) , (8.15) are compact, where l > is arbitrary and signifies a weakening of the decayrate (we will here only use | x | , so l = 1 ).Proof. It is a standard exercise to verify that these spaces are complete withthe norms we have defined.Since Ω is non-compact, it is for the compactness of the embeddings(8.13)-(8.15) to be true crucial that: (A) We have arranged that the weightfunctions on all derivatives are decaying, and (B) Cones are modeled byfunctions on a compact curve in S , here on ∂ Ω = S . Note that it is animportant special feature of the operator L P that the property (B) can bebrought into play (see the Liouville result in Proposition 8.9).Namely, for any bounded domain D ⊂⊂ R n the embeddings C k,α (cid:48) ( D ) (cid:44) → C k,α ( D ), of the usual H¨older spaces, are compact if 0 < α < α (cid:48) <
1, asfollows from the Arzel`a-Ascoli theorem. This fact along with a standardcut-off argument and the property (A) shows that the embeddings in (8.13)and (8.14) are compact.From the compactness of (8.14) and the property (B), i.e. compactnessof C k,α (cid:48) ( ∂ Ω) (cid:44) → C k,α ( ∂ Ω), it now finally follows that also(8.16) C ,α (cid:48) ( ∂ Ω) × C ,α (cid:48) an (Ω , | x | − ) (cid:44) → C ,α ( ∂ Ω) × C ,α an (Ω , | x | − l )is compact, completing the proof of (8.15). (cid:3) Homogeneously weighted H¨older estimates.
In this section weprove the second derivative Schauder estimates in the weighted H¨older spaces.Recall that we take Ω = R \ B R (0) to be a domain exterior to a disk. Proposition 8.8. If E ∈ C ,α hom (Ω , | x | − ) and v ∈ e | x | / H (Ω) ∩ C ,α loc (Ω) isa solution to L P v = ∆ v − ( (cid:126)X · ∇ − v = E , then D x i x j v ∈ C ,α hom (Ω , | x | − ) , and if v | ∂ Ω = 0 there is a constant C = C ( α ) > such that (8.17) (cid:107) D x i x j v (cid:107) C ,α hom (Ω , | x | − ) ≤ C (cid:107) E (cid:107) C ,α hom (Ω , | x | − ) . Proof.
There are several routes one may take to prove such a result, forexample the resolvents can be found in the form of contour integrals bysumming up the eigenfunctions via Mehler’s formula.However, using the well-known connection to parabolic equations (andwhence this problem came, of course) is less involved. Namely, the equation L P u = ∆ u − ( (cid:126)X · ∇ − u = E, s the elliptic equation describing a backwards self-similar solution to theflat space heat equation, but with a modified source term.It is convenient to consider a fixed extension map v (cid:55)→ ˜ v ∈ C ,α loc ( R ) withthe property(8.18) (cid:107) ˜ v (cid:107) C ,α ( B R (0)) ≤ C (cid:107) v (cid:107) C ,α ( B R +1 (0) \ B R (0)) , where the constant is independent of v . Then letting ˜ E = L P ˜ v we see that˜ E ∈ C ,α hom ( R , | x | − ) and (cid:107) ˜ E (cid:107) C ,α hom ( R , | x | − ) ≤ (cid:107) E (cid:107) C ,α hom (Ω , | x | − ) + C (cid:107) ˜ E (cid:107) C ,α ( B R (0)) ≤ (cid:107) E (cid:107) C ,α hom (Ω , | x | − ) + C (cid:107) v (cid:107) C ,α ( B R +1 (0) \ B R (0)) ≤ (cid:107) E (cid:107) C ,α hom (Ω , | x | − ) + C (cid:2) (cid:107) E (cid:107) C ,α ( B R +1 (0) \ B R (0)) + sup x ∈ Ω (1 + | x | ) | E ( x ) | (cid:3) ≤ C (cid:107) E (cid:107) C ,α hom (Ω , | x | − ) , where in the second to last estimate we used Schauder estimates (such asTheorem 10.2.1-10.2.2 in [Jo]), using the fact that v | ∂B R = 0 and the boundson v from the second part of Lemma 8.4. Now, since also automatically(8.19) (cid:107) D x i x j v (cid:107) C ,α hom (Ω , | x | − ) ≤ (cid:107) D x i x j ˜ v (cid:107) C ,α hom ( R , | x | − ) , we see that it is enough to prove the estimate (8.17) for ˜ v and ˜ E , so weassume without loss of generality that v and E are defined on R .The elliptic equation is now, as mentioned above, easily rewritten to thecondition(8.20) v ( x, t ) := √ − t u (cid:0) x √ − t (cid:1) solves the following heat equation(8.21) (cid:40) ∂ t v − ∆ v = F ( x, t ) , ( t, x ) ∈ (0 , × R ,v ( x,
0) = u ( x ) , x ∈ R , where the correspondingly transformed source term now reads:(8.22) F ( x, t ) := − E (cid:16) x √ − t (cid:17) √ − t . Now, recall the heat kernel in Euclidean space,Φ( x − y, t − s ) := 14 π ( t − s ) e − | x − y | t − s ) . Note that we have the following representation formula which allows us touse standard methods of proof (e.g. the standard, non-weighted Schauder heory for the heat equation. See for example [La]) D x i x j v ( x, t ) = (cid:90) R D x i x j Φ( x − y, t ) u ( y ) dy − (cid:90) t (cid:90) R D x i x j Φ( x − y, t − s ) (cid:104) F ( x, s ) − F ( y, s ) (cid:105) dyds, (8.23)where(8.24) D x i x j Φ( x − y, t − s ) = (cid:104) ( x i − y i )( x j − y j )4( t − s ) − δ ij t − s ) (cid:105) Φ( x − y, t − s ) , and we have subtracted a term which is zero. The expression is well-definedwhen F is H¨older in the x -variable, and justified by inserting a cut-off χ h ( t ),supported away from t = 1, then differentiating under the integral andfinally letting h → A, B > | D x i x j Φ( x − y, t − s ) | ≤ A ( t − s ) − e − B | x − y | t − s and note also that since E ∈ C ,α hom ( | x | − ) | F ( x, s ) − F ( y, s ) | ≤ (cid:107) E (cid:107) C ,α hom ( | x | − ) | x − y | α (cid:0) | x | − − α + | y | − − α (cid:1) , by the way we have defined C ,α hom ( | x | − ).Let us first prove that with E ∈ C ,α hom ( | x | − ), we have(8.26) sup x ∈ R \ B R (0) (1 + | x | ) | D x i x j u ( x ) | ≤ C (cid:107) E (cid:107) C ,α hom ( | x | − ) Note that by virtue of the scaling in the definition of v , it suffices for(8.26) to establish thatsup t ∈ ( t R , sup | x | =1 | D x i x j v ( x, t ) | ≤ C (cid:107) E (cid:107) C ,α hom ( | x | − ) , where t R := 1 − R . Let us fix R = 2, such that t R = . We see that for | x | = 1 we have from Equation (8.23) | D x i x j v ( x, t ) | ≤ C (cid:107) E (cid:107) C ,α hom ( | x | − ) (cid:90) R e − B (cid:48) | y | (1 + | y | ) dy + C (cid:48) (cid:107) E (cid:107) C ,α hom ( | x | − ) (cid:90) (cid:90) R | x − y | α (cid:0) | x | − − α + | y | − − α (cid:1) (1 − s ) − e − B | x − y | − s dyds = C (cid:48)(cid:48) (cid:107) E (cid:107) C ,α hom ( | x | − ) , where we used | u ( x − y ) | ≤ C (cid:107) E (cid:107) C ,α hom ( | x | − ) (1 + | y | ) as well as (8.25) toestimate the first term, and where of course the integral (cid:90) (cid:90) R | y | α (1 − s ) − e − B | y | − s dyds < ∞ , for any α > . gain, by the scaling in (8.20) and our definition of the weighted spaces,the desired estimate for the H¨older coefficients will follow if we can showthatsup t ∈ ( t R , sup | x |≤| x | =1 | D x i x j v ( x, t ) − D x i x j v ( x , t ) | ≤ C (cid:107) E (cid:107) C ,α hom ( | x | − ) | x − x | α . Hence we compute for | x | ≤ | x | ≤ D x i x j v ( x, t ) − D x i x j v ( x , t ) = (cid:90) R (cid:16) D x i x j Φ( x − y, t ) − D x i x j Φ( x − y, t ) (cid:17) u ( y √ ) dy + (cid:90) t (cid:90) | x − y |≤ | x − x | D x i x j Φ( x − y, t − s ) (cid:104) F ( y, s ) − F ( x, s ) (cid:105) dyds − (cid:90) t (cid:90) | x − y |≤ | x − x | D x i x j Φ( x − y, t − s ) (cid:104) F ( y, s ) − F ( x , s ) (cid:105) dyds + (cid:90) t (cid:90) | x − y |≥ | x − x | (cid:16) D x i x j Φ( x − y, t − s ) − D x i x j Φ( x − y, t − s ) (cid:17)(cid:104) F ( y, s ) − F ( x, s ) (cid:105) dyds − (cid:90) t (cid:90) | x − y |≥ | x − x | D x i x j Φ( x − y, t − s ) (cid:104) F ( x, s ) − F ( x , s ) (cid:105) dyds =: I + . . . + I . (8.27)In this expression, the first term is estimated using the mean value principle,such that for | D x i x j Φ( x − y, t ) − D x i x j Φ( x − y, t ) | ≤ | y + ξ | | x − x | t Φ( ξ + y, t ) ≤ C (cid:48) e − B (cid:48) | y + ξ | t | x − x | α , for some point ξ on the line between the points x and x , so | ξ | ≤
2, andsome constants B (cid:48) , C (cid:48) = C ( t R ) independent of | x | , | x | ≤
1. Hence one getsthe estimate(8.28) | I | ≤ C | x − x | α (cid:107) E (cid:107) C ,α hom ( | x | − ) (cid:90) R e − B (cid:48) | y | t ( | y | + 2) dy. The terms I and I are of course symmetric in x ↔ x and have similarestimates. For I we get: | I | ≤ C (cid:107) E (cid:107) C ,α hom ( | x | − ) (cid:90) t (cid:90) | x − y |≤ | x − x | ( t − s ) − e − B | x − y | t − s | x − y | α dyds ≤ C (cid:107) E (cid:107) C ,α hom ( | x | − ) (cid:90) | x − y |≤ | x − x | | x − y | − α ≤ C (cid:107) E (cid:107) C ,α hom ( | x | − ) | x − x | α . (8.29) or the term I we use the estimate | D x i x j Φ( x − y, t ) − D x i x j Φ( x − y, t ) | ≤ c | x − x | ( t − s ) − / e − B | x − y | t − s , which holds whenever | x − y | ≥ | x − x | . Hence we see | I | ≤ C (cid:107) E (cid:107) C ,α hom ( | x | − ) | x − x | (cid:90) t (cid:90) | x − y |≥ | x − x | ( t − s ) − / e − B | x − y | t − s | x − y | α dyds ≤ C (cid:107) E (cid:107) C ,α hom ( | x | − ) | x − x | α . (8.30)For the last term, we rewrite it as I = − (cid:90) t (cid:90) | x − y | =2 | x − x | ∂ Φ( x − y, t − s ) ∂x j (cid:104) F ( x, s ) − F ( x , s ) (cid:105) ( (cid:126)e i · (cid:126)ν ) dM ( y ) ds, where (cid:126)e i the i th unit vector in R , dM ( y ) is the line element and (cid:126)ν theoutward pointing unit normal to the disk of radius 2 | x − x | . Since(8.31) ∂ Φ( x − y, t − s ) ∂x j = − x j − y j π ( t − s ) e − | x − y | t − s ) , we finally get | I | ≤ (cid:107) E (cid:107) C ,α hom ( | x | − ) | x − x | α (cid:90) t (cid:90) | x − y | =2 | x − x | e −| x − x | t − s ) π ( t − s ) dM ( y ) ds = C (cid:107) E (cid:107) C ,α hom ( | x | − ) | x − x | α . (cid:3) Using the second derivative bound we can now proceed to our final propo-sition of this section, which is a Liouville-type structure theorem in that weprove solutions are homogeneous degree one polynomials in x plus a re-mainder belonging to the space C ,α an ( | x | − ). This detailed analysis of thesolutions – completing our separation of the conical part – is exactly whatwill make our construction work. Theorem 8.9 (Liouville-type result) . There is a constant
C > s.t. forany G τ -equivariant E ∈ C ,α hom (Ω , | x | − ) there exists a unique G τ -equivariant u = ( c, f ) ∈ CS ,α (Ω , | x | − ) such that L P (cid:2) c ( θ ) · | x | + f ( x ) (cid:3) = L P u = E, where u = u ( c,f ) and u = 0 on ∂ Ω . Furthermore we have the estimate (8.32) (cid:107) ( c, f ) : CS ,α (Ω , | x | − ) (cid:107) ≤ C (cid:107) E : C ,α hom (Ω , | x | − ) (cid:107) . Proof.
Let u ∈ C ,α loc (Ω) be a solution to L u = E . It follows from theweighted H¨older estimates in Proposition 8.8 that(8.33) D x i x j u ∈ C ,α hom (Ω , | x | − ) , nd hence we have ∆ u ∈ C ,α hom (Ω , | x | − ) and hence w := − (cid:126)X · ∇ u + u = E − ∆ u ∈ C ,α hom (Ω , | x | − ). Solving for u in polar coordinates ( | x | = r ), we getafter imposing initial conditions u | ∂ Ω = 0 that (normalize here for simplicitythe radius R of ∂ Ω to 1): u ( r, θ ) = c ( θ ) r + v , (8.34) c ( θ ) := − (cid:90) ∞ w ( s, θ ) s ds = lim r →∞ u ( r, θ ) r , (8.35) v ( x ) := r (cid:90) ∞ r w ( s, θ ) s ds. (8.36)By (8.33) and (8.34), and the lemma for homogeneous functions (8.8), wesee that also D x i x j v ∈ C ,α hom (Ω , | x | − ).Now, − (cid:126)X · ∇ v + v = − (cid:126)X · ∇ u + u = w ∈ C ,α hom (Ω , | x | − ) from above. Itfollows easily from the formula (8.36) for v that v ∈ C ,α hom (Ω , | x | − ), andhence we see that also (cid:126)X · ∇ v ∈ C ,α hom (Ω , | x | − ).It remains to show that the full gradient satisfies(8.37) ∇ v ∈ C ,α hom (Ω , | x | − ) , Note for this, that(8.38) ∆ v = E − ∆( c ( θ ) | x | ) + (cid:126)X · ∇ v − v ∈ C ,α hom (Ω , | x | − ) . Equation (8.37) follows now easily from this with v ∈ C ,α hom (Ω , | x | − ), bystandard use of the Green’s function for the ordinary flat Laplacian on theplane (see for example the estimate (10.1.30) in [Jo]).Hence we have shown that there is the desired Liouville decomposition,and the corresponding estimates follow. (cid:3) Linearized equation on the initial surface M [ τ, θ ]We let a := 8 | log τ | and then N ± y , N ± x are used to denote the connectedcomponents of {M [ τ, θ ] : s ≥ a } . Let also S := H (Σ), where we denote by H the homothety by a factor of τ . Definition 9.1.
Let v ∈ C k,α loc ( M ) . We identify v with its restrictions to Σ , N ± y and N ± x . Then for k = 0 , we define the norm (cid:107) v (cid:107) X S k,α to be the max-imum of the following quantities, where b = e − δ s /τ and b = e − δ s /τ /τ : (1) τ − k (cid:107) v ◦ H(cid:107) C k,α (Σ ,e − γs ,g Σ ) , and (2) b − k (cid:107) v (cid:107) CS k,α ( N + x \S , | x | − ) , as given in Definition 8.5. (3) b − k (cid:107) v (cid:107) C k,α ( N ± y \S ,g N± y ) , and b − k (cid:107) v (cid:107) C k,α ( N − x \S ,g N± y ) .We let be X S k,α ( M ) be the space of functions v for which (cid:107) v (cid:107) X S k,α < ∞ . Lemma 9.2.
Let N i stand for any of the ends N ± y , N ± x . Then for τ > sufficiently small the Dirichlet operator, for zero initial value on ∂ N i , L N i : X S ,α ( N i ) → X S ,α ( N i ) s invertible, with operator norm of the inverse bounded uniformly in τ > .Proof. For the exterior flat domain, this is what was proved in Section 8. Forthe flat disk and round spherical cap, we check the invertibility by computingthe Dirichlet spectrum of the stability operator L on these surfaces, using aperturbation argument to extend the property to the θ -family of sphericalcaps (by possibly taking δ θ smaller). These spectrum computations can befound in the Appendix.Note however that we are considering the region of τ Σ[ τ, θ ], very nearthe removed circle, and here the initial surface M [ τ, θ ] and hence the ends N , do not exactly coincide with the subsets of the configuration C [ θ ]. Thedifference is on each piece a small normal graph with compact support,coming from the function f ( s, z ) describing Scherk’s surface as a graph overits four asymptotic planes. But by construction and the estimates (5.3) weverify that the cut-off a = 8 log τ is appropriately large, since for the twoinduced metrics in question, (cid:107) g N i − g C [ θ ] : C ( { τ Σ[ τ, θ ] : s ≥ a } , g C [ θ ] ) (cid:107) ≤ Cτ − e − a = Cτ , and similarly for the induced second fundamental forms | A | , and hence thelemma follows for small enough τ > { τ Σ[ τ, θ ] : s ≥ a } , for the quantities used in the definition of L . (cid:3) Note that the property (8.12) extends so that also L M , the linearizedoperator of H − (cid:104) (cid:126)X, (cid:126)ν (cid:105) over M , is a bounded map from the H¨older conespace. Definition 9.3.
Let
Θ : [ − δ θ , δ θ ] → C ∞ ( M ) be given by Θ( θ ) = 1 τ H ∗ ( θw ) , where θ ∈ [ − δ θ , δ θ ] . Theorem 9.4.
Given E ∈ X S ,α ( M ) , there exist v E ∈ X S ,α ( M ) and θ E ∈ R , such that L M v E = E + Θ( θ E ) , and (cid:107) v E (cid:107) X S ,α ≤ C (cid:107) E (cid:107) X S ,α , | θ E | ≤ C (cid:107) E (cid:107) X S ,α , Proof.
Let the cut-off functions ψ := ψ [5 δ s /τ, δ s /τ − ◦ s as well as ψ (cid:48) := ψ [ a, a + 1] ◦ s be given on M , and let a = 8 | log τ | .The starting point of our iteration is E := E . Applying Proposition 7.3to Σ = Σ[ τ, θ ] = H − ( S ) with the cut-off source term E (cid:48) := τ ( ψE n − ) ◦ H .From the corresponding v E we get v := τ H ∗ ( v E ) and we let the θ n := θ E (cid:48) .By construction we have thus on S that L M v = ψE n − + Θ( θ n ) . We now feed the new source term E (cid:48)(cid:48) = (1 − ψ ) E n − − [ L M , ψ ] v intothe equation on the union of the ends N ∪ := N y ∪ N − x ∪ N + x (here the ommutator is by definition [ L M , ψ ] f := L M ( ψ ) f − ψ ( L M f )), and obtaina solution v E (cid:48)(cid:48) which we call v (cid:48) , L M v (cid:48) = (1 − ψ ) E n − − [ L M , ψ ] v. We then finally define v n := ψv + ψ (cid:48) v (cid:48) . By considering the supports of ψ, ψ (cid:48) and [ L M , ψ ], we see that(9.1) L M v n = E n − + [ L M , ψ (cid:48) ] v (cid:48) + Θ( θ n ) . We then also define the new source term E n = − [ L M , ψ (cid:48) ] v (cid:48) . Again, we usethe fact that [ L M , ψ (cid:48) ] is supported on [ a, a +1], use Lemma 9.2, and estimate(for τ sufficiently small), (cid:107) E n (cid:107) X S ,α = τ (cid:107) E n ◦ H : C ,α (Σ , g Σ , e − γs ) (cid:107)≤ τ e γ ( a +1) (cid:107) [ L M , ψ (cid:48) ] v (cid:48) ◦ H : C ,α (Σ [ a,a +1] , g Σ ) (cid:107)≤ Cτ − p e γ ( a +1) (cid:107) v (cid:48) ◦ H : C ,α (Σ [ a,a +1] , g Σ ) (cid:107)≤ Cτ − p (cid:48) e γ ( a +1) e − ( δsτ − ) (cid:107) (1 − ψ ) E n − − [ L M , ψ ] v (cid:107) X S ,α , where we used in the third line the uniform control of the geometry of Σ inthe strips s ∈ [ a, a + 1], and in the third line the Definition 9.1, and the factthat the term considered in the last line has support in s ∈ [ δ s τ − , δ s τ ], wethus get (cid:107) E n (cid:107) X S ,α ≤ Cτ − p (cid:48) e γ ( a +1) e − ( δsτ − ) (cid:107) E n − (cid:107) X S ,α ≤ Ce − δsτ (cid:107) E n − (cid:107) X S ,α . We define v E := (cid:80) ∞ n =1 v n and θ E := (cid:80) ∞ n =1 θ n . The first sum convergesin the Banach space X S ,α ( M ), the second converges to some real numberwhich is the θ E , with the desired estimates. The function v E then satisfies L M v E = E + Θ( θ E ). (cid:3) Definition 9.5.
Let S be a smooth surface (possibly with boundary). For afunction v ∈ C ,α ( S ) for which S v is a C ,α -surface, we define on S : F S ( v ) := H S v − (cid:104) (cid:126)X, (cid:126)ν S v (cid:105) , and denote F S := F S (0) . Corollary 9.6.
There are v F ∈ X S ,α ( M ) and θ F such that L M v F = F M + Θ( θ F ) w, | θ F − θ | ≤ Cτ, (cid:107) v F (cid:107) X S ,α ≤ Cτ, where M = M [ τ, θ ] . The nonlinear terms in H¨older cone spaces
Proposition 10.1.
Given v ∈ X S ,α ( M ) with (cid:107) v (cid:107) X S ,α smaller than a suit-able constant, we have that the graph M v over M , is a smooth immersionand moreover F M ( v ) − F M − L M v ∈ X S ,α ( M ) , with the quadratic improvement bounds: (10.1) (cid:107)F M ( v ) − F M − L M v (cid:107) X S ,α ≤ C (cid:107) v (cid:107) X S ,α . Proof.
We first deal with the argument needed on the exterior plane Ω = R \ B R (0). Note that (cid:126)ν ≡ (cid:126)e and the terms for the equation (4.1) read(10.2) F Ω ( v ) = − Hess v ( ∇ v, ∇ v )(1 + |∇ v | ) / + L P v (cid:112) |∇ v | , where L P is again the linearized operator from (8.1).Thus we see that for the exterior plane Ω we have F Ω ( v ) − F Ω − L Ω v = − Hess v ( ∇ v, ∇ v )(1 + |∇ v | ) / + (cid:104) (1 + |∇ v | ) − / − (cid:105) L P v =: T + T . Let us first estimate the weighted sup-norm. By the Bernoulli inequalitieswe have the quadratic bounds: (cid:12)(cid:12)(cid:12) (1 + |∇ v | ) − / − (cid:12)(cid:12)(cid:12) ≤ |∇ v | , (cid:12)(cid:12)(cid:12) (1 + |∇ v | ) − / − (cid:12)(cid:12)(cid:12) ≤ |∇ v | . (10.3)We can now estimate the supremum part of the weighted norms: | x | | ( F Ω ( v ) − F Ω (0) − L Ω v )( x ) | ≤ | x || Hess v ||∇ v | + | x ||∇ v | |L P v |≤ (cid:107) v (cid:107) CS ,α + (cid:107) v (cid:107) CS ,α = (cid:107) v (cid:107) CS ,α , on the exterior of the disk, where we used again the crucial mapping property(8.12) on the H¨older cone spaces.Similar but slightly more involved computations now show that the H¨oldercoefficients in the norm are also estimated as claimed. For example it followsby (10.3) that (cid:12)(cid:12) (1 + |∇ v ( x ) | ) − / − (1 + |∇ v ( y ) | ) − / (cid:12)(cid:12) | x − y | α ≤ |∇ v ( x ) | + |∇ v ( y ) | |∇ v ( y ) | ) / (cid:12)(cid:12) ∇ v ( x ) − ∇ v ( y ) (cid:12)(cid:12) | x − y | α ≤ (cid:2) | x | − α + | y | − α (cid:3) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) . It follows easily that(10.4) T := (cid:34) (cid:112) |∇ v | − (cid:35) L P v ∈ C ,α hom (Ω , | x | − ) , since by assumption L P v ∈ C ,α hom (Ω , | x | − ) and also ∇ v ∈ C ,α hom (Ω , | x | ). Weget the corresponding higher order bounds as follows. Assume without loss f generality that | x | ≥ | y | , which is reflected in how we distribute terms,and recall the estimates (8.8) in Lemma 8.4: | T ( x ) − T ( y ) || x − y | α ≤ (cid:12)(cid:12) (1 + |∇ v ( x ) | ) − / − (1 + |∇ v ( y ) | ) − / (cid:12)(cid:12) | x − y | α | ( L P v )( x ) | + |∇ v | | ( L P v )( x ) − ( L P v )( y ) || x − y | α ≤ (cid:2) | x | − α + | y | − α (cid:3) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) (cid:107)L P v (cid:107) C ,α hom (Ω , | x | − ) | x | + (cid:2) | x | − − α + | y | − − α (cid:3) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) (cid:107)L P v (cid:107) C ,α hom (Ω , | x | − ) ≤ (cid:2) | x | − − α + | y | − − α (cid:3) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) . As for the term T , we write Hess v ( ∇ v, ∇ v ) = (cid:80) i,j ( D x i x j v )( D x i v )( D x j v ).We again have an estimate (cid:12)(cid:12) (1 + |∇ v ( x ) | ) − / − (1 + |∇ v ( y ) | ) − / (cid:12)(cid:12) | x − y | α ≤ |∇ v ( x ) | + |∇ v ( y ) | (1 + |∇ v ( y ) | ) / (cid:12)(cid:12) ∇ v ( x ) − ∇ v ( y ) (cid:12)(cid:12) | x − y | α ≤ (cid:2) | x | − α + | y | − α (cid:3) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) . We find by the above, since we may again assume | x | ≥ | y | , | T ( x ) − T ( y ) || x − y | α ≤ | ( D x i x j v )( x ) − ( D x i x j v )( y ) || x − y | α |∇ v ( y ) | (1 + |∇ v ( y ) | ) / + | ( D x i x j v )( x ) | |∇ v ( x ) − ∇ v ( y ) || x − y | α |∇ v ( y ) | (1 + |∇ v ( y ) | ) / + | ( D x i x j v )( x ) | |∇ v ( x ) − ∇ v ( y ) || x − y | α |∇ v ( x ) | (1 + |∇ v ( y ) | ) / + | ( D x i x j v )( x ) ||∇ v ( x ) | (cid:12)(cid:12) (1 + |∇ v ( x ) | ) − / − (1 + |∇ v ( y ) | ) − / (cid:12)(cid:12) | x − y | α ≤ (cid:2) | x | − − α + | y | − − α (cid:3) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) + 2 | x | − (cid:107) v (cid:107) CS ,α (Ω , | x | − ) (cid:2) | x | − α + | y | − α (cid:3) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) + 3 | x | − (cid:107) v (cid:107) CS ,α (Ω , | x | − ) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) (cid:2) | x | − α + | y | − α (cid:3) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) ≤ (cid:2) | x | − − α + | y | − − α (cid:3) (cid:16) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) + (cid:107) v (cid:107) CS ,α (Ω , | x | − ) (cid:17) . Collecting these estimates, we have shown: (cid:107)F Ω ( v ) −F Ω (0) −L Ω v : C ,α hom (Ω , | x | − ) (cid:107) ≤ C (cid:0) (cid:107) v (cid:107) CS ,α (Ω , | x | − ) + (cid:107) v (cid:107) CS ,α (Ω , | x | − ) (cid:1) . Picking now τ > δ s to ensure b > b >
1) in the Definition 9.1 of
X S ,α ( M ), we see that taking (cid:107) v (cid:107) X S ,α ( M ) ≤
1, we finally obtain:(10.5) (cid:107)F Ω ( v ) − F Ω (0) − L Ω v : X S ,α ( M ) (cid:107) ≤ C (cid:107) v : X S ,α ( M ) (cid:107) . or the core piece Σ[ θ, τ ], the argument follows closely the one in [Ka97].Namely, using the uniform control on the geometry (cid:107) A : C (Σ , g Σ ) (cid:107) ≤ C and (cid:107) τ (cid:126)X · (cid:126)ν : C (Σ) (cid:107) ≤ τ with the expression for the quadratic term inEquation (11.12), one obtains again that when (cid:107) f : C ,α (Σ , g Σ , e − γs + b ) (cid:107) is small enough, (cid:107)F Σ ( f ) − F Σ − τ L Σ f : C ,α (Σ , g Σ , e − γs ) (cid:107)≤ C (cid:107) f : C ,α (Σ , g Σ , e − γs + b ) (cid:107) . For the central disk and the top and bottom spherical caps the proofs areagain the same, by uniform control of the geometry and (11.12). (cid:3)
Fixed point argument: Existence of the self-shrinkers
We consider for any fixed 0 < α (cid:48) < α <
X S ,α (cid:48) := X S ,α (cid:48) ( M [ τ, , from the family that we previously studied, and take the subsetsΞ = { ( θ, u ) ∈ [ − δ θ , δ θ ] × X S ,α (cid:48) : | θ | ≤ ζτ, (cid:107) u (cid:107) X S ,α ≤ ζτ } . We state the following lemma (as in [Ka97]), whose easy proof we omit.
Lemma 11.1.
There is for θ ∈ [ − δ θ , δ θ ] a smooth family of diffeomorphisms D θ : M [ τ, → M [ τ, θ ] , with (11.1) (cid:107) f ◦ D − θ (cid:107) X S ,α ≤ C (cid:107) f (cid:107) X S ,α , (cid:107) f ◦ D θ (cid:107) X S ,α ≤ C (cid:107) f (cid:107) X S ,α , for all f ∈ C ,α ( M [ τ, and f ∈ C ,α ( M [ τ, θ ]) . The problem stated in Theorem 9.4 is then continuous in τ and θ in thesense that, for fixed E ∈ X S ,α ( M [ τ, v E ◦ D − θ ◦ D θ , θ E ◦ D − θ ) ∈ X S ,α ( M [ τ, × R depends continuously on τ and θ .We define the map J : Ξ → [ − δ θ , δ θ ] × X S ,α (cid:48) as follows: Let ( θ, u ) ∈ Ξand let v := u ◦ D − θ − v F , where v F comes from an application of Corollary9.6, and the function F = F M [ τ,θ ] (0) as before is defined on M [ τ, θ ]. Wethus have (cid:107) v (cid:107) X S ,α ≤ C ( ζ + 1) τ. Now, we use Proposition 10.1 to get that M v is well-defined, and (cid:107)F M ( v ) − F M − L M v (cid:107) ≤ C ( ζ + 1) τ . Inserting therefore E = F M ( v ) − F M − L M v into Theorem 9.4 gives a v E and θ E . We obtain, for some appropriate constant C that: F M ( v ) = L M (cid:0) u ◦ D − θ + v E (cid:1) − Θ( θ F + θ E ) , (11.2) | θ − θ F − θ E | ≤ C (cid:0) τ + ( ζ + 1) τ ) , (11.3) (cid:107) v E (cid:107) X S ,α ≤ C ( ζ + 1) τ . (11.4) hen the definition of J is taken to be(11.5) J ( θ, u ) = (cid:0) θ − θ F − θ E , − v E ◦ D θ (cid:1) . Thus, by assuming ζ large enough and τ > J (Ξ) ⊆ Ξ. By the properties of our weighted spaces in Proposition 8.7and α (cid:48) < α , Ξ is a compact subset of [ − δ θ , δ θ ] × X S ,α (cid:48) and is also convex.The map J is continuous by Definition 11.1 and Proposition 9.4. Finally, bySchauder’s fixed point theorem, we get existence of the desired fixed point( θ ∗ , u ∗ ) ∈ Ξ, and we see that the corresponding M [ τ, θ ∗ ] u ∗ is an immersedself-shrinker. The rest of the proof of the main theorem now follows easily.For example, embeddedness is assured by our setup: By construction,there is some fixed ball B R such that the end of every Σ g is graphicaloutside that ball, and hence embedded. Now, above one could pick ζ =2 C independent of τ , one concludes for all τ > (cid:107) u ∗ (cid:107) X S ,α ≤ Cτ , and since also by construction the normalinjectivity radius of a compact piece the initial surface, say B R +1 (0), canbe assumed bounded below as inj ⊥ ( M [ τ, θ ] ∩ B R +1 (0)) ≥ cτ , for some c > τ >
0) the constructed surfaces Σ g are embedded. The Hausdorff convergence statement (v) in Theorem 1.1also follows immediately from the definitions of the norms.It also follows easily that each surface Σ g is geodesically complete. Namely,a curve that leaves every compact set must have infinite length, as followsby projecting it onto the plane and estimating the arc length from below,again since the ends are graphical outside some ball. ppendix A: The building blocks of the initial surfaces The proof of Proposition 5.5 follows from the following lemma.
Lemma 11.2.
There exist δ, ε , ε > such that there is a smooth map h (cid:55)→ ρ h for h ∈ (2 − δ, δ ) such that • For each h , the function ρ h is a generates a curve contained in theset { ( x, , z ) : x, z ≥ } . • ρ ( ϕ ) ≡ , with ϕ = arctan( x/z ) , and ρ h (0) = 2 + h, ρ (cid:48) h (0) = 0 . • The following are orientation-reversing diffeomorphisms: (1) h (cid:55)→ ρ h (0) : (2 − δ, δ ) → (2 − ε , ε ) , (2) h (cid:55)→ ρ (cid:48) h ( π/
2) : (2 − δ, δ ) → ( − ε , ε ) . • The graph ( ρ h ( ϕ ) , ϕ ) , ϕ ∈ (0 , π/ , in the xz -plane gives by revolu-tion a self-shrinker.Proof of Lemma 11.2. A curve ( ρ, ϕ ) , ϕ = arctan( x/z ) in the xz -plane gen-erated by a function ρ ( ϕ ) that generates a smooth solution to the self-shrinker equation (4.1) satisfies:(11.6) ρ (cid:48)(cid:48) ( ϕ ) = 1 ρ (cid:26) ρ + 2( ρ (cid:48) ) + (cid:104) − ρ − ρ (cid:48) ρ tan ϕ (cid:105)(cid:0) ρ + ( ρ (cid:48) ) (cid:1)(cid:27) . The Taylor-expansion in the Banach space of C functions of the solutionin the h -parameter is ρ h ( ϕ ) = 2 + ( h − w ( ϕ ) + ( h − w ( ϕ ) + O (cid:0) ( h − (cid:1) , where w i are smooth functions. The w i satisfy the conditions w (0) = 1 and w (cid:48) (0) = 0 (and w (0) = w (cid:48) (0) = 0 and similarly for higher corrections), andas is easily computed w satisfies the linear equation(11.7) w (cid:48)(cid:48) + 1tan ϕ w (cid:48) + 4 w = 0 , while w satisfies a linear equation where the w enters into the coefficients.The claims (1) and (2) follow from the following two properties w ( π ) < , (11.8) w (cid:48) ( π ) < , (11.9)for the solution to (11.7) having w (0) = 1 and w (cid:48) (0) = 0.In fact since if we subsitute x = cos( ϕ ) in the equation (11.7) to obtainLegendre’s differential equation, the explicit general solution to this initialvalue problem is of course well-understood, namely w ( ϕ ) = C P l (cos ϕ ) + C Q l (cos ϕ ) , where P l and Q l are respectively the Legendre functions of the first andsecond kind, and l = ( √ − / l ( l + 1) = 4.Here we see C = 0, since Q l (cos ϕ ) has a pole at ϕ = 0, and C = 1 since P l (1) = 1. Thus the properties are easily verified and the lemma follows. (cid:3) roof of Proposition 5.5. Given ρ h constructed above, set θ = tan { ρ (cid:48) h ( π/ } .We then take K θ to be the surface immersed by the map κ θ given by κ θ ( s, z ) = r ( ϕ ( s ))(cos z, sin z,
0) + (0 , , z ( ϕ ( s )))where r ( ϕ ) = ρ h ( ϕ ) sin( ϕ ), z ( ϕ ) = ρ h cos( ϕ ), and the map s (cid:55)→ ϕ ( s ) satis-fying s ϕ = (cid:113) r ϕ + x ,ϕ r ( ϕ ) , s ( π/
2) = 0 . That (0)-(3) are satisfied by the family K θ are clear by construction. Like-wise, once it is checked that s ( ϕ ) is a conformal parameter, (4)i - iv are easyto verify. (cid:3) Appendix B: Variation formulae
Let (cid:126)X : M → R be a C -immersion of a surface. We denote by (cid:126)X u : M → R the surface (cid:126)X u = (cid:126)X + u(cid:126)ν . Then denoting by H u and (cid:126)ν u etc. thequantities for (cid:126)X u , we get (see [Ka97] and [Ng1]) (cid:126)ν u = (cid:126)ν − ∇ u + (cid:126)Q νu , (11.10) H u = H − (∆ u + | A | u ) + Q u , (11.11) H u − τ (cid:126)X u · (cid:126)ν u = H − τ (cid:126)X · (cid:126)ν (11.12) − (cid:2) ∆ u + | A | u − τ ( (cid:126)X · ∇ u − u ) (cid:3) + Q u + τ (cid:126)X · (cid:126)Q u , where the quantities Q u and (cid:126)Q u are quadratic. Appendix C: Stability operators
Let M (cid:44) → N := ( R , h = e ω h ) be an immersion into a conformallychanged Euclidean space, where ω : N → R . Here h will denote thestandard metric h = δ ij . Denote by g and g the metrics induced on M from respectively h and g by the immersion.Here we have the conventions:∆ f = div( ∇ f ) = tr( ∇ i ∂ j f ) ,R ( X, Y ) Z = ∇ Y ∇ X Z − ∇ X ∇ Y Z + ∇ [ X,Y ] Z Ric(
X, Y ) = tr( Z (cid:55)→ R ( X, Z ) Y ) , so that the Ricci curvature of the standard round sphere is positive.Then we have the following Lemma. Lemma 11.3.
Assume M is an oriented minimal surface in N . Then thestability operator of M is the operator on functions on M given by: L g = ∆ g + | A h | g + Ric h ( (cid:126)ν, (cid:126)ν )= e − ω (cid:104) ∆ g + e − ω | A h | g − Hess h ( (cid:126)ν , (cid:126)ν ) ω + ( (cid:126)ν .ω ) − ∆ h ω − (cid:107)∇ h ω (cid:107) h (cid:105) , here (cid:126)ν is the unit normal vector w.r.t the metric h . From this formula, we get the stability operators:
Proposition 11.4. (i)
The stability operator of the sphere S of radius in R as a minimalsurface in the metric g = e −| x | δ is: (11.13) L = e (cid:16) ∆ S + 1 (cid:17) . In particular ker( L ) = { } on S , as well as on the hemispheres ofradius 2 with Dirichlet boundary conditions. (ii) The stability operator on a flat plane through the origin is (11.14) L = e | x | (cid:16) ∆ R − | x |
16 + 1 (cid:17) , where ∆ R is the usual flat Laplacian in ( R , δ ij ) . In particular, onboth the disk of radius √ , and of radius 2, ker( L ) = { } when weimpose Dirichlet boundary conditions.Proof. Recall that by definition A ( X, Y ) = ¯ ∇ ¯ X ¯ Y − ∇ X Y , where ¯ · means asmooth extension to a neighborhood in N , and | A | = g ij g kl a ik a jl = e − ω g ij g kl a ik a jl . We also recall the conformal changes of the Levi-Civitas: ∇ h ¯ X ¯ Y = ∇ h ¯ X ¯ Y + ( ¯ X.ω ) Y + ( ¯ Y .ω ) ¯ X − h ( ¯ X, ¯ Y ) ∇ h ω (11.15) ∇ gX Y = ∇ g X Y + ( X.ω ) Y + ( Y.ω ) X − g ( X, Y ) ∇ g ω. (11.16)This gives that(11.17) A h ( X, Y ) = A h ( X, Y ) − g ( X, Y ) (cid:8) ∇ h ω − ∇ g ω (cid:9) The Ricci curvature changes in dimension n = 3 when h = δ accordingto:Ric h = Ric h − ( n − (cid:104) ∇ h d ω − d ω ⊗ d ω (cid:105) + (cid:104) − ∆ h ω − ( n − (cid:107)∇ h ω (cid:107) h (cid:105) h , = − Hess h ω + d ω ⊗ d ω − (cid:104) ∆ h ω + (cid:107)∇ h ω (cid:107) h (cid:105) h . Here we have used that for h = δ we have ∇ h d ω = Hess ω, and that (cid:126)ν = e − ω ν is the new unit normal.Recall also that in 2 dimensions the Laplacian is conformally covariant:(11.18) ∆ g = e − ω ∆ g . Using these formulae, the Lemma follows. o prove the proposition, we need ω = − | x | . Thus we have ∇ h ω = − x, Hess ω ( ∂ i , ∂ j ) = − δ ij , ∆ ω = − , (cid:107)∇ R ω (cid:107) R = 116 | x | . And we get on R that(Ric h ) ij = − Hess h ω ( ∂ i , ∂ j ) + ( ∂ i ω )( ∂ j ω ) − (cid:104) ∆ R ω + (cid:107)∇ R ω (cid:107) (cid:105) δ ij = 14 δ ij + 116 x i x j + 34 δ ij − | x | δ ij = δ ij + 116 x i x j − | x | δ ij . Thus we get(11.19) Ric h ( (cid:126)ν, (cid:126)ν ) = e − ω (cid:20) | x · ν | − | x | (cid:21) , so that on the round sphere of radius 2,(11.20) Ric h ( (cid:126)ν, (cid:126)ν ) = e − ω . Now, we pull back the induced metric g on S of radius 2 by the mapΦ( x ) = 2 x taking S → S to get the isometry ( S , Φ ∗ g ) (cid:39) ( S , g ). Thennote that for X, Y ∈ T S we have Φ ∗ g ( X, Y ) = g R ( d Φ( X ) , d Φ( Y )) =4 g R ( X, Y ) = 4 g . Thus by the covariance in Equation (11.18), the spectrumof the operator L is the same as that of ∆ S + 4 on the sphere of radius 1.Now, since the eigenvalues of ∆ on the unit sphere S = S are λ k = − k ( k + 1) , we see that ∆ S + 4 is invertible on the sphere. The eigenvalues for theDirichlet problem for ∆ on the hemispheres are the same, but with smallermultiplicity (and in particular 0 is not an eigenvalue). Thus ∆ S + 4 is alsoinvertible there.Considering the plane { z = 0 } , one gets similarly A = 0, and(11.21) Ric h ( (cid:126)ν, (cid:126)ν ) = e − ω (cid:16) − | x | (cid:17) . Recall that for the Dirichlet problem for the harmonic oscillator on theunit disk, we have that λ k = − k , where k = 1 , , , . . . are the integers.Thus λ k = − k on the disk of radius √
2, while λ k = − k on the diskof radius 2. Thus in either case the corresponding stability operator L isinvertible. (cid:3) Appendix D: Laplacians on flat cylinders
We here recall a simple analytical result on (Ω , g ) the flat cylinder Ω = H + ≤ l /G equipped with the standard metric g = ds + dz , where G is thegroup generated by ( s, z ) → ( s, z + 2 π ), and l ∈ (10 , ∞ ) is called the length f the cylinder. We have ∂ Ω = ∂ ∪ ∂ l where ∂ and ∂ l are the boundarycircles { s = 0 } and { s = l } respectively.Let L on the flat cylinder (Ω , g ) be given by(11.22) L v = ∆ χ v + A · ∇ v + B · v, where χ is a C Riemannian metric, A ∈ C (Ω , R ) is a vector field, and B ∈ C (Ω). We define N ( L ) := (cid:8) (cid:107) χ − g : C (Ω , g ) (cid:107) + (cid:107) A : C (Ω , g ) (cid:107) + (cid:107) B : C (Ω , g ) (cid:107) (cid:9) Proposition 11.5.
Given γ ∈ (0 , and ε > , if N ( L ) is small enough interms of α , γ and ε , then there is a bounded linear map R : C ,α ( ∂ , g ) × C ,α (Ω , g , e − γs ) → C ,α (Ω , g , e − γs ) such that for ( f, E ) in the domain of R and v = R ( f, E ) , the followingproperties are true, where the constants C depend only on α and γ : (1) L v = E on Ω . (2) v = f − avg ∂ f + B ( f, E ) on ∂ , where B ( f, E ) is a constant on ∂ and avg ∂ f denotes the average of f over ∂ . (3) v ≡ on ∂ l . (4) (cid:107) v : C ,α (Ω , g , e − γs ) (cid:107)≤ C (cid:107) f − avg ∂ f : C ,α ( ∂ , g ) (cid:107) + C (cid:107) E : C ,α (Ω , g , e − γs ) (cid:107) . (5) | B ( f, E ) | ≤ ε (cid:107) f − avg ∂ f : C ,α ( ∂ , g ) (cid:107) + C (cid:107) E : C ,α (Ω , g , e − γs ) (cid:107) . (6) If E vanishes, then (cid:107) v : C (Ω) (cid:107) ≤ (cid:107) v : C ( ∂ ) (cid:107) . Moreover, the function v depends continuously on L .Proof. For a metric χ = ( χ ij ) in local coordinates, we can write the Laplaceoperator ∆ χ as(11.23) ∆ χ = χ ij ∂ ij + χ ij,i ∂ j + χ kj Γ iik ∂ j where χ − = ( χ ij ) is the inverse matrix for ( χ ij ), and where(11.24) Γ kij = 12 χ lk ( χ li,j + χ jl,i − χ ij,l )are the Christoffel symbols for the Riemannian connection for χ . By (11.23)we have that (cid:107)L − ∆ g (cid:107) ≤ C N ( L )where (cid:107)L − ∆ g (cid:107) denotes the operator norm of L − ∆ g as a map from C ,α (Ω , g , e − γs ) to C ,α (Ω , g , e − γs ). Thus by taking N ( L ) sufficiently smallwe can arrange so that(11.25) (cid:107)L − ∆ g (cid:107) < δ for any δ >
0. Despite the presence of small L eigenvalues for the flatlaplacian ∆ g on a long cylinder we can still define a uniformly boundedinverse as follows: Given a function E ∈ C ,α (Ω , g , e − γs ), write E ( s, z ) = ( s, z ) + e ( z ), with e ( z ) = π (cid:82) σ = z E ( s, σ ) ds the radial average of E .Then for any function f ∈ C ,α ( ∂ ) we can solve∆ g U = E U = f − avg ∂ f on ∂ U = 0 on ∂ l with (cid:107) U : C ,α (Ω , g , e − γs ) (cid:107) ≤ C (cid:107) E : C ,α (Ω , g , e − γs ) (cid:107) + C (cid:107) f − avg ∂ f : C ,α ( ∂ ) (cid:107) . The radial part which projects onto the small eigenvalues is then directlyintegrated by setting u ( z ) = (cid:82) lz (cid:82) ls e ( t ) dtds . We then have L ( U + u ) = E + ( L − ∆ g )( U + u ) := E + E U + u = c on ∂ U + u = 0 on ∂ l where E is defined by the equality above and satisfies (cid:107) E : C ,α (Ω , g , e − γs ) (cid:107) ≤ δ (cid:107) U + u : C ,α (Ω , g , e − γs ) (cid:107)≤ δ ( C + 1) (cid:107) E : C ,α (Ω , g , e − γs ) (cid:107) where C denotes the operator norm of ∆ − g in the space of L functionswith zero radial average. The process is then iterated to obtain a sequence { ( U k , u k ) } ∞ k =1 satisfying∆ g U k = ( L − ∆ g )( U k − + u k − ) − e k ,U k = 0 on ∂ Ω , with e k ( z ) = (cid:90) σ = z ( L − ∆ g )( U k − + u k +1 )( s, σ ) ds and(11.26) u k = (cid:90) lz (cid:90) ls e k ( t ) dtds Choosing δ so that δC = (cid:15) (cid:48) <
1, we than have that(11.27) (cid:107) U k : C ,α (Ω , g , e − γs ) (cid:107) , (cid:107) u k : C ,α (Ω , g , e − γs ) (cid:107) < (cid:15) (cid:48) k (cid:107) E : C ,α (Ω , g , e − γs ) (cid:107) The alternating partial sums v k = Σ ki =0 ( − i ( U i + u i ) then converge to afunction v satisfying (1) − (6) above. The continuous dependence on L follows directly by construction. (cid:3) Remark 11.6.
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Nikolaos Kapouleas, Department of Mathematics, Brown University, Prov-idence, RI 02912.
E-mail address : [email protected] Stephen James Kleene, MIT, Cambridge, MA 02139.
E-mail address : [email protected] Niels Martin Møller, MIT, Cambridge, MA 02139.
E-mail address : [email protected]@math.mit.edu