CMC hypersurfaces with bounded Morse index
aa r X i v : . [ m a t h . DG ] F e b CMC hypersurfaces with bounded Morse index
Theodora Bourni ∗ Ben Sharp Giuseppe TinagliaFebruary 10, 2021
Abstract
We develop a bubble-compactness theory for embedded CMC hypersurfaceswith bounded index and area inside closed Riemannian manifolds in low dimen-sions. In particular we show that convergence always occurs with multiplicity one,which implies that the minimal blow-ups (bubbles) are all catenoids. We also pro-vide bounds on the area of separating CMC surfaces of bounded (Morse) indexand use this, together with the previous results, to bound their genus.
Mathematics Subject Classification:
Primary 53A10, Secondary 49Q05, 53C42.
Key words and phrases:
Minimal surface, constant mean curvature, finite index,curvature estimates, area estimates.
Throughout this paper, N will be a closed (compact and without boundary) Rieman-nian n -manifold of dimension n ≤ H -hypersurface M ⊂ N will be a closedconnected hypersurface embedded in N with constant mean curvature (CMC) H > H -hypersurfaces in N : this study isinspired by the result by Choi and Schoen [11] that the moduli space of fixed genusclosed minimal surfaces embedded in ( S , h ) with a metric h of positive Ricci curvaturehas the structure of a compact real analytic variety, see Theorem 3.5. Theorem 1.1.
Let ≤ n ≤ . Given H > , let { M k } k ∈ N be a sequence of H -hypersurfaces in N n satisfying sup k H n − ( M k ) < ∞ and sup k Ind ( M k ) < ∞ . Then, there exists a hypersurface M ∞ effectively embedded in N with constant meancurvature H and a finite set of points ∆ ⊂ N such that, after passing to a subse-quence, { M k } k ∈ N converges smoothly and with multiplicity one , to M ∞ away from ∆ . Furthermore ∆ is contained in the non-embedded part of M ∞ . ∗ The first author was supported by grant 707699 of the Simons Foundation H -hypersurfaces ( H >
0) converges to a limit which is itself not embedded.For instance a sequence of degenerating Delaunay surfaces converges to a string ofpearls - CMC spheres which self-intersect tangentially. We refer to connected collectionsof H -hypersurfaces which meet tangentially as “effectively embedded” (see Definition2.8). Here Ind refers to the number of negative eigenvalues of the Jacobi operatorwhen restricted to volume-preserving deformations (see Section 2). Our compactnesstheorem guarantees that any weak-limit of a sequence of H -hypersurfaces with boundedMorse index (Ind ) and area is effectively embedded and obtained via multiplicity onegraphical convergence away from finitely many points. Remark 1.2.
Notice that the convergence always happens with multiplicity one as aresult of the strict positivity of the mean curvature
H > and not by an assumption onthe ambient manifold. If { M k } are all separating and stable ( Ind = 0 ), multiplicity oneconvergence has been obtained in [32, Theorem 2.11 (ii)] where in this case ∆ = ∅ bythe regularity theory for stable CMC hypersurfaces (see e.g. Lemma 2.3). The theoremabove shows that multiplicity one convergence continues to hold under bounded index,regardless of whether ∆ is empty or not. These facts are in sharp contrast to the settingof minimal hypersurfaces where higher multiplicity convergence is guaranteed if ∆ = ∅ ,or ruled out altogether (for instance) under the assumption that Ric N > [23].Furthermore when n = 3 , in view of Theorem 3.1, we can replace the assumed areabound with the topological condition that each M k separates N . In [6] the authors develop an extensive regularity and compactness theory for codi-mension 1 integral varifolds with constant mean curvature and finite index in a Rie-mannian manifold of any dimension. They in fact deal with a much larger class ofvarifolds with appropriately bounded first variation.Inspired by the bubbling analysis carried out in [8], and the works of Ros [18] andWhite [31], we are able to capture shrinking regions of instability along a convergentsequence M k to provide a more refined picture close to ∆. As in [8] we can blow-upthese regions to obtain complete embedded minimal hypersurfaces in R n (“bubbles”)which themselves have finite index and Euclidean volume growth. A key feature in thesetting of H -hypersurfaces ( H >
Theorem 1.3 (Corollary 5.3) . Let ≤ n ≤ and H > . Then there exists G = G ( N, Λ , I , H ) so that the collection of H -hypersurfaces with index bounded by I andvolume bounded by Λ has at most G distinct diffeomorphism types. Furthermore for any H -hypersurface M with the above index and volume bounds we have uniform controlon the total curvature Z M | A | n − ≤ G . n = 3 we provide bounds on the areaof separating CMC surfaces of bounded (Morse) index and use this, together with theprevious results, to bound their genus as well. Theorem 1.4.
Given
I ∈ N and H > , let M be an H -surface in N with indexbounded by I . If we furthermore assume that either M is separating in N , or N has finite fundamental group (e.g. if N has positive Ricci curvature)then there exists a constant A := A ( I , H, N ) such that genus( M ) + area( M ) ≤ A . Since there exist examples of connected closed minimal surfaces embedded in andseparating a flat 3-torus with arbitrarily large area but bounded index [27], see Re-mark 3.3, having
H > N with positive scalar curvature R N >
0, an analogous result has been obtained in [10]. Indeed, Theorem 1.4 has beenproven independently by Saturnino [19] using techniques developed in [10]. Finally, forarbitrary three-manifolds N and immersed CMC surfaces Σ ⊂ N with sufficiently largemean curvature H Σ > H , an effective (and linear) genus bound in terms of index hasbeen obtained in [1].We will prove the area estimate in Section 3 (See Theorem 3.1 and Corollary 3.4),the genus bound will then follow from a general bubble-compactness argument for H -hypersurfaces with bounded index and area, the full details of which appear inSection 5. Let N n be a closed (compact and without boundary) Riemannian n -manifold, wherehere and throughout we restrict 3 ≤ n ≤ Definition 2.1. An H -hypersurface M ⊂ N will be a closed connected hypersurfaceembedded in N with constant mean curvature H > . When n = 3 we will often referto M as an H -surface. Let µ be the canonical measure corresponding to the metric on M (inherited by themetric on N ), ν a choice for its unit normal and A the second fundamental form of theembedding. We consider Q , the quadratic form associated to the Jacobi operator: Q ( u, u ) = Z M |∇ u | − ( | A | + Ric N ( ν, ν )) u dµ , u ∈ W , ( M ) , where Ric N is the Ricci curvature of N . 3ecall that for an open set U ⊂ N , the index of M in U , Ind( M ∩ U ), is defined asthe index of Q over W , ( M ∩ U ), that is, by the minimax classification of eigenvalues,the maximal dimension of the vector subspaces E ⊂ { u ∈ W , ( M ∩ U ) : Q ( u, u ) < } .Constant mean curvature (CMC) hypersurfaces are critical points of the area ( H n − -measure) functional for variations which preserve the signed volume ( H n -measure).This can be characterised infinitesimally as all variations whose initial normal speed u satisfies R M u dµ = 0. Thus it makes sense to define a new index Ind ( M ∩ U ) as theindex of Q over ˙ W , ( M ∩ U ) = { u ∈ W , ( M ∩ U ) : Z M ∩ U u dµ = 0 } that is the maximal dimension of the vector subspaces ˜ E ⊂ { u ∈ ˙ W , ( M ∩ U ) : Q ( u, u ) < } . We will call the CMC surface M stable (in U ) if Ind ( M ) = 0 (Ind ( M ∩ U ) = 0) and strongly stable (in U ) if Ind( M ) = 0 (Ind( M ∩ U ) = 0). Note that if U ⊂ W ⊂ N are open sets, then Ind( M ∩ W ) ≥ Ind( M ∩ U ) and Ind ( M ∩ W ) ≥ Ind ( M ∩ U ) and the two indices satisfy the following relation. Lemma 2.2.
For any k ∈ N ∪ { } we have Ind ( M ) = k = ⇒ k ≤ Ind( M ) ≤ k + 1 . Proof.
It follows trivially from the definition of our indices that Ind ( M ) ≤ Ind( M ).So suppose that the lemma is not true and instead we have Ind( M ) ≥ k + 2. Thusthere exists a k + 2-dimensional vector subspace E ⊂ W , ( M ) with Q ( f, f ) < f ∈ E . Let E ⊤ = { f ∈ E : R M f = 0 } ⊂ ˙ W , ( M ), then dim E ⊤ ≥ k + 1 and we stillhave Q ( f, f ) < f ∈ E ⊤ giving Ind ( M ) ≥ k + 1, a contradiction.Next we remind the reader of the curvature estimates available for stable H -hypersurfaces via the work of Lopez–Ros [15] when n = 3 and Schoen-Simon [21]when n ≥ Lemma 2.3.
Let
H > be fixed and M n − ⊂ N n an H -hypersurface. Given p ∈ M and ρ > , assume that M B Nρ ( p ) and that either(i) n = 3 , Ind ( M ∩ B Nρ ( p )) = 0 or(ii) n ≤ , Ind( M ∩ B Nρ ( p )) = 0 and ρ − ( n − H n − ( M ∩ B Nρ ( p )) ≤ µ .Then, | A | ( p ) ≤ Cρ , where C is a constant that depends on N , the value of the mean curvature and, in case(ii), also on µ . roof. The proof is by contradiction, so we suppose that we have a sequence of H -hypersurfaces { M k } k ∈ N , p k ∈ M k and ρ k > M k B Nρ k ( p k ) and ρ k | A k | ( p k ) ≥ k, where | A k | is the norm of the second fundamental form of M k . Abusing the notation,let M k denote the connected component of M k ∩ B Nρ k ( p k ) containing p k and let a k := | A k | ( q k ) dist N ( q k , ∂B Nρ k ( p k )) = max q ∈ M k | A k ( q ) | dist N ( q, ∂B Nρ k ( p k )) ≥ | A k | ( p k ) ρ k ≥ k. Using the notation d k = dist N ( q k , ∂B ρ k ( p k )), we rescale B Nd k ( q k ) by | A k | ( q k ) and denoteby f M k the scaled connected component of M k ∩ B Nd k ( q k ) containing q k , where the scalingis done in geodesic coordinates with origin at q k . Note that d k is bounded and since | A k | ( q k ) → ∞ and a k := d k | A k | ( q k ) → ∞ , then f M k is a sequence of CMC hypersurfacesin B a k (0) equipped with metrics g k which converge in C to the Euclidean metric andwhose mean curvature e H k = | A k | ( p k ) − H converges to 0. Moreover, | e A k (0) | ≡ k and for z ∈ B ak (0) we have that | e A k ( z ) | ≤
2. Furthermore Ind ( f M k ∩ B ak (0)) = 0when n = 3 and Ind( f M k ∩ B ak (0)) = 0 when 3 < n ≤ f M k converges (locally uniformly) in C tosome complete minimal surface f M ∞ embedded in R n with Ind ( f M ∞ ) = 0 in case n = 3 and Ind( M ∞ ) = 0 in case 3 < n ≤
7. For the case n = 3, by Lopez–Ros[15], M ∞ is a plane, contradicting that | A ∞ (0) | = 1. In case 3 < n ≤ f M ∞ isa stable minimal surface which, by the monotonicity formula (applied to each f M k )and the assumption on the H n − -measure, has Euclidean volume growth. Therefore,the curvature estimates of Schoen–Simon [21] imply that M ∞ must be a plane whichcontradicts that | A ∞ (0) | = 1. Remark 2.4.
The estimates for the norm of the second fundamental form in (ii) ofLemma 2.3 also hold when
Ind ( M ) = 0 [5]. The proof follows from the same scalingargument once the authors prove that the hyperplane is the only complete connectedoriented stable minimal hypersurface embedded in R n that has Euclidean area growthand no singularities. We note that in [5] our notion of being stable with respect tovolume preserving variations is referred to as weak stability. We also note that a keyingredient in proving this characterization of the hyperplane is the fact that a completeconnected oriented stable minimal hypersurface immersed in R n is one ended [9]. Definition 2.5.
Let U be an open set in N and let { M k } k ∈ N be a sequence of H -hypersurfaces in N . We say that the sequence { M k } k ∈ N has locally bounded norm ofthe second fundamental form in U if for each compact set B in U , sup k sup M k ∩ B | A M k | < ∞ where | A M k | is the norm of the second fundamental form of M k . efinition 2.6. Let { M k } k ∈ N be a sequence of H -hypersurfaces in N . A closed set ∆ ⊂ N is called a singular set of convergence if, after passing to a subsequence andreindexing, we have the following. • For any q ∈ ∆ , ρ > and n ∈ N , sup k sup M k ∩ B Nρ ( q ) | A M k | > n ; • { M k } k ∈ N has locally bounded norm of the second fundamental form in N \ ∆ .A point q ∈ ∆ will then be called a singular point of convergence . Note that ∆, as in Definition 2.6, is not uniquely defined. However, when { M k } k ∈ N does not have locally bounded norm of the second fundamental form in N , we canalways construct a singular set, for instance as follows. For each k ∈ N , let the maximumof the norm of the second fundamental form | A M k | of M k be achieved at a point p ,k ∈ M k . After choosing a subsequence and reindexing, we obtain a sequence M ,k such that the points p ,k ∈ M ,k converge to a point q ∈ N . Suppose the sequenceof hypersurfaces M ,k fails to have locally bounded norm of the second fundamentalform in N \ { q } . Let q ∈ N \ { q } be a point that is furthest away from q and suchthat, after passing to a subsequence M ,k , there exists a sequence of points p ,k ∈ M ,k converging to q with lim k →∞ A M k, ( p ,k ) = ∞ . If the sequence of hypersurfaces M ,k fails to have locally bounded norm of the second fundamental form in N \ { q , q } , thenlet q ∈ N \ { q , q } be a point in N that is furthest away from { q , q } and such that,after passing to a subsequence, there exists a sequence of points p ,k ∈ M ,k convergingto q with lim n →∞ A M k, ( p ,k ) = ∞ . Continuing inductively in this manner and using adiagonal-type argument, we obtain after reindexing, a new subsequence M k (denoted inthe same way) and a countable (possibly finite) non-empty set ∆ ′ := { q , q , q , . . . } ⊂ N such that the following holds. For every i ∈ N , there exists an integer N ( i ) such thatfor all k ≥ N ( i ) there exist points p ( k, q i ) ∈ M k ∩ B N /k ( q i ) where A M k ( p ( k, q i )) > k .We let ∆ denote the closure of ∆ ′ in N . It follows from the construction of ∆ that thesequence M n has locally bounded norm of the second fundamental form in N \ ∆.In light of the previous discussion, given a sequence { M k } k ∈ N of H -hypersurfaces in N , after possibly replacing it with a subsequence, we will consider ∆ to be a well-definedsingular set of convergence, as in Definition 2.6. Lemma 2.7.
Let { M k } k ∈ N be a sequence of H -hypersurfaces with sup k Ind ( M k ) < ∞ and assume that either n = 3 or n ≤ and for any open B ⊂⊂ N there exists aconstant µ B such that sup k H n − ( M k ∩ B ) < µ B . Then, up to subsequence there existsa finite singular set of convergence ∆ with | ∆ | ≤ sup k Ind ( M k ) + 1 . Moreover, thereexists a constant C such that for any open B ⊂⊂ N \ ∆lim k →∞ sup M k ∩ B | A M k | ≤ C dist N ( B, ∆) . Proof.
The proof is similar to that of [23, Claims 1 and 2]. Let I ∈ N be such thatInd ( M k ) + 1 ≤ I for all k and assume that ∆ has at least I + 1 distinct points { q , . . . q I +1 } . Let ε <
12 min { min i = j dist N ( q i , q j ) , σ N } , σ N is a lower bound for the injectivity radius of N . By Lemma 2.3, after passingto a subsequence, Ind( B Nε ( q i ) ∩ M k ) >
0, for all 1 ≤ i ≤ I + 1. Since { B Nε ( q i ) } I +1 i =1 arepairwise disjoint we obtain that Ind( M k ) ≥ I +1 and by Lemma 2.2 Ind ( M k )+1 ≥ I +1,which is a contradiction.To prove the curvature estimate, it suffices to show that there exists ε > < ε ≤ ε lim k →∞ Ind(( B Nε ( q i ) \ B Nε/ ( q i )) ∩ M k ) = 0 for all q i ∈ ∆. (1)This is indeed sufficient, because M k has locally bounded norm of the second funda-mental norm in N \ ∆ and (1) combined with Lemma 2.3 yields the required curvatureestimate.To prove (1) we argue by contradiction: suppose there exists q i ∈ ∆ so that for all ε >
0, there exists ε ≤ ε with lim inf Ind(( B Nε ( q i ) \ B Nε / ( q i )) ∩ M k ) ≥
1. We cansuccessively apply this statement (setting ε = ε l / I + 1 timesto find a sequence ε , ε , ε , . . . ε I +1 satisfying ε l +1 ≤ ε l / B Nε l ( q i ) \ B Nε l / ( q i )) ∩ M k ) ≥
1. Once again we have found I + 1 disjoint sets for which each M k is unstable and shown Ind ( M k ) ≥ I + 1 for all large k , a contradiction.To study the limiting behaviour of CMC surfaces, we will need the following defi-nition. Definition 2.8.
A connected subset V ⊂ N will be called an effectively embedded H -hypersurface if V is a finite union of smoothly immersed compact connected constantmean curvature hypersurfaces and at any point p ∈ V , there exists ε > such thateither1. B Nε ( p ) ∩ V is a smooth embedded disk, or2. B Nε ( p ) ∩ V is the union of two embedded disks, meeting tangentially and whosemean curvature vectors point in opposite directions. Let V be an effectively embedded H -hypersurface as in Definition 2.8. We will referto the set of points p ∈ V satisfying 1 . of Definition 2.8 as the regular part of V andwe will denote it by e ( V ) . Note that e ( V ) is relatively open and splits into a finitenumber of (mutually disjoint) connected components e ( V ) = ∪ Li =1 V i , each of which is a smooth embedded CMC hypersurface having the same size meancurvature H . The set of points satisfying 2 . of Definition 2.8 is the singular set of V ,denoted by t ( V ) which is relatively closed, and t ( V ) := ∪ Li =1 V i \ V i . e ( V ) standing for the embedded part of V t ( V ) for touching set V i self-intersecting, however, with this no-tation we have that if p ∈ t ( V ) then there exists ε > e ( V ) ∩ B Nε ( p ) splitsinto two disjoint components C i , C j with C i ⊂ V i , C j ⊂ V j and { C i } i =1 , are the twosmooth embedded CMC disks touching tangentially at p with opposite mean curvaturevectors. It might happen that i = j if one component V i self-intersects. It is notdifficult to check that each V i is individually an immersed, smooth, connected CMChypersurface which is embedded unless it is self-intersecting.Below is a definition of convergence that we will be using often in this paper andwe will be refering to as H-convergence . Definition 2.9.
A sequence { M k } k ∈ N of H -hypersurfaces H -converges to V = ∪ Li =1 V i ,an effectively embedded H -hypersurface, with finite multiplicity ( m , . . . , m L ) ∈ N L if d H ( M k , V ) → as k → ∞ and if its singular set of convergence ∆ ⊂ V is finite andwhenever p ∈ V \ ∆ the following holds. • If p ∈ V i , then there exists an ε > so that B Nε ( p ) ∩ M k converges smoothly andgraphically (normal grpahs) with multiplicity m i , to B Nε ( p ) ∩ V . • If p ∈ t ( V ) , then there exists an ε > so that B Nε ( p ) ∩ M k uniquely partitions intotwo parts. The first part converges smoothly and graphically, with multiplicity m i ,to C i , and the second converges smoothly and graphically, with multiplicity m j ,to C j , where C i , C j are as discussed in the previous paragraph. Remark 2.10. If ∆ = ∅ then V = V i for some fixed i and the multiplicity of con-vergence is one, contrary to what happens if we allow the limit to be minimal . Thisfollows from the fact that all H -hypersurfaces are two-sided. Thus over each V i we canwrite the approaching M k ’s globally as graphs - if the multiplicity is larger than one, orthere is more than one V i , the M k ’s must have been disconnected. Finally, in the next sections, we will also use the following notation. We let S , I , V > N . Given H >
0, we fix J H ∈ (0 , I ) so that for any ρ ≤ J H , thegeodesic balls B Nρ ( p ) are H -convex, that is their boundaries are hypersurfaces whosemean curvature is bigger than or equal to H , independently of p ∈ N . When n = 3, we use the results in Section 2 to prove the following area estimate for H -surfaces, H > For instance in the standard S = { x ∈ R : | x | = 1 } , if S = { x = 0 } ∩ S is a great sphere, theequidistant surfaces M k defined by M k = { x := 1 /k } are CMC spheres converging smoothly to S .If we project this picture to R P then we have a sequence of CMC spheres converging smoothly (so∆ = ∅ ) with multiplicity two to a great R P . heorem 3.1. Given
I ∈ N and H > , there exists a constant A := A ( I , N ) suchthat if M is an H -surface separating N with Ind ( M ) ≤ I , we have that H ( M ) ≤ A . Proof.
We first prove a local area estimate when the norm of the second fundamentalform is bounded.
Claim 3.2.
Given α > there exists ω := ω ( α, N ) such that the following holds. Given p ∈ M and ρ < J H , if sup B Nρ ( p ) | A | < α then H ( M ∩ B Nρ/ ( p )) < ω H ( N ) . Proof of Claim 3.2.
Given ρ < J H , the techniques used to prove Lemma 3.1 in [17] givethat there exists β := β ( α, J H , S ) > M ∩ B Nρ ( p ) bounds an H -convexdomain, then M ∩ B Nρ/ ( p ) has a one-sided regular neighbourhood of fixed size β . Thismeans that the collection of geodesics of length β starting at x ∈ M ∩ B Nρ/ ( p ) and withinitial velocity given by H ( x ) / | H ( x ) | are pairwise-disjoint, only intersect M at x andtherefore foliate a one-sided neighbourhood of M . The result is mainly a consequenceof the observation that two H -surfaces with bounded norm of the second fundamentalform which are oppositely oriented and such that one lies on the mean convex sideof the other, cannot be too close away from their boundary and this is essentially aconsequence of the maximum principle for quasi-linear uniformly elliptic PDEs. Notethat this is not true when H = 0.Since M is separating in N we do have that M ∩ B Nρ ( p ) bounds an H -convexdomain. Let U β denote the 1-sided regular neighbourhood of M ∩ B Nρ/ ( p ) as above.Then, since the norm of second fundamental form of M is uniformly bounded we candirectly relate the area of M ∩ B Nρ/ ( p ) with the volume of U β : there exists a constant ω := ω ( β ) > ω H ( M ∩ B Nρ/ ( p )) ≤ H ( U β ) ≤ H ( N ) . This finishes the proof of the claim.We now begin the proof of the area estimate. Arguing by contradiction, assumethat there exist
I ∈ N , H >
0, and a sequence of H -surfaces { M k } k ∈ N such that for all k ∈ N , the H -surface M k separates N , Ind ( M k ) ≤ I and H ( M k ) > k. By Lemma 2.7, after passing to a subsequence, there exists a finite set of points∆ := { p , . . . , p l } , l ≤ I + 1, such that the sequence { M k } k ∈ N has locally bounded normof the second fundamental form in N \ ∆. Since N is compact, applying Claim 3.2 anda covering argument gives that for any ε >
0, there exists a constant V ( ε ) such that H ( M k ∩ [ N \ l [ i =1 B Nε ( p i )]) < V ( ε ) .
9n order to obtain a contradiction, it remains to show that the area of M k ∩ B Nε ( p i ), i = 1 , . . . , l is also bounded, uniformly in k . To that end, we will use the monotonicityformula for the area. After isometrically embedding the ambient space N in an Eu-clidean space R m , the submanifolds M k ⊂ N ⊂ R m have mean curvature vector fields H k = H Nk + H N ⊥ k , where H Nk and H N ⊥ k are the projections of H k (the mean curvaturevector of M k ⊂ R m ) onto the tangent and the normal space of N respectively. Notethat | H Nk | = H and H N ⊥ k depends only on the embedding of N and thus its norm isuniformly, in k , bounded. We thus have a sequence of submanifolds with uniformlybounded mean curvature, | H k | ≤ c . Therefore, the area monotonicity, see for example[24, 17.6], yields, for any p ∈ R m and 0 < σ < ρ , e cσ σ − H ( M k ∩ { x : | x − p | < σ } ) ≤ e cρ ρ − H ( M k ∩ { x : | x − p | < ρ } ) . Since M k ⊂ N and the embedding is isometric we obtain e cσ σ − H ( M k ∩ B Nσ ( p )) ≤ e cρ ρ − H ( M k ∩ B Nρ ( p )) . Take now p to be a point in the singular set. Then for small ε we have ε − H ( M k ∩ B Nε ( p )) ≤ e cε (2 ε ) − H ( M k ∩ B N ε ( p )) ≤ ε − H ( M k ∩ B N ε ( p )) , which yields H ( M k ∩ B Nε ( p )) ≤ H ( M k ∩ ( B N ε ( p ) \ B Nε ( p ))) . But now, choosing ε small enough so that B N ε ( p ) \ B Nε ( p ) is away from ∆, the right handside is uniformly bounded by V ( ε ) and thus H ( M k ) < ( l + 1) V ( ε ). This contradictsthe assumption that H ( M k ) > k and finishes the proof of the area estimate. Remark 3.3.
In [27], Traizet proved for any positive integer g , g = 2 , every flat3-torus admits connected closed embedded and separating minimal surfaces of genus g with arbitrarily large area. Fix g = 2 and let M k be a sequence of such minimal surfaceswhose area is becoming arbitrarily large. Since the genus is fixed, by the Gauss-Bonnettheorem, the total curvature of M k is uniformly bounded in k . And this gives that theindex of M k is also uniformly bounded in k [28]. Thus, these examples show that thearea estimates do not hold when H = 0 . As a corollary of the proof above, if the ambient manifold N has finite fundamentalgroup (e.g. if it has positive Ricci curvature), then the area bound is true withoutassuming that the H -surface M is separating. Corollary 3.4.
Given
I ∈ N and H > , there exists a constant A := A ( I , N ) suchthat if M is an H -surface in N with Ind ( M ) ≤ I and N has finite fundamental group,we have that H ( M ) ≤ A . roof. Since N has finite fundamental group its universal cover Π : e N → N is a finitecovering. Π − ( M ) is a disjoint collection of H -hypersurfaces in e N and we denote by f M a connected component of Π − ( M ). Then f M is an H -surface separating e N , because e N is simply-connected. We may now reduce to the setting of Theorem 3.1: let { M k } ⊂ N be a sequence of H -hypersurfaces with index uniformly bounded by I . By Lemma2.7, after passing to a subsequence, there exists a finite set of points ∆ := { p , . . . , p l } , l ≤ I + 1, such that the sequence { M k } k ∈ N has locally bounded norm of the secondfundamental form in N \ ∆. Thus picking connected lifts f M k ⊂ e N we have that f M k areseparating and there exists a finite set of points e ∆ := { e p , . . . , e p L } , L ≤ | π ( N ) | ( I + 1),such that the sequence { f M k } k ∈ N has locally bounded norm of the second fundamentalform in e N \ e ∆. We can now apply Claim 3.2 to f M k ⊂ e N and follow the remaining partsof the proof of Theorem 3.1 to conclude the proof of the corollary.Thanks to the area estimate, an elegant compactness result for H -surfaces separat-ing N now follows. Theorem 3.5.
Given
H > , let { M k } k ∈ N be a sequence of H -surfaces such that,for all k ∈ N , M k separates N (or not necessarily separating if | π ( N ) | < ∞ ) and sup k Ind ( M k ) < ∞ . Then, there exists an effectively embedded H -surface M ∞ suchthat, after passing to a subsequence, { M k } k ∈ N H-converges with multiplicity one to M ∞ ,where the convergence is as in Definition 2.9.Proof. Using the curvature estimate of Lemma 2.7 and the area estimate of Theorem 1.4(or Corollary 3.4 if N has finite fundamental group), a standard argument yields thataway from a finite set of points ∆ ⊂ N , that is the singular set of convergence (seeDefinition 2.6), a subsequence H -converges with finite multiplicity to a surface M ∞ effectively embedded in N \ ∆ with constant mean curvature H .We next show that M ∞ ∪ ∆ is in fact effectively embedded in N , which will implythat { M k } k ∈ N H -converges with finite multiplicity to M ∞ ∪ ∆ with ∆ being the singularset of convergence. For this we will need the following claim. We let ∆ = { q , . . . q l } and ε := inf i,j =1 ,...,l ; i = j dist N ( q i , q j ). Claim 3.6.
Given δ > , there exists < ρ ≤ ε such that for any q i ∈ ∆ and p ∈ M ∞ ∩ B Nρ ( q i ) | A M ∞ | ( p ) ≤ δ dist N ( p, q i ) . Proof of Claim 3.6.
Note first that, by the nature of the convergence and Lemma 2.7,for any q i ∈ ∆ and p ∈ M ∞ ∩ B Nε ( q i ) we have | A M ∞ | ( p ) ≤ C dist N ( p, q i ) . (2)Moreover, arguing as in [23, Claim 2] taking ε even smaller if necessary we have thateach connected component of M ∞ ∩ ( B Nε ( q i ) \ { q i } ) for all q i ∈ ∆ is strongly stable.11o prove the claim we argue by contradiction and suppose that for some δ > q ∈ ∆ and a sequence of points p k ∈ M ∞ such that lim k →∞ p k = q and | A M ∞ | ( p k ) > δ dist N ( p k , q ) . Consider now scaling M ∞ by N ( p k ,q ) , with the scaling performed in geodesic coor-dinates and with origin at q . Letting k → ∞ , and since dist N ( p k , q ) →
0, after passingto a subsequence, the scaled surfaces converge to a tangent cone of M ∞ ∪ { q } at q .The convergence is in general weak convergence, however, by the curvature estimate(2) and the comments following it, it is in fact smooth away from the origin and thelimit is strongly stable away from the origin. Since the limit is also a stationary cone itmust be a plane. This contradicts the fact that there exists a point at distance 1 fromthe origin with | A | ≥ δ > M ∞ ∪ ∆ is effectively embedded following the ideas of [30](see also [25, Theorem 4.3]). Let p ∈ ∆ and r > B N r ( p ) ∩ ∆ = { p } .Consider a sequence r i → f M i the scaling of M ∞ ∩ B Nr ( p ) by 1 /r i .Then, the curvature estimates of Claim 3.6 yield that, after passing to a subsequence f M i converge to a union of planes. This in turn implies that M ∞ is a union of disksand punctured disks. We can thus argue exactly as in [25, Theorem 4.3] to show that M ∞ ∪ ∆ is indeed effectively embedded.That the multiplicity of convergence is 1 will be a consequence of the results inSection 4.The curvature estimates discussed in Section 2 and that were used to prove The-orem 1.4 and Theorem 3.5, crucially depend on a bound for the volume of the H -hypersurface when 3 < n ≤ Theorem 1.1.
Given
H > , let { M k } k ∈ N be a sequence of H -hypersurfaces in N satisfying sup k H n − ( M k ) < ∞ and sup k Ind ( M k ) < ∞ . Then, there exists a hypersurface M ∞ effectively embedded in N with constant meancurvature H , such that, after passing to a subsequence, { M k } k ∈ N H-converges withmultiplicity one to M ∞ , where the convergence is as in Definition 2.9. The main goal of this section, is to show that under certain hypotheses, a sequence of H -hypersurfaces that converges to an effectively embedded surface, will in fact converge12ith multiplicity one to its limit. This result will complete the proofs of Theorems 3.5and 1.1.We first recall that I > N . Andthat given H >
0, we have fixed J H ∈ (0 , I ) so that for any ρ ≤ J H , the ambientgeodesic balls B Nρ ( p ) are H -convex, independently of p ∈ N . Throughout this section,we will always assume that the radius of an ambient geodesic ball is less than J H .We will show that even if ∆ = ∅ we must always have multiplicity one convergence: Theorem 4.1.
Let V = ∪ Lℓ =1 V ℓ be a hypersurface effectively embedded in N withconstant mean curvature H > and let { M k } k ∈ N be a sequence of H -hypersurfacesthat H-converges to V with multiplicity ( m , . . . , m L ) ∈ N L . Then the singular set ofconvergence ∆ lies inside t ( V ) and m ℓ = 1 for all ℓ = 1 , . . . , L .Proof. Since M k is embedded with uniformly bounded volume and the number of pointsin ∆ is finite, there exist 0 < ε < δ < J H such that for k sufficiently large and y ∈ ∆, B Nδ ( y ) \ B Nε ( y ) ∩ M k is a collection of m ( y ) ≥ u yi , i := 1 , . . . , m ( y ),over V which converge smoothly to zero in k (where for simplicity we have omitted theindex k ). If y / ∈ t ( V ), let n y = H/ | H | be the unit normal to V at y , otherwise let n y be a choice of unit normal. The graphs of u yi , i := 1 , . . . , m ( y ), converge smoothly to B Nδ ( y ) \ B Nε ( y ) ∩ V as k → ∞ and can be ordered by height, say with respect to n y ,so that u yi is above u yi +1 for i := 1 , . . . , m ( y ) −
1. Let Q yi be the connected componentof B Nδ ( y ) ∩ M k that contains graph u yi . Claim 4.2. ∆ ⊂ t ( V ) .Proof of Claim 4.2. Arguing by contradiction, suppose that y ∈ ∆ ∩ e ( V ) - so that y lies on an embedded part of the limit. Then, by definition, V ∩ B Nδ ( y ) ⊂ V ℓ isan embedded CMC disc and the collection of graph u yi , i := 1 , . . . , m ( y ), converges to V ∩ [ B Nδ ( y ) \ B Nε ( y )].If for all i := 1 , . . . , m ( y ), Q yi ∩ [ B Nδ ( y ) \ B Nε ( y )] = graph u yi (i.e. Q yi ∩ [ B Nδ ( y ) \ B Nε ( y )]is connected) then since Q yi converges to the disc V ∩ B Nδ ( y ) as Radon measures withmultiplicity one, by Allard’s regularity theorem [3] the convergence is smooth and y ∆. Therefore, there exists i ∈ { , . . . m ( y ) } , such Q yi ∩ [ B Nδ ( y ) \ B Nε ( y )] consists ofmore than one connected components. However, note that because Q yi separates B Nδ ( y ),the sign of the inner product between the unit normal to Q yi and n y must change aswe alternate components of Q yi ∩ [ B Nδ ( y ) \ B Nε ( y )]. This contradicts the fact that suchcomponents must converge to a single CMC disc V ∩ B Nδ ( y ). This contradiction provesthat ∆ ⊂ t ( V ).It remains to prove that the convergence to V is with multiplicity one. Let y ∈ ∆ ⊂ t ( V ), then B Nδ ( y ) ∩ V is the union of two embedded discs, C ± meeting tangentially andwhose mean curvature vectors point in opposite directions. Without loss of generality,we pick n y = H + / | H + | where H + is the mean curvature of C + and thus so that C + lies above C − , in the sense discussed in the first paragraph of the proof. The collectiongraph u yi , i := 1 , . . . , m ( y ), converging smoothly to B Nδ ( y ) \ B Nε ( y ) ∩ V as k → ∞ can13e divided into two distinct finite collections of graphs ∆ + and ∆ − that satisfy thefollowing properties: • the graphs in ∆ + are above the graphs in ∆ − ; • the collection ∆ + := { graph u yi, + , i := 1 , . . . , m + ( y ) } , converges smoothly to C + ∩ [ B Nδ ( y ) \ B Nε ( y )] as k → ∞ ; • the collection ∆ − := { graph u yi, − , i := m + ( y )+1 , . . . m − ( y ) } , converges smoothlyto C − ∩ [ B Nδ ( y ) \ B Nε ( y )] as k → ∞ .Recall that Q yi is the connected component of B Nδ ( y ) ∩ M k that contains graph u yi .Just like we observed before, if Q yi ∩ [ B Nδ ( y ) \ B Nε ( y )] consists of more than one connectedcomponent, since Q yi separates B Nδ ( y ), then the sign of the inner product between theunit normal to Q yi and n y must change as we alternate component of Q yi ∩ [ B Nδ ( y ) \ B Nε ( y )]. This implies that alternating components must alternating convergence to u y + and u y − . This gives that if Q yi ∩ [ B Nδ ( y ) \ B Nε ( y )] consists of more than one connectedcomponent, then it consists of exactly two components, one in ∆ + and the other in ∆ − .And Q yi converges to B Nδ ( y ) ∩ V on compact subsets of B Nδ ( y ) \ { y } with multiplicity1. Claim 4.3.
There is only one Q yi such that Q yi ∩ [ B Nδ ( y ) \ B Nε ( y )] is disconnected.Proof of Claim 4.3. Arguing by contradiction, assume that Q yj , i = j also has theproperty that Q yj ∩ [ B Nδ ( y ) \ B Nε ( y )] consists of exactly two components. Let Q yi ∩ [ B Nδ ( y ) \ B Nε ( y )] = graph u yi, + ∪ graph u yl i , − and let Q yj ∩ [ B Nδ ( y ) \ B Nε ( y )] = graph u yj, + ∪ graph u yl j , − . Then, because of the convergence and separation properties, we can assumethat j < i < l i < l j .Let W be the connected component of B Nδ ( y ) \ Q yi ∪ Q yj such that Q yi ∪ Q yj ⊂ ∂W .The convergence and elementary separation properties yield that the mean curvaturevector of M k is pointing outside W on Q yj and inside W on Q yi . Moreover, as k → ∞ ,we have that W → C + ∪ C − in Hausdorff distance. The argument described in [2] canbe modified to prove the following claim. Claim 4.4. If Q yi is not strongly stable, then there exists a compact, oriented, stablehypersurface Γ embedded in W with constant mean curvature H and such that ∂ Γ = ∂Q yi and Γ is homologous to Q yi in W .Proof of Claim 4.4. Let F be the family of subsets Q ⊂ W of finite perimeter whoseboundary ∂Q is a rectifiable integer multiplicity current such that Q yi ⊂ ∂Q and letΣ = ∂Q \ Q yi , so that ∂ Σ = ∂Q yi . Given µ >
0, let F µ : F → R be the functional F µ ( Q ) = H n − (Σ) + ( H + µ ) H n ( Q ) . Let W be the mean convex component of B Nδ ( y ) \ Q yi , let S min ⊂ W be a volumeminimizing hypersurface with ∂S min = ∂Q yi and homologous to Q yi , and let Q min W enclosed by Q yi ∪ S min [12, 13, 14]. Recall that since n ≤ Q ρ := { x ∈ W : dist N ( x, Q yj ) ≤ ρ } and note that if ρ is chosen sufficientlysmall, then the sets S t := { x ∈ Q ρ such that dist N ( x, Q yj ) = t } , ≤ t ≤ ρ are smooth hypersurfaces parallel to Q yj and foliating Q ρ . Let Y be the the unit vectorfield normal to the foliation and pointing toward Q yj . Let H t denote the mean curvatureof S t as it is oriented by Y . Then ddt H t | t =0 = | A | + Ric N ( n j , n j )where n j is the unit normal vector field to Q yj . Thus, for any µ > ρ µ > N ( n j , n j ), such for t ∈ [0 , ρ µ ] we have that H t < H + µ and at a point p ∈ S t div N Y = div S t Y = − H t = ⇒ − H − µ < div N Y. Let Q par := Q ρ µ and S par = S ρ µ .Next we are going to work on Q yi . Let φ be the first eigenfunction of the stabilityoperator of Q yi . The eigenfunction φ is positive in the interior of Q yi and since Q yi isnot stable, then ∆ φ + | A | φ + Ric N ( n i , n i ) φ + λ φ = 0 , where λ is a negative constant and n i is the unit normal vector field to Q yi . Andpossibly after a small perturbation of δ , we can assume that 0 is not an eigenvalue of∆ + | A | + Ric N ( n i , n i ). Thus there is a smooth function v vanishing on ∂Q yi , such that∆ v + | A | v + Ric N ( n i , n i ) v = 1 in Q yi . By Hopf’s maximum principle the derivative of φ with respect to the outer pointing normal vector to ∂Q yi is strictly negative. Therefore,there exists a > u = φ + av is positive in the interior of Q yi .Let e S t := { x ∈ W such that dist N ( x, Q yi ) = tu } , ≤ t ≤ e ρ. If e ρ is sufficiently small, the sets e S t are smooth hypersurfaces foliating a closed neigh-bourhood e Q e ρ of Q yi in W .Let X be the the unit vector field normal to the foliation and pointing away from Q yi . Let H t denote the mean curvature of e S t as it is oriented by X . Then ddt H t | t =0 = ∆ u + | A | u + Ric N ( n i , n i ) u = − λ φ + a > , where n i is the unit normal vector field to Q yi . Therefore, if e ρ is taken sufficiently small,for t ∈ (0 , e ρ ] we have that H t > H and at a point p ∈ e S t we havediv N X < − H. Let Q uns := e Q e ρ and S uns := e S e ρ . 15 laim 4.5. Let Q ∈ F with Σ smooth and transverse to S min , S par , and S uns . Thefollowing statements hold.1. If Q Q min then F µ ( Q ∩ Q min ) ≤ F µ ( Q ) ;2. If Q ∩ Q par = ∅ then F µ ( Q \ Q par ) ≤ F µ ( Q ) ;3. If Q uns Q then F µ ( Q ∪ Q uns ) ≤ F µ ( Q ) .Proof of Claim 4.5. We first prove that if Q Q min then F µ ( Q ∩ Q min ) ≤ F µ ( Q ).Since Q ∩ Q min ⊂ Q , we have that H n ( Q ∩ Q min ) ≤ H n ( Q ) and, by construction, H n − (Σ ′ ) ≤ H n − (Σ) where Σ ′ := ∂ ( Q ∩ Q min ) \ Q yi .We now prove that if Q ∩ Q par = ∅ then F µ ( Q \ Q par ) ≤ F µ ( Q ). Recall that in Q par , − H − µ < div N Y , therefore( − H − µ ) H n ( Q ∩ Q par ) < Z Q ∩ Q par div N Y = Z ∂ ( Q ∩ Q par ) Y · ν where ν is the outer pointing unit normal to ∂ ( Q ∩ Q par ) and Z ∂ ( Q ∩ Q par ) Y · ν = Z Q ∩ S par Y · ν + Z Σ ∩ Q par Y · ν. Since, by construction, Y · ν = − S par and Y · ν ≤ ∩ Q par , we have that( − H − µ ) H n ( Q ∩ Q par ) < −H n − ( Q ∩ S par ) + H n − (Σ ∩ Q par )and F µ ( Q \ Q par ) = ( H + µ )( H n ( Q ) − H n ( Q ∩ Q par )) + H n − (Σ \ Q par ) + H n − ( Q ∩ S par ) < ( H + µ ) H n ( Q ) + H n − (Σ ∩ Q par ) + H n − (Σ \ Q par ) = F µ ( Q )We finally prove that if Q uns Q then F µ ( Q ∪ Q uns ) ≤ F µ ( Q ). We argue similarlyto the previous claim. Recall that in Q uns , div N X < − H . Therefore − H H n ( Q uns \ Q ) > Z Q uns \ Q div N X = Z ∂ ( Q uns \ Q ) X · ν where ν is the outer pointing unit normal to ∂ ( Q uns \ Q ) and Z ∂ ( Q uns \ Q ) X · ν = Z S uns \ Q X · ν + Z Σ ∩ Q uns X · ν. Since, by construction, X · ν = 1 on S uns and X · ν ≥ − ∩ Q uns , we have that − H H n ( Q uns \ Q ) > H n − ( S uns \ Q ) − H n − (Σ ∩ Q uns )and F µ ( Q ∪ Q uns ) = ( H + µ )( H n ( Q ) + H n ( Q uns \ Q )) + H n − (Σ \ Q uns ) + H n − ( S uns \ Q ) < ( H + µ ) H n ( Q ) + µ H n ( Q uns \ Q ) + H n − (Σ ∩ Q uns ) + H n − (Σ \ Q uns ) < F µ ( Q ) . This finishes the proof of Claim 4.5. 16n order to find a minimizer for the functional F µ we consider a minimizing sequence Q m and, since they have uniformly bounded areas, we can apply the compactness resultsof [13] to extract a converging subsequence. Note that by Claim 4.5, we can assumethat Q m ⊂ Q min , Q m ∩ Q par = ∅ , and Q uns ⊂ Q m . It is known that a minimizer of F µ is smooth [4, 7, 22] and thus we obtain a compact, embedded, oriented minimizerΓ µ ⊂ W of the functional F µ such that ∂ Γ µ = ∂Q yi and Γ µ is homologous to Q yi in W .In particular, Γ µ has constant mean curvature equal to H + µ .We can also assume that H + µ < H and H n − (Γ µ ) ≤ H n − ( Q yi ) ≤ H n − ( C + ∪ C − ) . The first inequality above follows because H n − (Γ µ ) ≤ F µ (Γ µ ) ≤ F µ ( Q yi ) = H n − ( Q yi ).The second inequality holds because away from the singular point of convergence y , thevolume can be bounded by the volume of the limit, and nearby y it can be boundedby using the monotonicity formula for the volume, exactly like we have done to finishthe proof of Theorem 1.4. Then the results in [5] (see Lemma 2.3 and Remark 2.4)give that Γ µ has norm of the second fundamental form uniformly bounded on compactsets of B Nδ ( y ). And taking the limit of Γ µ as µ goes to zero, we obtain in the limit thedesired Γ and finish the proof of Claim 4.4.We can now finish the proof of Claim 4.3. Since y is a singular point of convergence, Q yi cannot be strongly stable and thus cannot have norm of the second fundamentalform bounded nearby y . Therefore Claim 4.4 gives a compact, oriented, stable hyper-surface Γ embedded in W with constant mean curvature H and such that ∂ Γ = ∂Q yi and Γ is homologous to Q yi in W .We now recall that while we have omitted the index k , we have in fact a sequenceof domains W ( k ) and stable hypersurfaces Γ( k ) ⊂ W ( k ). By the previous discussion,Γ( k ) has norm of the second fundamental form uniformly bounded on compact sets of B Nδ ( y ), uniform in k . And by construction, since Γ( k ) is homologous to Q yi in W , forany ρ > k > k ) ∩ B Nρ ( y ) = ∅ . Using the uniform boundon the norm of the second fundamental form gives that Γ( k ) must converge smoothlyto C + or C − or both. Elementary separation properties give that Q yj cannot convergesmoothly to [ C + ∪ C − ] \ { y } . This contradiction proves that there is only one Q yi suchthat [ Q yi ∩ B Nδ ( y )] \ B Nε ( y ) is disconnected.We now prove that the convergence to V is with multiplicity one and finish the proofof the theorem. Arguing by contradiction, assume that the multiplicity of convergencealong some V ℓ is m ℓ ≥
2. Recall that the convergence is smooth on compact subsets K ⊂⊂ V ℓ \ ∆. Observe that we must have ∆ ∩ V ℓ = ∅ : if not, since V ℓ is connected, wecan write the approaching M k ’s globally as graphs over V ℓ (since CMC hypersurfacesare always two-sided). And if there were more than one graph, then the M k ’s aredisconnected.Let ∆ ∩ V ℓ = { y , . . . , y g ( ℓ ) } . Since the convergence is smooth on V ℓ \ S g ( ℓ ) j =1 B Nε ( y j )and with finite multiplicity, we can write the approaching surfaces M k \ S g ( ℓ ) j =1 B Nε ( y j )17lobally as graphs over V ℓ \ S g ( ℓ ) j =1 B Nε ( y j ) and order such graphs by height with respectto the mean curvature vector ~H ℓ of V ℓ . This gives ordered sheets S k , . . . , S m ℓ k eachconverging smoothly to V ℓ \ S g ( ℓ ) j =1 B Nε ( y j ). Note that this ordering is different from theprevious local ordering established nearby a singular point. Let y j ∈ ∆ ∩ V ℓ ⊂ t ( V )and recall that V ∩ [ B Nδ ( y j ) \ B Nε ( y j )] consists of two oppositely oriented componentswhich we denote by Γ + j and Γ − j . Assume that Γ + j ⊂ V ℓ and let Q j denote the con-nected component of M k ∩ B Nδ ( y j ) containing the component of S k ∩ [ B Nδ ( y j ) \ B Nε ( y j )]converging to Γ + j . If Γ − j ⊂ V ℓ , let Q j − denote the connected component of M k ∩ B Nδ ( y j )containing the component of S k ∩ [ B Nδ ( y j ) \ B Nε ( y j )] converging to Γ − j . Recall that if Q j ∩ [ B Nδ ( y ) \ B Nε ( y )] consists of more than one connected component, then it consistsof exactly two components, one converging to Γ + j and the other to Γ − j . And the sameis true of Q j − . If for each y j ∈ ∆ ∩ V ℓ , Q j ∩ [ B Nδ ( y ) \ B Nε ( y )] and Q j − ∩ [ B Nδ ( y ) \ B Nε ( y )]each consists of exactly one component, then S k would correspond to a single connectedcomponent of M k converging smoothly with multiplicity one to V ℓ , and in particular M k would be disconnected. Therefore, after possibly relabelling, there exists y j ∈ ∆ ∩ V ℓ ,such that Q j ∩ [ B Nδ ( y ) \ B Nε ( y )] consists of exactly two components, one converging toΓ + j and the other to Γ − j .Notice that by the previous claim, Q j must be the unique such component. Thatis, if Λ is another connected component of M k ∩ B Nδ ( y j ), then Λ ∩ [ B Nδ ( y ) \ B Nε ( y )] isconnected. In particular, if Λ is the connected component of M k ∩ B Nδ ( y j ) containingthe component of S k ∩ [ B Nδ ( y j ) \ B Nε ( y j )] converging to Γ + j , then Λ ∩ [ B Nδ ( y ) \ B Nε ( y )]is connected. Note that by our choice of S k , Λ ∩ [ B Nδ ( y ) \ B Nε ( y )] must be below thecomponent component of Q j ∩ [ B Nδ ( y ) \ B Nε ( y )] that converges to Γ + j and above thecomponent of Q j ∩ [ B Nδ ( y ) \ B Nε ( y )] that converges to Γ − j . By elementary separationproperty, we obtain a contradiction. This proves that m ℓ = 1, l = 1 , . . . , L , and finishesthe proof of the theorem. The goal of this section is to prove the bubble-compactness theorem for H -hypersurfaceswhen H > C n − ⊂ R n is arotationally symmetric complete minimal hypersurface with Ind( C ) = lim R →∞ ( C ∩ B R (0)) = 1 and finite total curvature (see e.g. [26] for further details). In the sequel C will denote any catenoid up to scaling, rotations and translations, without re-labelling.We first recall a result of Schoen [20, Theorem 3] which states that for each n ≥ M n − ⊂ R n which are regular at infinity andhave two ends are either catenoids C n − or a pair of hyperplanes. Combining a resultof Tysk [29, Lemma 4] with [20, Proposition 3] we see in particular that this implies18 emma 5.1. When ≤ n ≤ the only embedded, complete minimal hypersurfaces M n − ⊂ R n with Euclidean volume growth, finite index and at most two ends, areeither one or two parallel planes , or a catenoid. The total curvature of a hypersurface is denoted by T = R | A | n − and T ( C n − )denotes the total curvature of the catenoid. When n = 3 we have T ( C ) = 8 π .The main result of this section is as follows. Theorem 5.2.
With the same hypotheses as Theorem 1.1, for each y ∈ ∆ there existsa finite number < J y ∈ N of point-scale sequences (see Definition 5.4) { ( p y,ℓk , r y,ℓk ) } J y ℓ =1 so that:1. these point-scale sequences are distinct, in the sense that for all ≤ i = j ≤ J y dist g ( p y,ik , p y,jk ) r y,ik + r y,jk → ∞ . Taking normal coordinates centred at p y,ℓk and letting f M y,ℓk := M k /r y,ℓk ⊂ R n then f M y,ℓk converges smoothly on compact subsets to a catenoid C n − with multiplicityone, for all ℓ .2. There exist δ , R > so that for all y ∈ ∆ , δ ≤ δ , R ≥ R and k sufficientlylarge M k ∩ (cid:16) B δ ( y ) \ ∪ J y ℓ =1 B Rr y,ℓk ( p y,ℓk ) (cid:17) can be written as two smooth graphs over T y V = { x n = 0 } with mean curvaturevectors pointing in opposite directions (in suitable normal coordinates { x i } centredat y ) with slope η = η ( k, R, δ ) satisfying lim δ → lim R →∞ lim k →∞ η = 0 .
3. The number of catenoid bubbles P y ∈ ∆ J y = J ≤ I , and index ( V ) := P Li =1 index ( V i ) ≤I − J .4. There is no loss of total curvature: lim k →∞ T ( M k ) = L X i =1 T ( V i ) + J T ( C n − ) where we have denoted by T ( V i ) and T ( M k ) the total curvature in ( N n , g ) of thehypersurfaces V i and M k , respectively. In particular, when n = 3 we have, forall k sufficiently large χ ( M k ) = L X i =1 χ ( V i ) − J. two parallel planes may include a single plane of multiplicity two . When k is sufficiently large, the surfaces M k of this subsequence are pair-wisediffeomorphic to one another. By a contradiction argument we immediately obtain the following
Corollary 5.3.
Given
H > there exists C = C ( N, Λ , I , H ) so that the collectionof H -hypersurfaces with index bounded by I and volume bounded by Λ has at most C distinct diffeomorphism types. Furthermore for any H -hypersurface M with the aboveindex and volume bounds we have Z M | A | n − ≤ C. In order to prove Theorem 5.2 we will repeatedly blow-up a sequence of H -hypersurfacesaccording to a given shrinking scale centred at a sequence of points. We first introducesome terminology for this, where here and throughout this section δ > < δ < inj N : Definition 5.4.
Let { M k } be a sequence of H -hypersurfaces in some closed Rieman-nian manifold N . Given x ∈ N we say that { ( x k , r k ) } ⊂ N × R > is a point-scalesequence for { M k } , based at x , if x k ∈ M k ∩ B δ ( x ) , x k → x and r k → .Given normal coordinates based at B δ ( x k ) we say that f M k ⊂ B R n δ/r k defined by f M k = M k /r k in these coordinates, is a blow up at scale ( x k , r k ) .We furthermore say that f M k converges non-smoothly to a plane of multiplicitytwo if there exists at least one, but finitely many points, where the convergence is smoothand graphical away from these points but not smooth and graphical across them. With Lemma 5.1 and this terminology we are now able to prove
Lemma 5.5.
Let V = ∪ Lℓ =1 V ℓ be a hypersurface effectively embedded in N with con-stant mean curvature H > and let { M k } k ∈ N be a sequence of H -hypersurfaces with sup k Ind ( M k ) < ∞ that H-converges to V with multiplicity one and let x ∈ t ( V ) . Let ( x k , r k ) be a point-scale sequence for { M k } based at x and f M k := M k /r k ⊂ R n a blowup along this scale. Then up to subsequence and on compact subsets, f M k converges toa limit f M ∞ , which must pass through the origin. This happens in one of three distinctways:1. smoothly and graphically to a catenoid2. non-smoothly to a plane of multiplicity two3. smoothly and graphically to a single plane or two parallel planes.In case 1 above, if ( z k , ρ k ) is another point-scale sequence based at x with r k ≤ ρ k and dist g ( x k , z k ) r k + ρ k ≤ C then taking a blow up c M k at scale ( z k , ρ k ) yields two further distinct possibilities (a) there exists some K with ρ k /r k ≤ K and c M k converges smoothly to a catenoid or1(b) ρ k /r k → ∞ and c M k converges non-smoothly to a plane with multiplicity two.Again in either case the limit c M ∞ of the c M k ’s passes through the origin.Proof. Since x ∈ t ( V ) then lim r → k V k ( B Nr ( x )) ω n − r n − = 2 . Now by varifold convergence, coupled with the monotonicity formula for CMC hyper-surfaces (see e.g. [24]), we know that for all ε > η > r > z k ∈ M k ∩ B Nη ( x ) and k sufficiently large then k M k k ( B Nr ( z k )) ≤ (2 + ε ) ω n − r n − for all r ≤ r .In particular if { ( z k , ρ k ) } is any point-scale sequence based at x thenlim sup k →∞ k M k k ( B Nρ k ( z k )) ω n − ρ n − k ≤ . (3)Now considering ( x k , r k ) and M k as in the statement of the lemma we will performa blow-up at this scale in normal coordinates centred at x k . Note that the metric on N in these coordinates can be written g k = g + O k ( | x | ), and we may suppress thedependance on k and simply write g k = g + O ( | x | ) where g denotes the Euclideanmetric. We can therefore consider f M k ⊂ (cid:16) B R n δ/r k (0) , e g k (cid:17) where e g k = g + r k O ( | x | ). We have that f M k is a potentially disconnected CMChypersurface with mean curvature H k = r k H →
0. Moreover by (3), for any
R > k H n − ( f M k ∩ B R ) ω n − R n − = lim sup k k M k k ( B NRr k ( z k )) ω n − ( Rr k ) n − ≤ . (4)It follows from a standard argument using Lemma 2.7 (following along the lines of e.g.[8, Theorem 2.4, Corollary 2.5]) that each component of f M k converges smoothly, awayfrom finitely many points, to a minimal limit M ∞ which has Euclidean volume growthand finite index by construction, and if the convergence is of multiplicity one then it issmooth everywhere. M ∞ has at most two ends by taking the limit as R → ∞ in (4) soby Lemma 5.1 it must be a catenoid or at most two parallel planes. Finally, appealingagain to the arguments in e.g. [8, Theorem 2.4, Corollary 2.5], if the convergence is notmultiplicity one (equivalently not smooth), then the limit must be (stable in compactsubsets, and therefore) a plane of multiplicity two.For the second part of the lemma, we first note that r k ≤ ρ k and dist g ( x k ,z k ) r k + ρ k ≤ C implies that B r k ( x k ) ⊂ B Cρ k ( z k ). We leave the final details to the reader as the21rguments are standard, noting that in case 1( b ) there must exist a sequence of pointsconverging to the origin in c M k where the second fundamental form blows up, and thusit cannot converge smoothly and graphically near the origin. Lemma 5.6.
Let V = ∪ Lℓ =1 V ℓ be a hypersurface effectively embedded in N with con-stant mean curvature H > and let { M k } k ∈ N be a sequence of H -hypersurfaces with sup k Ind ( M k ) < ∞ that H-converges to V with multiplicity one and let x ∈ t ( V ) . Sup-pose ( x k , r k ) is a point-scale sequence for { M k } based at x so that the blow-up at thisscale converges smoothly locally to a catenoid. Suppose further that there is a positivesequence ρ k → with ρ k /r k → ∞ and so that f M k := M k /ρ k converges smoothly to thedouble plane { x n = 0 } on B \ B η for all η > . Then there exists R < ∞ so that forall R ≥ R f M k ∩ ( B \ B Rs k ) where s k = r k /ρ k → can be written as a pair of graphs over { x n = 0 } with mean curvatures pointing inopposite directions and the graphs converge to zero in C as first k → ∞ and then R → ∞ .Proof. We will show that if t k → s k /t k → c M k = f M k /t k converges smoothly and graphically to { x n = 0 } on compact subsetsaway from the origin - in fact we need only check this in the region B \ B . Since theslope of the graph is scale-invariant, this will complete the proof.Lemma 5.5 tells us that (up to subsequence) c M k converges to some plane passingthrough the origin. By the hypotheses of the lemma and the choice of t k , this con-vergence happens smoothly with multiplicity two in compact subsets away from theorigin. In particular there is some ( n − E of R n so that c M k ∩ B \ B can be written as two graphs over E which are uniformly converging tozero as k → ∞ . We will prove below that E = { x n = 0 } ; this fact will be independentof the choice of sequence t k as above, and any subsequence.Without loss of generality we will prove what we need only for the top sheet, whosemean curvature points upwards. Denote by D ξ the closed ball of radius ξ centred at theorigin in { x n = 0 } . Let u k : D \ D / → R describe the top sheet of f M k (whose meancurvature points upwards) and notice that k u k k C l → l , and H k = ρ k H → f M k . Thus, using Proposition 5.7 and Remark 5.8, we can foliatea region of D / × [ − δ, δ ] by CMC graphs v hk : D / → R with boundary values givenby u k + h , h ∈ R . Notice that as k → ∞ we have that g k → g and u k → C l forall l which tells us that k v hk − h k C ,α → k → ∞ which follows from Proposition 5.7.Similarly as is [31, Lemma 3.1] (cf [8]) we can define a diffeomorphism of thiscylindrical region (via its inverse) F − k ( x , . . . , x n − , y ) = ( x , . . . , x n − , v y − h k k ( x , . . . , x n − ))22here h k → v − h k k (0 , . . . ,
0) = 0 (so that F k (0) = 0). Noticethat F k → Id as k → ∞ in C , so in particular the metric g k in these coordinates isalso converging to the Euclidean metric.We now work with these new coordinates ( x , . . . , x n − , y ), on which horizontalslices { y = c } provide a CMC foliation, and furthermore in these coordinates, the partof f M k described by u k takes a constant value h k at the boundary of D / . Withoutloss of generality (by perhaps choosing a sub-sequence) we assume that h k ≥ k (if h k ≤ /t k , and let c M k = f M k /t k ⊂ D / (2 t k ) × [ − δ/t k , δ/t k ] . Strictly speaking this is not the same c M k as before (which was a blow-up of f M k in adifferent coordinate system) but since our two choices of coordinates are asymptoticallyequivalent (as k → ∞ ), their limits are equal. In particular we still have that c M k ∩ B /B is uniformly graphical over E (equivalently defined in either coordinates), andour goal is to prove that E = { y = 0 } = { x n = 0 } . Notice that over ∂D / (2 t k ) , thetop sheet of c M k is described by a constant function of value b h k = h k /t k ≥
0, and thehorizontal slices { y = c } still provide a CMC foliation where the mean curvature of thefoliation equals that of the top sheet of c M k .For a contradiction suppose that E = { y = 0 } , which means thatmin c M k ∩ (( D / (2 tk ) \ D ) × R ) y < c M k is globally a horizontal slice { y = − c } , for some c <
0, which contradicts b h k ≥
0. Thus we must have E = { y = 0 } .Thus we have that, for k , R sufficiently large f M k ∩ ( B \ B Rs k ) is graphical over { x n = 0 } with slope η = η ( k, R ) → k → ∞ then R → ∞ . Proof of Theorem 5.2.
To begin we choose δ sufficiently small so that2 δ < min (cid:26) min ∆ ∋ y i = y j ∈ ∆ d g ( y i , y j ) , inj N (cid:27) and furthermore that B Nδ ( x ) ∩ V is stable for all x ∈ V . Towards the end of the proofwe will consider δ →
0, but for the majority of the proof we work with some fixed δ satisfying the above.From now on we work with a single y ∈ ∆ since we only need check the conclusionof the theorem for one such point chosen arbitrarily. Picking the smallest scale
Let r k = inf n r > | M k ∩ B r ( p ) is unstable for some p ∈ B δ ( y ) ∩ M k o . r k defined above, we can pick p k ∈ B δ ( y ) ∩ M k and δ > r k > M k ∩ B r k / ( p k ) is unstable.We must have p k → y since if not, we know that M k ∩ B d g ( p k ,y ) / ( p k ) convergessmoothly to V and thus is eventually stable inside all such balls by the choice of δ .Furthermore r k → L ∞ estimate on the second fundamentalform for M k ∩ B δ/ ( y ) and we reach a contradiction to the fact that y is a point of badconvergence.Thus ( p k , r k ) is a point scale sequence based at y and we let f M k be the blow-up atthis scale (see Definition 5.4).The metric on N in these coordinates can be written g k = g + O k ( | x | ), andwe may suppress the dependance on k and simply write g k = g + O ( | x | ) where g denotes the Euclidean metric. Thus we may consider f M k ⊂ ( B R n +1 δ/r k (0) , e g k ) where e g k = g +( r k ) O ( | x | ). By the choice of r k we have that f M k is a potentially disconnectedCMC hypersurface with mean curvature H k = r k H → f M k is stable inside every (Euclidean) ball of radius in ( B R n δ/r k , e g k ), byLemma 2.3 , it converges (up to subsequence) smoothly with multiplicity one to someminimal limit M ∞ in R n equipped with the Euclidean metric and by Lemma 5.5 M ∞ is either at most two planes or a catenoid. M ∞ cannot be a collection of one or two planes, as this would contradict theinstability hypothesis on balls of radius 2 centred at the origin: if M ∞ were a collectionof planes it would be strictly stable in any compact set, and this strict stability wouldeventually pass to f M k for large k . Thus we must have that M ∞ is a catenoid. Finally,since index ( M k ∩ B r k / ( p k )) ≥
1, for all large k and any ξ > index ( M k \ B ξ ( y )) ≤ index ( M k \ B r k / ( p k )) ≤ I − index ( V ) ≤ I −
1. This lat step follows since there exists ξ > k index ( M k \ ∪ y ∈ ∆ B ξ ( y )) ≥ index ( V \ ∪ y ∈ ∆ B ξ ( y )) = index ( V ) . (5)Here the index of any domain is computed with respect to Dirichlet boundary condi-tions. Picking further scales
Now let r k = inf n r > | B r ( p ) ∩ ( M k \ B r k ( p k )) is unstable for some p ∈ B δ ( y ) ∩ M k o . If lim inf k →∞ r k > r k → p k ∈ M k ∩ B δ ( y ) so that ( p k , r k ) is a point scale sequence based at y and24 M k ∩ B r k / ( p k )) \ B r k ( p k )) is unstable. As before, let f M k be the blow-up at thisscale which by Lemma 5.1 converges to at most two planes or a catenoid.There are two distinct cases:1. dist g ( p k ,p k ) r k + r k ≤ C < ∞ (i.e. B r k ( p k ) ⊂ B Cr k ( p k )) and f M k converges non-smoothlyto a double plane2. dist g ( p k ,p k ) r k + r k → ∞ and f M k converges smoothly to a catenoid.Indeed, in the first case we claim that the limit is attained non-smoothly and istherefore a double plane by Lemma 5.5. For a contradiction if the limit is attainedsmoothly we must have r k /r k ≤ K for some K and the limit is a catenoid by Case1( a ) of Lemma 5.5. However, by definition of r k we have λ ( M k ∩ B r k / ( p k )) < λ ( M k ∩ B r k / ( p k )) \ B r k ( p k )) <
0. These disjoint open regions of M k remainstrictly unstable for all k and thus, after blowing up at scale ( p k , r k ) pass to two non-empty disjoint open regions of the limiting catenoid Ω , Ω for which λ (Ω ) ≤ λ (Ω ) ≤
0. This contradicts the fact that the catenoid has index one.In the second case we invite the reader to blow up precisely as we did for ( r k , p k )and see that f M k converges smoothly to a catenoid: at this blow up scale we once againhave that, on compact subsets, f M k is stable on all balls of radius and the first formingcatenoid is disappearing at infinity.We wish to keep track of this point-scale sequence in either scenario, but in caseone, the blow-up procedure produces no extra catenoid so we mark this sequence forremoval later. In either case we conclude similarly as before that index ( V ) ≤ I − j − { ( r ik , p ik ) } j − i =1 satisfyinga) for each 2 ≤ i ≤ j − r ik → p ik → y b) Denoting U i − = ∪ i − s =1 B r sk ( p sk )( M k ∩ B r ik / ( p ik )) \ U i − is unstablec) index ( M k \ U j − ) ≤ I − ( j −
1) and thus index ( V ) ≤ I − ( j −
1) by (5)Furthermore we suppose there are two distinct cases:1. There exists
C < ∞ and m < i so that B r mk ( p mk ) ⊂ B Cr ik ( p ik ) and blowing up atthis scale we converge non-smoothly to a double plane2. min m | B r ( p ) ∩ ( M k \ U j − ) is unstable for some p ∈ B δ ( y ) ∩ M k o .
25f lim inf k →∞ r jk > r jk → p jk ∈ M k ∩ B δ ( y ) so that( M k ∩ B r jk / ( p jk )) \ U j − is unstableand show that once again we are in case 1. or 2. above (we leave the details to thereader) and this time index ( M k \ U j ) ≤ I − j implying index ( V ) ≤ I − j . In short, wesatisfy conditions a ) − c ) and the j th sequence also satisfies condition 1. or 2.This process must stop eventually (after at most I iterations) and we can moveon to the neck analysis, noting that if J y is the total number of distinct point-scalesequences forming at y (distinct in the sense that we have removed all point-scalesequences satisfying case 1), then in particular have index ( V ) ≤ I − J y which is part3 of the theorem.Before we move on let us now throw away all the marked sequences (those satisfyingcondition 1 above), since blowing up at these scales means that we see only a doubleplane passing through the origin as a weak limit, and we have finished proving part 1of the theorem. Part 2 of the theorem
If there is only one catenoid forming at y (i.e. J y = 1) wefirst pick an arbitrary ρ k → ρ k /r k → ∞ and we first apply Lemma 5.6 to theblow up f M k at scale ( p k , ρ k ) to conclude that f M k ∩ ( B \ B Rr k /ρ k ) is uniformly graphicalover a fixed plane E (in these coordinates) with slope converging to zero as k → ∞ and then R → ∞ .We now consider the point scale sequence given by ( p k , δ ) and the correspondingblow up ˇ M k = M k /δ . Notice that, for any δ > T y V is parallel to { x n = 0 } and that for any fixed µ <
1, ˇ M k ∩ B \ B µ canbe written as two graphs over { x n = 0 } with slope η → k → ∞ and δ →
0. The reader can check that (by following the steps in the proof of Lemma5.6) ˇ M k ∩ B \ B ρ k /δ is uniformly graphical over { x n = 0 } with slope converging to zeroas k → ∞ and δ →
0. Thus the orientation of the plane { x n = 0 } is passed downto the next scale (so E = { x n = 0 } above), and we recover that ˇ M k ∩ B \ B Rr k /δ isuniformly graphical over { x n = 0 } (equivalently over T y V ) with slope converging tozero as k → ∞ , R → ∞ and finally δ → M k ∩ ( B δ ( p k ) \ B Rr k ( p k )) is uniformly graphicalover T y V with slope η ( k, R, δ ) converging to zero as k → ∞ , R → ∞ and δ → T y V )is passed down to each smaller scale: the ends of the catenoids are always parallel to T y V . 26 he neck analysis when J y > ρ k = 2 max j> dist( p k , p jk ) which gives inparticular that ρ k /r k → ∞ and Lemma 5.5 guarantees that by blowing up at scale( p k , ρ k ) we see weak convergence of f M k = M k /ρ k to a double plane. Furthermorethere are J y catenoid bubbles forming inside the ball of radius 1 / R n \ B .In exactly the same fashion as above we now consider ˇ M k = M k /δ the blow up atscale ( p k , δ ). After rotating our coordinates so that T y V is parallel to { x n = 0 } , (andagain following the steps in the proof of Lemma 5.6) we have that ˇ M k ∩ B \ B ρ k /δ isuniformly graphical over { x n = 0 } .Going back to f M k we now successively apply Lemma 5.6 to each bubble forminginside B at scale ( p k , ρ k ) to conclude part 2 of the theorem. No loss of total curvature, part 4 of the theorem
By smooth, multiplicity oneconvergence away from ∆ we know thatlim δ → lim k →∞ Z M k \∪ y ∈ ∆ B δ ( y ) | A k | n − → X i Z V i | A | n − = Z V | A | n − . (6)Furthermore, by the scale invariance of the total curvature, given any point-scale se-quence ( p ℓ,yk , r ℓ,yk ) corresponding to a catenoid we havelim R →∞ lim k →∞ X y ∈ ∆ J y X ℓ =1 Z M k ∩ B Rrℓ,yk ( pℓ,yk ) | A k | n − = J T ( C n − ) . (7)It thus remains to check that, in each degenerating neck region between the bubblescales we havelim δ → lim R →∞ lim k →∞ Z M k ∩ ( ∪ y ∈ ∆ ( B δ ( y ) \∪ Jyℓ =1 B Rrℓ,yk ( p ℓ,yk )) | A k | n − = 0 . (8)Given that we know such regions are uniformly graphical over the limit, with slope η → | ∆ b g k u k | = (cid:12)(cid:12)(cid:12)b g αβk Γ k ( b u k ) n +1 jl ∂ b u jk ∂x α ∂ b u lk ∂x β + b g αβk ( g k ) ij ∂ b u jk ∂x α ∂ b u lk ∂x β H (cid:12)(cid:12)(cid:12) ≤ Cη ( | b u k | + H ) , since we are working with CMC H = 0. This makes no difference to the remainder ofthe argument so we leave it to the interested reader to follow up. Finite diffeomorphism type, part 5 of the theorem
Notice that we have im-plicitly constructed a finite open cover of ∪ k M k so that in each element of the cover the M k ’s are pair-wise graphical over one-another, for sufficiently large k . Thus the M k ’sare globally graphical over one-another and have the same diffeomorphism type.27 .1 Local CMC foliations Here we wish to show the existence of local CMC foliations by disks for metrics suf-ficiently close to the Euclidean metric, and mean curvature sufficiently small. Let D ⊂ R n − be the closed unit (Euclidean) ball and C = D × R ⊂ R n . For anyfixed α ∈ (0 ,
1) denote by G the collection of C ,α Riemannian metrics on C so thatwe can view G = C ,α ( C, R ) where R is the open set of symmetric, positive-definite n × n -matrices. Let W = C ,α ( D ) and U = C ,α ( D ) = { u ∈ W : u ≡ ∂D } .For ( t, g, w, u ) ∈ R × G × W × U we denote H g ( t + w + u ) the g -mean curvature ofthe graph t + w + u with respect to the upward pointing unit normal N g ( t + w + u ).We consider Φ : R × G × W × U × C ,α ( D ) → C ,α ( D ) defined byΦ( t, g, w, u, H ) = H g ( t + w + u ) − H (9)and notice that Φ is C with Φ( t, g E , , ,
0) = 0 . Here g E ∈ G denotes the Euclidean metric on C . We now consider the derivative withrespect to u at u = 0, D Φ( t, g E , , ,
0) : C ,α ( D ) → C ,α ( D ) where for v ∈ C ,α ( D )we have D Φ( t, g E , , , v ] = ∂∂h h =0 H g E ( t + hv ) . This is equivalent to considering an infinitesimal variation of the flat disc by the am-bient vector field V ( x , . . . , x n ) = (0 , . . . , , v ( x , . . . , x n − )) ∈ C ,α ( C ), whose normalcomponent is given by h V, N g E ( t + u H ) i = v . Thus we have D Φ( t, g E , , , v ] = ∆ v (10)which is a Banach space isomorphism, noting that by Schauder theory we have k D Φ( t, g E , , , − [ f ] k C ,α ( D ) ≤ C k f k C ,α ( D ) . In particular for each fixed t there exists ε > C mapping U : ( t − ε, t + ε ) × B R ε ( g E ) × B Wε (0) × B C ,α ε (0) → B Uδ (0) (11)so that whenever( s, g, w ) ∈ ( t − ε, t + ε ) × B R ε ( g E ) × B Wε (0) × B C ,α ε (0)then Φ( s, g, w, U ( s, g, w, H ) , H ) = 0. In particular when g, w, H are fixed, s + w + U is a graphical foliation with mean curvatures given by the function H with boundaryvalues given by s + w . By uniqueness of such H -graphs we can carry out this localfoliation for any t noting that whenever two leaves have the same boundary values,they must coincide. Thus we have proven:28 roposition 5.7. Let D ⊂ R n − be the closed unit (Euclidean) ball and C = D × R ⊂ R n . Then there exists ε > so that for any w ∈ C ,α ( D ) , H ∈ C ,α ( B ) andRiemannian metric g on C satisfying k w k C ,α + k g − g E k C ,α + k H k C ,α < ε there exists a C ,α foliation of graphs u : R → C ,α ( D ) with g -mean curvature H pointing upwards, and for each t ∈ R , u ( t ) has boundary values t + w . Furthermore k u k C ,α depends on t, w, g and H in a C way. Remark 5.8.
If we consider g , w and H to have higher regularity we can pass thisonto the foliation by the usual regularity results: in particular if g , is C l,α for l ≥ then Φ H is C l − and we can find a C l − CMC foliation, i.e. u : R → C ,α is C l − in t . Theodora Bourni, [email protected] of Mathematics, University of Tennessee, U.S.A.Ben Sharp, [email protected] of Mathematics, University of Leeds, U.K.Giuseppe Tinaglia, [email protected] of Mathematics, King’s College London, U.K.
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