A dichotomy for minimal hypersurfaces in manifolds thick at infinity
aa r X i v : . [ m a t h . DG ] F e b A DICHOTOMY FOR MINIMAL HYPERSURFACES INMANIFOLDS THICK AT INFINITY
ANTOINE SONG
Abstract.
Let ( M n +1 , g ) be a complete ( n + 1)-dimensional Riemann-ian manifold with 2 ≤ n ≤
6. Our main theorem generalizes the solutionof Yau’s conjecture for minimal surfaces and builds on a result of Gro-mov. Suppose that (
M, g ) is thick at infinity, i.e. any connected finitevolume complete minimal hypersurface is closed. Then the following di-chotomy holds for the space of closed hypersurfaces in M : either thereare infinitely many saddle points of the n -volume functional, or there isnone.Additionally, we give a new short proof of the existence of a finitevolume minimal hypersurface if ( M, g ) has finite volume, we check Yau’sconjecture for finite volume hyperbolic 3-manifolds and we extend thedensity result of Irie-Marques-Neves when (
M, g ) is shrinking to zero atinfinity.
Introduction
The search for minimal hypersurfaces in compact manifolds has enjoyedsignificant progress recently, thanks to the development of various min-maxmethods, such as the systematic extension of Almgren-Pitts’ min-max the-ory [37] led by Marques and Neves [27, 29, 28, 30], the Allen-Cahn approach[18, 13, 5], or others [39, 23, 8, 48, 38, 36]. One central motivation was thefollowing conjecture of S.-T. Yau:
Yau’s conjecture [46]: In any closed three-dimensional manifold, thereare infinitely many minimal surfaces.Strong results implying the conjecture were obtained for generic met-rics by Irie-Marques-Neves [22] (see [31] for a quantified version), Chodosh-Mantoulidis [5], X. Zhou [49], Y. Li [24]. Concurrently to these results, theconjecture for non-generic metrics was treated with a different line of argu-ments. When the manifold satisfies the “Frankel property”, it was solved byMarques-Neves in [29]. We recently settled the general case in [42], wherewe localized min-max constructions appearing in [29] to some compact man-ifold with stable minimal boundary by introducing a non-compact manifoldwith cylindrical ends.
The author was partially supported by NSF-DMS-1509027.
On the other hand, results about minimal hypersurfaces in complete non-compact manifolds are comparatively few and far between, and most ofthem are existence results. We give here a non exhaustive list. In [9, 10],Collin-Hauswirth-Mazet-Rosenberg constructed one closed embedded min-imal surfaces in any finite volume hyperbolic 3-manifold; there is also thework of Z. Huang and B. Wang [20], and of Coskunuzer [11]. In [33], Mon-tezuma showed that a strictly mean concave compact domain in a completemanifold intersects a finite volume embedded minimal hypersurface. In [17],Gromov proved the following existence theorem that we interpret as theanalogue of Almgren-Pitts existence result [37] for non-compact manifolds:
Gromov’s result [17]: In a complete non-compact manifold M , eitherthere is an embedded finite volume complete minimal hypersurface, or thereis a possibly singular strictly mean convex foliation of any compact domainof M .In [2], Chambers and Liokumovich showed the existence of a finite volumeembedded minimal hypersurface in finite volume complete manifolds; in factthey proved the existence of such a minimal hypersurface if there is a regionwhose boundary is, say, ten times smaller than its width. In asymptoticallyflat 3-manifolds, Chodosh and Ketover constructed minimal planes in [4],using a degree argument (see Mazet-Rosenberg [32] for generalizations).The goal of this paper is to propose a relevant generalization of the so-lution of Yau’s conjecture to non-compact manifolds, by building on Gro-mov’s result. Motivations came from our solution of the conjecture when theFrankel property is not satisfied [42], where we perform min-max in a non-compact manifold with cylindrical ends. Besides some classes of manifoldsnaturally contain non-compact manifolds, for instance finite volume hyper-bolic 3-manifolds. The non-compact situation substantially differs from thecompact case: there are many non-compact manifolds without any closed(or finite volume) minimal hypersurfaces. For any integer m >
0, it is easyto construct a metric on say S × R with exactly m closed minimal surfaces.That metric can look like a long tube which gets thinner around S × { } ,and the minimal surfaces are S × { } and some other slices S × { t } whichare degenerate stable. At first sight, it seems hard to come up with essen-tially different examples of manifolds that would contain only finitely manyclosed minimal hypersurfaces. In our main result, we confirm this intuitionfor manifolds called “thick at infinity”, which we define next. The class T ∞ of manifolds thick at infinity. Before stating our maintheorem, we recall the notion of “thickness at infinity” introduced by Gro-mov [17]. Unless mentioned, we consider hypersurfaces without boundary.
Definition 0.1.
Let ( X n +1 , g ) be a complete ( n + 1) -dimensional Riemann-ian manifold. ( X, g ) is said to be thick at infinity (in the weak sense) if any INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 3 connected finite volume complete minimal hypersurface in ( X, g ) is closed.We denote by T ∞ the class of manifolds thick at infinity. In [17], Gromov actually uses a slightly stronger notion of thickness atinfinity, since he asks that any connected finite volume minimal hypersurfacewith maybe non-empty compact boundary is compact.The property of “thickness at infinity” is checkable. Instances are givenin [17, Section 1.3], another more general example is the condition ⋆ k in [33].Special cases of the previous conditions include coverings of closed manifoldsand asymptotically flat manifolds.Note that M can be thick at infinity and at the same time “thin” in acertain sense. Indeed, using the monotonicity formula, it is easy to constructa warped product metric g t ⊕ dt on a cylinder N n × R (where N is any closed n -dimensional manifold) such that ( N n × R , g t ⊕ dt ) is thick at infinity, hasfinite volume, and the n -volume of the cross section N × { t } decreases tozero as t → ±∞ . A zero-infinity dichotomy for manifolds thick at infinity.
Almost bydefinition, closed minimal hypersurfaces are critical points of the n -volumefunctional. By the properties of the Jacobi operator, which encodes thesecond variation of the n -volume at a minimal hypersurface, the space ofdeformations that do not increase the area at second order is finite dimen-sional. It is natural to define saddle points of the n -volume functional (orsimply saddle point minimal hypersurfaces ) as follows. Consider a connectedclosed embedded minimal hypersurface Γ. If it is 2-sided then we call it asaddle point minimal hypersurface if there is a smooth family of hypersur-faces { Γ t } t ∈ ( − ǫ,ǫ ) ( ǫ >
0) which are small graphical perturbations of Γ = Γ so that { Γ t } t ∈ ( − ǫ, and { Γ t } t ∈ (0 ,ǫ ) are on different sides of Γ and distinctfrom Γ, and Vol n (Γ) = max t ∈ ( − ǫ,ǫ ) Vol n (Γ t ) . If Γ is 1-sided, we call it a saddle point minimal hypersurface if its connecteddouble cover is a saddle point minimal hypersurface in a double cover of theambient manifold. Note that if the metric is bumpy (i.e. no closed minimalhypersurface has a non-trivial Jacobi field), then saddle point minimal hy-persurfaces are exactly unstable 2-sided closed minimal hypersurfaces and1-sided closed minimal hypersurfaces with unstable double cover. By “com-pact domain”, we mean a compact ( n + 1)-dimensional submanifold of M with smooth boundary.Our main theorem is a dichotomy for the space of closed hypersurfacesembedded in a manifold thick at infinity. It says that either this space hasinfinite complexity from a Morse theoretic point of view, or its structure islocally simple. Theorem 1.
Let ( M, g ) be an ( n + 1) -dimensional complete manifold with ≤ n ≤ , thick at infinity. Then the following dichotomy holds true: ANTOINE SONG (1) either ( M, g ) contains infinitely many saddle point minimal hyper-surfaces,(2) or there is none; in that case for any compact domain B , there is anembedded closed area minimizing hypersurface Σ B such that B \ Σ B has a singular weakly mean convex foliation. We make the following comments, which will be developed in Sections 2,3 and 4: • In the second case, the minimal hypersurface Σ B may be empty ordisconnected. The foliation comes from the mean curvature flow soit has the corresponding regularity [44], and it is shrinking towardsΣ B . • This theorem still holds if M has minimal boundary and if eachcomponent of ∂M is compact. Closed manifolds are trivially in T ∞ .We will see in Section 2 that 1-parameter min-max produces a saddlepoint minimal hypersurface, thus Theorem 1 implies the existence ofinfinitely many saddle point minimal hypersurfaces. This fact doesnot follow directly from the solution of Yau’s conjecture [29][42]. • Finite volume hyperbolic 3-manifolds do not belong to T ∞ in gen-eral. Nevertheless we will show in Section 4 that they satisfy Yau’sconjecture since they contain infinitely many saddle point minimalhypersurfaces, extending the existence result of Collin-Hauswirth-Mazet-Rosenberg [9, 10]. • The situation for geodesics in surfaces is different: some 2-spherescontain only three simple closed geodesics, and immersed closedgeodesics in hyperbolic surfaces are all strictly stable.In the process of proving the Theorem 1, we will explain the following localversion of Gromov’s result for complete manifolds that are not necessarilyin T ∞ . Theorem 2.
Let ( M, g ) be an ( n + 1) -dimensional complete manifold with ≤ n ≤ , and let B be a compact domain. Then(1) either M contains a complete embedded minimal hypersurface inter-secting B and with finite n -volume,(2) or B has a singular strictly mean-convex foliation. This theorem is essentially already contained in [17], but we find it usefulfor the reader to present a detailed proof with some new arguments, forinstance the use of Marques-Neves lower index bound [28] and the meancurvature flow. We think that after some technical improvements, our proofshould also cover higher dimensions (the minimal hypersurface then mayhave a codimension at least 7 singularity set).Let us list a few corollaries (see Corollary 5 for more details). If M hasfinite volume, then a complete finite volume embedded minimal hypersurfaceexists (that was known since [2]). Moreover, if there is a compact subset X ⊂ M whose boundary is mean concave in the sense that the mean curvature INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 5 vector is pointing outside of X , then such a hypersurface also exists andintersects X . In particular this recovers a result of Montezuma [33]. A density result `a la Irie-Marques-Neves.
A complete ( n +1)-dimensionalmanifold ( M, g ) is said to has a thin foliation at infinity if there is a properMorse function f : M → [0 , ∞ ) so that the n -volume of the level sets f − ( t )converges to zero as t goes to infinity. The relevant topology on the spaceof complete metrics on M is the strong (Whitney) C ∞ -topology. Let F thin be the family of complete metrics on M with a thin foliation at infinity;it is an open subset for the strong topology. Similarly, the intersection F thin ∩ Int( T ∞ ) is a non-empty open subset for that topology (here Int( T ∞ )denotes the interior of T ∞ in the space of complete metrics). The followingtheorem generalizes the density theorem of Irie, Marques and Neves to thesemetrics: Theorem 3.
Let M be an ( n + 1) -dimensional manifold with ≤ n ≤ .(1) For any metric g in a C ∞ -dense subset of F thin , the union of completefinite volume embedded minimal hypersurfaces in ( M, g ) is dense.(2) For any metric g ′ in a C ∞ -generic subset of F thin ∩ Int( T ∞ ) , theunion of closed embedded minimal hypersurfaces in ( M, g ′ ) is dense. The proof borrows an idea of Irie, Marques and Neves in [22], where theyuse an elegant argument based on the Weyl law for the volume spectrumproved by Liokumovich, Marques and Neves [25]. Many non-compact man-ifolds of finite volume do not obey the Weyl law, even if all the min-maxwidths are finite (see Remark 5.1 for an informal justification). Thus, wehave to find a more robust property of the min-max widths which in factgives an alternative argument even in the compact case, not based on theWeyl law. On the other hand, the Weyl law seems essential in the quantifiedresult we obtained with Marques and Neves [31] about the generic equidis-tribution of a sequence of minimal hypersurfaces. Another remark is thatwe cannot prove the result for a C ∞ -generic subset of F thin , but only for a C ∞ -dense subset, because non-compact minimal hypersurfaces may appearand structural results like White’s bumpy metric theorems [43, 45] becomefalse. Organisation.
In Section 1, we reprove a local version of Gromov’s result[17] and derive a few corollaries. After explaining how to construct saddlepoint minimal hypersurfaces with 1-parameter min-max in Section 2, weshow the zero-infinity dichotomy for manifolds thick at infinity in Section3. We also check that finite volume hyperbolic 3-manifolds satisfy Yau’sconjecture in Section 4, and extend in the last section the density result ofIrie-Marques-Neves.
Acknowledgement.
I am grateful to my advisor Fernando Cod´a Marquesfor his crucial guidance. I thank Yevgeny Liokumovich for explaining [2] tome and mentioning [17], [35]. I am thankful to Misha Gromov for exchanges
ANTOINE SONG about [17]. I also want to thank Franco Vargas Pallete for discussing withme Yau’s conjecture for finite volume hyperbolic 3-manifolds, a result ofwhich he was also aware.1.
Local existence of finite volume minimal hypersurfaces
Let ( M n +1 , g ) be a complete possibly non-compact ( n + 1)-dimensionalRiemannian manifold. We will use in this section a local version of Almgren-Pitts theory; definitions and some relevant results are stated in Appendix B.We will also need the mean curvature flow, in its level set flow formulation,whose basic properties are recalled in Appendix C. Strict and weak meanconvexity in the sense of level set flow are also defined there. By “compactdomain”, we mean a compact ( n + 1)-dimensional submanifold of M withsmooth boundary.We say that B has a singular (strictly) mean convex foliation if there isa bigger compact domain B containing B and endowed with a metric g ′ coinciding with g on B , so that there is a family of closed subsets of B , { K t } t ∈ [0 , , satisfying: • K = B , K ∩ B = ∅ , • ∂K t has the regularity of the level set flow, • ∂K t is (strictly) mean convex for g ′ in the sense of mean curvatureflow.The main theorem of this section is essentially proved in [17]. We givehere a detailed proof with new arguments, especially for the min-max part.Another more formal difference is the alternative use of level set flow insteadof constructing foliations “by hand” (already suggested by B. Kleiner, see[17]). Here M can be compact or non-compact. Theorem 4 (Local version of Gromov’s theorem) . Let ( M, g ) be an ( n + 1) -dimensional complete manifold with ≤ n ≤ , and let B be a compactdomain. Then(1) either M contains a complete embedded minimal hypersurface inter-secting B and with finite n -volume,(2) or B has a singular strictly mean-convex foliation. That theorem immediately implies the following corollaries which wereproved in [2] and [33] respectively (it seems that the relation with [17] wasnot in the literature).
Corollary 5.
Let M be as in the previous theorem.(1) If M has finite volume or more generally if there is an exhaustion X ⊂ ... ⊂ X i ⊂ ... of M be compact subsets with smooth boundariessuch that lim i →∞ Vol n ( ∂X i ) = 0 , then there is a complete finite volume embedded minimal hypersur-face. INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 7 (2) If M contains a compact subset X with mean concave smooth bound-ary (the mean curvature vector is non-zero pointing outwards), thenthere a complete finite volume embedded minimal hypersurface inter-secting X . We will say that a hypersurface S embedded in ( N, g ) is locally (resp. globally ) area minimizing if S is a minimal hypersurface and if any hyper-surface isotopic to S in a neighborhood of S in N (resp. any hypersurfacein the same Z -homology class as S ⊂ N ) has n -volume at least Vol n ( S ).Before giving the proof of Theorem 4 and Corollary 5, we need the followinglocal min-max theorem for non-bumpy metrics. In our setting, the width W of a compact manifold ( N, g ) is defined in Appendix B, (23). Condition[M] and Type I, II, III stable minimal hypersurfaces are introduced in thediscussion of Appendix A
Proposition 6.
Let ( N n +1 , g ) be a compact manifold with minimal bound-ary, with ≤ n ≤ , such that ∂N is locally area minimizing inside N .Then there is a closed embedded minimal hypersurface Γ inside the interior Int( N ) whose index is at most one and whose n -volume is bounded by thewidth W of ( N, g ) .Proof. Since from [34] the width W of ( N, g ) is larger than the n -volumeof any connected component of ∂N , by Lemma 22 in Appendix A and thediscussion following it, we can suppose that each component of ∂N satisfiesCondition [M] and is either strictly stable or degenerate stable of Type II.Then by Lemma 23 (1) in Appendix A, there is a sequence of metrics { h ( q ) } converging to g such that for each h ( q ) , ∂N is strictly stable and has aneighborhood N q foliated by hypersurfaces which are strictly mean convex(except ∂N of course). The thickness of this neighborhood N q essentiallydoes not depend on q . Consider a sequence of bumpy metrics g q convergingto g , so that g q is close to h q and ( N q , g q ) contains no minimal hypersurfaceexcept ∂N . Now we apply the local min-max theorem for bumpy metricsto ( N, g q ), see Theorem 24 in Appendix B. For each q we have a closedembedded minimal hypersurface Γ q inside the interior Int( N ) with index atmost one and n -volume bounded by the width of ( N, g q ), intersecting N \N q .Sending q to infinity, Γ q converges (in the varifold sense) subsequently by[41] to a minimal hypersurface embedded in N , with index at most one and n -volume at most W . It cannot be contained in the boundary ∂N by themonotonicity formula and the fact that Γ q ∩ N \N q = ∅ for all q . (cid:3) Proof of Theorem 4.
Fix p ∈ M . Let B be a compact domain which wecan assume to be connected, and let D be a compact domain containing ageodesic ball B r ( p ) which itself contains the closure of B in M . We fix r temporarily but at the end of the argument we will make r go to ∞ . Wemodify the metric in a thin neighborhood of ∂D to make it mean convex(the mean curvature vector points inwards). Let D ′ ⊂ D be a domain ANTOINE SONG containing B with C , weakly mean convex boundary (the mean curvatureis nonnegative), consider a closed minimal hypersurface S embedded insideInt( D ′ ) \ B and let D be the metric completion of D ′ \ S . B is isometricallyembedded inside D .We consider the following constrained minimization problem: minimizethe n -volume of ∂B ′ over open sets with rectifiable boundary B ′ containing B , such that there is a family of integral currents { b t } t ∈ [0 , ⊂ I n +1 ( D ; Z )verifying: • { ∂b t } t ∈ [0 , ⊂ Z n ( D ; Z ) is continuous in the F -topology, • spt( ∂b ) = ∂B , spt( ∂b ) = ∂B ′ , • for all t ∈ [0 , B ⊂ spt( b t ) ⊂ D • for all t ∈ [0 , M ( ∂b t ) ≤ Vol n ( ∂B ) + 1.This is a constrained Plateau problem since there is an n -volume constraint(which does not affect regularity) and a geometric constraint given by B , D (which does). A solution of this minimization problem exists by weak meanconvexity of ∂ D , by compactness of cycles for the flat topology and by inter-polation results [28, Proposition A.2]: it gives a C , -hypersurface Γ smoothexcept maybe at points touching ∂B and with n -volume at most Vol n ( ∂B ).Let B ′′ be the metric completion of the component of D \ Γ containing B .Given B ⊂ B ′′ as above, two possibilities can occur:(i) either one component A of ∂B ′′ is strictly mean convex in the senseof mean curvature flow,(ii) or the boundary ∂B ′′ is a smooth minimal hypersurface.In the first case, the strictly mean convex component A has to touch ∂B .We can run the mean curvature flow starting from A (see Appendix C) andget { A t } t ≥ : either the level sets A t are all strictly mean convex and sweepout the whole domain B (the conclusion of the theorem is then true) orit converges to a non empty stable closed embedded minimal hypersurface S intersecting B and having n -volume less than that of ∂B . In the lattersituation occurs for every r and some D , B ′′ , then by sending r to infinity(i.e. D covers larger and larger regions of M ), we get subsequently a lim-iting complete embedded finite volume minimal hypersurface ([41]) whichintersects B . This proves the theorem in the first case.In the second case we can assume that ∂B ′′ does not touch B (otherwisethe theorem is proved). It means we can suppose that ∂B ′′ is locally areaminimizing inside B ′′ . If for any choice of D and B ′′ , this situation occursthen we can consider such a B ′′ of minimal volume, where the minimum istaken over all B ′′ obtained from the constrained minimization problem in achoice of D constructed as previously. Such a minimizer exists by compact-ness because of the n -volume bound on Γ and the stability of ∂B ′′ (see [41]).This manifold B ′′ is compact. Given such a minimizer B ′′ , we remove a max-imal number of disjoint 1-sided minimal hypersurfaces S , ..., S q and 2-sidednon-separating minimal hypersurfaces T , ..., T r contained in the interior of INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 9 B ′′ \ B so thatVol n ( ∂B ′′ ) + q X i =1 n ( S i ) + r X i =1 Vol n ( T i ) ≤ Vol n ( ∂B ) + 1and consider the metric completion B of B ′′ \ (cid:0) S qi =1 S i ∪ S ri =1 T i (cid:1) . This man-ifold B is compact and has a non empty boundary. Besides, the originaldomain B is isometrically embedded in B . We used similar ideas of consid-ering a “core” in [42].We can apply local min-max to B : by Proposition 6, we get a closed con-nected embedded minimal hypersurface S inside the interior Int( B ), whichhas Morse index at most one. Moreover the width of B and thus the n -volume of S are bounded in terms of B only. To see this, it suffices to un-derstand that one can deform ∂B inside B continuously in the F -topologyto ∂ B , such that along the deformation the n -volume of the hypersurfaces issay less than Vol n ( ∂B ) + 1; this in turn follows from the volume minimalityproperty of B and [28, Proposition A.2]. We wish to show that S intersects B (for any choice of radius r ) because then the theorem is proved by takinga limit of such hypersurfaces S using [41], as r → ∞ . To argue towardsa contradiction, assume that S ⊂ Int( B ) does not intersect B . Supposefirst that S is 2-sided: then consider the metric completion of B \ S . If S is 2-sided separating B into two components then it is clear that one canfind a competitor ˆ B to B contradicting the minimality of its volume. If S is 1-sided or 2-sided non-separating, then by removing S from B and tak-ing the metric completion B ′ , we could run again the minimization processdescribed earlier to ∂B inside B ′ . Either we find a minimizer not entirelycontained in the boundary ∂ B ′ or a minimizer is contained in ∂ B ′ (thenwe have Vol n ( ∂ B ′ ) ≤ Vol n ( ∂B ) + 1). In any case, this would contradicteither the minimality of the volume of B or the maximality of the numberof boundary components of ∂ B , since ∂ B ′ has more boundary componentsthan ∂ B . (cid:3) Proof of Corollary 5.
The first item (1) follows from the following argument.Note that the T i X i contains a small fixed ball b . By Theorem 4, if theconclusion we want is not true then for each i there is a family of closed sets { K t } ≤ t ≤ such that X i ⊂ K , X i ∩ K = ∅ and { ∂K t } is a mean convexfoliation of K for a metric that coincides with g near X i . By a first variationcomputation, the n -volume of ∂ ( K t ∩ X i ) decreases in t , for all i . But since ∂ ( K t ∩ X i ) sweepouts the ball b , the maximum n -volume of ∂ ( K t ∩ X i ) when t ∈ [0 , i . Thiscontradicts the facts that lim i →∞ Vol n ( ∂X i ) = 0 and that Vol n ( ∂ ( K t ∩ X i ))decreases in t .Similarly, for the second item, if the desired conclusion does not holdthen by Theorem 4 (2), there is a family of closed sets { K t } ≤ t ≤ such that X ⊂ K , X ∩ K = ∅ and { ∂K t } is a mean convex foliation of K for a metric that coincides with g near X . But this cannot happen if X is strictlymean concave by the maximum principle. (cid:3) Local min-max and saddle point minimal hypersurfaces
Consider a complete manifold ( M n +1 , g ). We define saddle points of the n -volume functional (or simply saddle point minimal hypersurfaces ) as follows.Let Γ be a connected closed embedded minimal hypersurface. If it is 2-sidedthen we call it a saddle point if there is a smooth family of hypersurfaces { Γ t } t ∈ ( − ǫ,ǫ ) ( ǫ >
0) which are small graphical perturbations of Γ = Γ sothat { Γ t } t ∈ ( − ǫ, and { Γ t } t ∈ (0 ,ǫ ) are on different sides of Γ and distinct fromΓ, and Vol n (Γ) = max t ∈ ( − ǫ,ǫ ) Vol n (Γ t ) . If Γ is 1-sided, we call it a saddle point if its connected double cover is asaddle point in a double cover of the ambient manifold. Note that if the met-ric is bumpy (i.e. no closed minimal hypersurface has a non-trivial Jacobifield), then saddle point minimal hypersurfaces are exactly 2-sided unsta-ble closed embedded minimal hypersurfaces and 1-sided closed embeddedminimal hypersurfaces with unstable double cover.In this section, we explain how to construct saddle points for generalmetrics from a localized 1-parameter min-max procedure. When the metricis bumpy, this was achieved by the index bounds of Marques-Neves in [28].Most notations are recalled in Appendix B.2.1.
Deformation Theorems.
In this subsection, we consider a compactmanifold (
N, g ). Let { Φ i } i ∈ N be a sequence of continuous maps from [0 , Z n ( N ; F ; Z ). Set L = L ( { Φ i } i ∈ N ) := lim sup i →∞ sup x ∈ [0 , M (Φ i ( x ))and suppose that lim inf i →∞ ( L − max j =0 , M (Φ i ( j ))) >
0. To simplify thepresentation, let us assume that each component of ∂N is minimal and eitherstrictly stable or degenerate stable of Type II.We first need a 1-parameter version of Deformation Theorem A of [28] fornon-bumpy metrics. Let S ( L ) be the family of stationary integral varifoldsin V n ( N ) of total mass L with support a smooth closed embedded minimalhypersurface in N , which are 2-unstable (see [28, Definitions 4.1, 4.2]). Theunstable components of spt( V ), where V ∈ S ( L ), are inside the interior of N by assumption on the boundary ∂N . Theorem 7 (Deformation Theorem A, [28]) . Given { Φ i ( x ) } i ∈ N and a com-pact set K ⊂ V n ( N ) which is at positive F -distance of S ( L ) ∪ | Φ i | ([0 , forall i large, there exist another sequence { Ψ i } i ∈ N such that(i) Ψ i is homotopic to Φ i with fixed endpoints in the F -topology for all i ∈ N , INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 11 (ii) L ( { Ψ i } i ∈ N ) ≤ L ,(iii) for any Σ ∈ S ( L ) , there exists ¯ ǫ > , j ∈ N , so that for all i ≥ j , | Ψ i | ([0 , ∩ ( ¯ B F ¯ ǫ (Σ) ∪ K ) = ∅ .Proof. Let K := K ∪{ V ; || V || ( N ) ≤ (cid:0) L +lim inf i →∞ max j =0 , M (Φ i ( j )) (cid:1) / } and let d := min { F ( S ( L ) , Z ); Z ∈ K } >
0. Since S ( L ) can be written asa countable union of compact subsets of V n ( N ), we can find a sequence ofvarifolds Σ , Σ , ... ∈ S ( L ) and a sequence of positive numbers ǫ , ǫ , ... goingto 0 such that • for any pair k = k , B F ǫ k (Σ k ) is not included in B F ǫ k (Σ k ), • the union of B F ǫ k (Σ k ) covers S ( L ), • each Σ k is 2-unstable in an ǫ k -neighborhood for some family { F kv } v ∈ ¯ B ⊂ Diff( N ) and c ,k > k , B F ǫ k (Σ k ) intersects finitelymany other balls B F ǫ k ′ (Σ k ′ ) (start with any covering satisfying ǫ k → ǫ k one by one). Without loss of generality(by changing ǫ k , { F kv } , c ,k ) we can also assume that(1) min { F (( F kv ) ♯ V, Z ); Z ∈ K , v ∈ ¯ B } > d/ k , V ∈ ¯ B F ǫ k (Σ k )(2) and if the sequence { ǫ k } is infinite, lim k →∞ sup v ∈ ¯ B || F kv − Id || C ( N ) = 0Given a postive function u : S ( L ) → R , let N u := [ k ≥ { ¯ B F u (Σ) (Σ); Σ ∈ S ( L ) } . Consider a positive function η : S ( L ) → R such that N η ⊂ [ k ≥ B F ǫ k (Σ k ) . We require that η satisfies the following: for all V ∈ N η , for any k such that V ∈ B F ǫ k (Σ k ), there is a vector v ∈ ¯ B so that(i) for all s ∈ [0 ,
1] and k ′ with V ∈ B F ǫ k ′ (Σ k ′ ), ( F ksv ) ♯ V ∈ B F ǫ k ′ (Σ k ′ ),(ii) and ( F kv ) ♯ V / ∈ N η .To see that such a choice is possible, let Σ ∈ S ( L ) ∩ B F ǫ ¯ k (Σ ¯ k ) where ¯ k is thefirst integer with Σ ∈ B F ǫ ¯ k (Σ ¯ k ). There are only finitely many integers k , ..., k q such that B F ǫ ¯ k (Σ ¯ k ) ∩ B F ǫ kl (Σ k l ) = ∅ . For any k ∈ { k , ..., k q } with Σ ∈ ¯ B F ǫ k ( ǫ k ),we can choose v = 0 so that || ( F ksv ) ♯ Σ || ( N ) is decreasing (by the properties ofthe diffeomorphisms F kv ) and ( F ksv ) ♯ Σ ∈ B F . ǫ k ′ (Σ k ′ ) for any k ′ ∈ { k , ..., k q } such that Σ ∈ ¯ B F ǫ k ′ ( ǫ k ′ ) (in particular for k ′ = ¯ k ). By compactness of¯ B F ǫ ¯ k (Σ ¯ k ), v can be chosen bounded away from 0 as Σ ∈ B F ǫ ¯ k (Σ ¯ k ) ∩ S ( L ), andone can ensure that L − || ( F kv ) ♯ Σ || ( N ) > c ′ for a constant c ′ = c ′ (¯ k ) > ∈ B F ǫ ¯ k (Σ ¯ k ) ∩ S ( L ) and k ∈{ k , ..., k q } .We can now define η as follows: let Σ ∈ S ( L ), and let ¯ k be the firstinteger with Σ ∈ B F ǫ ¯ k (Σ ¯ k ). First we define an intermediate function η (Σ)small enough so that(3) B F η (Σ) (Σ) ⊂ B F ǫ ¯ k (Σ ¯ k ) . Secondly we can choose 0 < η (Σ) ≤ η (Σ) ( η is not continous in general),so that for all k ∈ { k , ..., k q } with Σ ∈ B F ǫ ¯ k (Σ ¯ k ) ∩ ¯ B F ǫ k (Σ k ), for all v aspreviously chosen and for all V ∈ B F η (Σ) (Σ): • L − || V || ( N ) < min { c ′ ( k ′ ) / k ′ such that B F η (Σ) (Σ) ∩ B F ǫ k ′ (Σ k ′ ) = ∅ } , • L − || ( F kv ) ♯ V || ( N ) > c ′ (¯ k ) / > • for all s ∈ [0 , F ksv ) ♯ V ∈ B F ǫ k ′ (Σ k ′ ) for any k ′ ∈ { k , ..., k q } with V ∈ B F ǫ k ′ (Σ k ′ ).Note that since ǫ k →
0, since no two balls B F ǫ k (Σ k ) are included one into theother and by (3), there are only finitely many balls B F ǫ k (Σ k ) intersecting B F η (Σ) (Σ). This finishes the definition of η . The two required properties areindeed satisfied: if V ∈ N η ∩ B F ǫ k (Σ k ), there is a Σ ∈ S ( L ), V ∈ B F η (Σ) (Σ) ⊂ B F ǫ ¯ k (Σ ¯ k ), where ¯ k is the first integer with Σ ∈ B F ǫ ¯ k (Σ ¯ k ), then B F ǫ ¯ k (Σ ¯ k ) ∩ B F ǫ k (Σ k ) = ∅ and by construction there exists v ∈ ¯ B , for all s ∈ [0 , L − || ( F kv ) ♯ V || ( N ) > c ′ (¯ k ) / F ksv ) ♯ V ∈ B F ǫ k ′ (Σ k ′ ) for any k ′ ∈ { k , ..., k q } with V ∈ B F ǫ k ′ (Σ k ′ ) .. which in particular gives the first property. Next, if by contradiction we sup-pose that V ′ := ( F kv ) ♯ V ∈ N η , then there is a Σ ′ ∈ S ( L ), V ′ ∈ B F η (Σ ′ ) (Σ ′ ) ∩ B F ǫ ¯ k (Σ ¯ k ) (by (5)), so L − || V ′ || ( N ) < c ′ (¯ k ) / N η , let U i,η = [ a , a ] ∪ ... ∪ [ a p − , a p ] be aunion of closed disjoint intervals in [0 ,
1] (the choice is not unique) so thatfor all x ∈ U i,η = p [ l =1 [ a l − , a l ] , | Φ i ( x ) | ∈ N η and for all x ∈ [0 , \ p [ l =1 ( a l − , a l ) , | Φ i ( x ) | ∈ V n ( N ) \N η/ . INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 13
For each i large, we will modify Φ i on each of the intervals [ a l − , a l ] into amap Ψ i (Φ i is left unmodified outside of U i,η ). We now describe the changeson [ a , a ], the modifications for the other intervals are similar. First wecan decompose this interval into [ a , a ] = [ b , b ] ∪ [ b , b ] ∪ ... [ b m − , b m ]( a = b , a = b m ) so that for every l = 1 , ..., m −
1, there exists a k = k ( l )such that | Φ i | ([ b l , b l +1 ]) ⊂ B F ǫ k (Σ k ). By construction | Φ i | ( a ) and | Φ i | ( a )are not in N η/ . We move each | Φ i | ( b l ) so that it becomes also true forthem, but keeping it inside B F ǫ k ( l ) (Σ k ( l ) ) as follows. By construction of η ,for all l = 2 , ..., m − v l ∈ ¯ B so that the mass || ( F sv l ) ♯ | Φ i | ( b l ) || ( N ) is decreasing in s , and for all s ∈ [0 , F sv l ) ♯ | Φ i | ( b l ) ∈ B F ǫ k ( l ) (Σ k ( l ) ) ∩ B F ǫ k ( l − (Σ k ( l − ) , ( F v l ) ♯ | Φ i | ( b l ) / ∈ N η . For l = 2 , ..., m −
1, let A l : [0 , → Z n ( N ; F ; Z ) be the path ∀ s ∈ [0 , , A l ( s ) := ( F sv l ) ♯ Φ i ( b l ) and let A − l be the same path but with reverseparametrization. Up to reparametrization, Φ i (cid:12)(cid:12) [ a ,a ] is clearly homotopic inthe F -topology to the following concatenation (where + stands for concate-nation):(6) (Φ i (cid:12)(cid:12) [ b ,b ] + A )+( A − +Φ i (cid:12)(cid:12) [ b ,b ] + A )+( A − + ... )+ ... +( ... +Φ i (cid:12)(cid:12) [ b m − ,b m ] ) . Each subsum in parentheses is a path P : [0 , → Z n ( N ; F , Z ) so that theimage (in the space of varifolds) of | P | is included in a ball B F ǫ k (Σ k ), andwhose endpoints satisfy | P (0) | , | P (1) | / ∈ N η/ . Let us explain how to deformany such path P into Q , fixing the endpoints, so that the varifold image | Q | ([0 , ∈ S ( L ). A similar moregeneral deformation is the object of the proof of [28, Deformation TheoremA], we will use the same notations and explain the few modifications neededhere. Let m i : [0 , → ¯ B be defined by m i ( s ) := m ( | P | ( s )). Fix theparameter i , there is a continuous homotopyˆ H i : [0 , × [0 , → ¯ B / i (0) so that ˆ H i ( s,
0) = 0 ∀ s ∈ [0 , , inf s ∈ [0 , | m i ( s ) − ˆ H i ( s, | ≥ µ i > . and for j = 0 , || ( F ˆ H i ( j,t ) ) ♯ | P | ( j ) || ( N ) decreases in t ∈ [0 , . Let T i = T ( µ i , Σ k , ǫ k , { F k v } , c ,k ) > Q is then obtained as the (reparametrized) concatenation of thefollowing paths: • Q ( t ) := ( F ˆ H i (0 ,t ) ) ♯ P (0) for t ∈ [0 , • Q ( t ) := ( F φ P (0) ( ˆ H i (0 , ,tT i ) ) ♯ P (0) for t ∈ [0 , • Q ( t ) := ( F φ P ( t ) ( ˆ H i ( t, ,T i ) ) ♯ P ( t ) for t ∈ [0 , • Q ( t ) := ( F φ P (1) ( ˆ H i (1 , , (1 − t ) T i ) ) ♯ P (1) for t ∈ [0 , • Q ( t ) := ( F ˆ H i (1 , (1 − t )) ) ♯ P (1) for t ∈ [0 , By replacing each path P in parentheses in (6) by the corresponding Q with parameter i constructed above for Φ i (cid:12)(cid:12) [ a ,a ] , and similarly for eachother restriction Φ i (cid:12)(cid:12) [ a l − ,a l ] ( l = 1 , ..., p ), we get a new map Ψ i : [0 , →Z n ( N ; F ; Z ) homotopic in the F -topology to Φ i . It is clear that L ( { Ψ i } i ∈ N ) ≤ L. To check item ( iii ), consider a sequence of maps P i : [0 , → Z n ( N ; F ; Z )with image included in a ball B F ǫ ki (Σ k i ), and whose endpoints satisfy | P i | (0) , | P i | (1) / ∈N η/ . Suppose that L ( { P i } ) ≤ L . Let Q i : [0 , → Z n ( N ; F ; Z )be the deformations corresponding to P i ( i ∈ N ) constructed as above withparameter i . Let Σ ∈ S ( L ) and suppose towards a contradiction that thereis a sequence of times t i ∈ [0 ,
1] so thatlim i →∞ F ( | Q i | ( t i ) , Σ) = 0 . Recall that the mass of Σ is L . Two cases can occur: either lim sup i →∞ k i < ∞ then we can use the mass decreasing properties of || ( F ˆ H i ( j,. ) ) ♯ | P | ( j ) || ( N )and φ P ( j ) , j = 0 ,
1, (see [28]), or lim sup i →∞ k i = ∞ and we can use (2); inany case we obtain that necessarily there exists j = 0 or 1 such thatlim i →∞ F ( | Q i | ( j ) , | Q i | ( t i )) = lim i →∞ F ( | Q i | ( j ) , Σ) = 0 . But this cannot happen since | Q i | ( j ) = | P i | ( j ) is at F -distance at least η (Σ) / ǫ = ¯ ǫ (Σ) > | Ψ i | ([0 , ∩ ¯ B F ¯ ǫ (Σ) = ∅ for i large. Finally (1) shows that | Ψ i | ([0 , ∩ K = ∅ for i large. (cid:3) We will not need Deformation Theorem B of [28]. However, DeformationTheorem C of [28] will be useful. Before stating it, consider a minimalhypersurface S which is degenerate stable of Type II. Then we associateto S some squeezing maps like the ones in [28, Proposition 5.7]. Supposethat S is 2-sided embedded in the interior of N , the other cases (1-sidedor boundary component) are similar. A neighborhood of S is foliated byhypersurfaces with mean curvature vector pointing towards S when non-zero (Appendix A, Lemma 22). Let f be a real function defined on such aneighborhood with ∇ f = 0 such that S ( s ) = f − ( s ) ( s ∈ [ − , S (0) = S (in particular, s h∇ f, −→ H ( S ( s )) i ≤ X = ∇ f / |∇ f | , and let φ : S × [ − , → N such that ∂φ∂s ( x, s ) = X ( φ ( x, s )) and φ ( S, s ) = S ( s ) for all s ∈ [ − , r = φ ( S × ( − r, r )) and define the maps P t : Ω → Ω such that P t ( φ ( x, s )) = φ ( x, s (1 − t )) for t ∈ [0 , . INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 15
Lemma 8.
Let S be as above. There exists r > such that(1) P t satisfies items (i), (ii), (iii) of [28, Proposition 5.7] ,(2) for all V ∈ V n (Ω r ) and every connected component Ω of Ω r , thefunction t
7→ || ( P t ) ♯ V || (Ω) is a strictly decreasing function of t , unless spt( V ) ∩ Ω ⊂ S ∩ Ω , in which case it is constant.Proof. The only minor change compared to [28, Proposition 5.7] is that t
7→ || ( P t ) ♯ V || (Ω) does not have strictly negative derivative. However, byinspecting the computations in its proof, we see that t
7→ || ( P t ) ♯ V || (Ω) al-ways has nonpositive derivative and for any a < b ∈ [0 , t
7→ || ( P t ) ♯ V || (Ω)must have negative derivative at some time t ′ ∈ ( a, b ) because S is of TypeII, except when spt( V ) ∩ Ω ⊂ S ∩ Ω. (cid:3) We observe that if S is 1-sided, or a boundary component of N , theprevious discussion still applies to a neighborhood of S , on which one candefine squeezing maps P t . Theorem 9 (Deformation Theorem C, [28]) . Suppose the sequence { Φ i } ispulled-tight (every varifold in Λ ( { Φ i } ) with || V || ( N ) = L is stationary). Let { Σ (1) , ..., Σ ( Q ) } ⊂ V n ( N ) be a collection of stationary integral varifolds suchthat for every ≤ q ≤ Q : • the support of (Σ ( q ) ) for ≤ q ≤ Q is a closed embedded minimalhypersurface S ( q ) whose components are either in the interior of N or a component of ∂N , • each component of S ( q ) (its double cover if not 2-sided) is strictlystable or degenerate stable of Type II, • L = || Σ ( q ) || ( N ) .Then there exist ξ > , j ∈ N so that for all i ≥ j one can find Ψ i : [0 , →Z n ( N ; F ; Z ) such that(i) Ψ i is homotopic to Φ i with fixed endpoints in the flat topology,(ii) L ( { Ψ i } i ∈ N ) ≤ L ,(iii) Λ ( { Ψ i } ) ⊂ (cid:0) Λ ( { Φ i } ) \ ∪ Qq =1 B F ξ (Σ ( q ) ) (cid:1) ∪ (cid:0) V n ( N ) \ B F ξ (Γ) (cid:1) , where Γ is the collection of all stationary integral varifolds V ∈V n ( N ) with L = || V || ( N ) .Proof. The proof is almost identical to the case of bumpy metrics [28]. Theonly difference is that we use Lemma 8 instead of [28, Proposition 5.7] fordegenerate stable minimal hypersurfaces of Type II. (cid:3)
Finally we need a last deformation theorem in order to deal with degen-erate stable minimal hypersurfaces of Type I. Let S be such a connectedhypersurface (it is necessarily 2-sided and contained in the interior of N ).Note that one can again define squeezing maps here. A neighborhood insideInt( N ) of S is foliated by hypersurfaces with mean curvature vector pointingtowards a fixed direction when non-zero. Let f be a real function defined on such a neighborhood with ∇ f = ∅ such that S ( s ) = f − ( s ) ( s ∈ [ − , S (0) = S and h∇ f, −→ H ( S ( s )) i ≤ X = ∇ f / |∇ f | , and let φ : S × [ − , → N such that ∂φ∂s ( x, s ) = X ( φ ( x, s )) and φ ( S, s ) = S ( s ) for all s ∈ [ − , ′ r = φ ( S × [ − r, r )) and define the maps P ′ r,t : Ω r → Ω r such that P ′ r,t ( φ ( x, s )) = φ ( x, ( s + r )(1 − t ) − r ) for t ∈ [0 , . The following lemma is proved as Lemma 8 and [28, Proposition 5.7].
Lemma 10.
Let S be as above. There exists r > such that P ′ r ,t : Ω ′ r → Ω ′ r satisfies:(1) P ′ r , ( x ) = x for all x ∈ Ω ′ r , P r ,t ( y ) = y for all y ∈ S ( − r ) , t ∈ [0 , ,(2) P ′ r ,t ( φ ( S × [ − r , r )) ⊂ φ ( S × [ − r , r ) for all t ∈ [0 , , r ≤ r , and P ′ r , (Ω ′ r ) = S ( − r ) ,(3) the map P ′ r ,t : Ω ′ r → Ω ′ r is a diffeomorphism onto its image for ≤ t < ,(4) for all V ∈ V n (Ω ′ r ) , the function t
7→ || ( P ′ r ,t ) ♯ V || (Ω ′ r ) is a strictlydecreasing function of t , unless spt( V ) ⊂ S ( − r ) , in which case it isconstant,(5) if V is a stationary integral varifold with support S , then for all ǫ > , there are κ = κ ( V , ǫ ) > , ǫ ′ = ǫ ′ ( V , ǫ ) > and t ′ = t ′ ( V , ǫ ) ∈ (0 , so that for all V ∈ V n (Ω ′ r ) ∩ B F ǫ ′ ( V ) and for all s ∈ [0 , t ′ ] , ( P ′ r ,s ) ♯ V ∈ B F ǫ ( V ) and || ( P ′ r ,t ′ ) ♯ V || (Ω ′ r ) ≤ || V || (Ω ′ r ) − κ. Proof.
The only new point is bullet (5), which follows from the continuityof ( P ′ r ,s ) ♯ in the F -topology. (cid:3) For the last deformation theorem, we will assume that[ ⋆ ] L : any degenerate stable minimal hypersurface in N of n -volume at most L is of Type I or II.Let T ( L ) be the family of stationary integral varifolds in V n ( N ) of total mass L whose support is a stable smooth closed embedded minimal hypersurfaceand for which at least one of the components is degenerate stable of Type I. Remark 2.1. By [40] for any sequence V i ∈ T ( L ) , the supports spt( V i ) converge subsequently smoothly to a stable minimal hypersurface S . As-suming [ ⋆ ] L , the components of S are either strictly stable, or degeneratestable of Type I, II. At least one of the components of S is degeneratestable of Type I. Indeed, we can write V i = a ,i | S i | + ... + a k i ,i | S k i i | where S i , ..., S k i i are disjoint minimal hypersurfaces, k i , a ,i ,..., a k i ,i are boundedindependently of i and we can suppose that these sequence of integers allstabilize to respectively k , a ,..., a k . Then V i = a ,i | S i | + ... + a k i ,i | S k i i | con-verges to V = a | S | + ... + a k | S k | where S l are the components of S and are INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 17 the smooth limit of S li . But if S l are all strictly stable or degenerate stableof Type II, [28, Proposition 5.7] and Lemma 8 would imply that S li = S l for l large since the mass || V i || ( N ) is constant equal to L . That contradicts thefact that V i ∈ T ( L ) . As a consequence of this discussion, assuming [ ⋆ ] L , T ( L ) is a compact subset of V n ( N ) and the number r in Lemma 10 can bechosen independently of V ∈ T ( L ) . Here is the last deformation theorem.
Theorem 11 (Deformation Theorem D) . Assuming [ ⋆ ] L , given { Φ i } i ∈ N ,there is another sequence { Ψ i } i ∈ N such that(i) { Ψ i } i ∈ N is homotopic to { Φ i } i ∈ N with fixed endpoints in the flat topol-ogy for all i ∈ N ,(ii) L ( { Ψ i } i ∈ N ) ≤ L, (iii) there exists ˆ ǫ > , j ∈ N so that for all i ≥ j , inf { F (Σ , | Ψ i | ([0 , ∈ T ( L ) } ≥ ˆ ǫ. Proof.
For any Σ ∈ T ( L ), one component S of spt(Σ) is degenerate stableof Type I. Recall that S has a neighborhood Ω ′ r and associated squeezingmaps P ′ r ,t : Ω ′ r → Ω ′ r . Let ˜Ω be a neighborhood of Σ such that one of thecomponent of ˜Ω is Ω ′ r . By abuse of notations, we denote by P ′ r ,t the mapfrom ˜Ω to itself, equal to P ′ r ,t on Ω ′ r and equal to the identity map on theother components. By Remark 2.1, r of Lemma 10 can and will be chosenuniformly in Σ ∈ T ( L ). Similarly for a given ǫ >
0, the quantities κ , ǫ ′ and t ′ of Lemma 10 can be chosen uniformly.Let ǫ > ∈ T ( L ) and all path p :[0 , → B F ǫ (Σ), we can apply the three constructions 5.9, 5.11, 5.13 in [28],where p replaces the connected components of V i,ǫ , P ′ r ,t replaces their maps P t . It exists by compactness of T ( L ). We also suppose ǫ small enough sothat for all V in an ǫ -neighborhood of T ( L ), || V || ( N ) ≥ ( L + lim inf i →∞ max j =0 , M (Φ i ( j ))) / i (0), Φ i (1)).For this ǫ , let ǫ ′ < ǫ , t ′ , κ be given by Lemma 10, which can be chosenuniformly in V ∈ T ( L ) by compactness. Choose also an ǫ ′′ < ǫ ′ so thatfor all Σ , Σ ′ ∈ T ( L ), V ∈ B F ǫ ′′ (Σ) ∩ B F ǫ ′′ (Σ ′ ), using the techniques in thefirst construction 5.9 in [28] (cf [33, Lemma 7.1]), there is a path { C t } t ∈ [0 , continuous in the mass topology from C = V to a cycle C with • spt( C ) ⊂ ˜Ω, • for all t ∈ [0 , | C t | ∈ B F ǫ ′ (Σ) ∩ B F ǫ ′ (Σ ′ ).We can cover T ( L ) with a finite number of balls B F ǫ ′′ (Σ k ), where Σ , ..., Σ K ∈T ( L ). If u >
0, let ˆ N u be the u -neighborhood of T ( L ) in the F topology. Let µ > N µ ⊂ K [ k =1 B F ǫ ′′ (Σ k ) and(8) ∀ V ∈ ˆ N µ , || V || ( N ) > L − κ. Consider Φ i : [0 , → Z n ( N ; F , Z ) and let U i,µ = [ a , a ] ∪ ... ∪ [ a p − , a p ]be a union of closed disjoint intervals in [0 ,
1] so thatfor all x ∈ U i,µ , | Φ i ( x ) | ∈ ˆ N µ and for all x ∈ [0 , \ p [ l =1 ( a l − , a l ) , | Φ i ( x ) | ∈ V n ( N ) \ ˆ N µ . For each i large, we want to modify Φ i on each interval [ a l − , a l ] into amap Ψ i which coincide with Φ i outside of U i,µ . Let us first focus on [ a , a ],the other intervals will be treated in the same way. By (7), we can write[ a , a ] as a union [ b , b ] ∪ [ b , b ] ∪ ... ∪ [ b m − , b m ] ( a = b and a = b m )such that for each l = 1 , ..., m , there exists a k = k ( l ) ∈ { , ..., K } with | Φ | ([ b l , b l +1 ]) ⊂ B F ǫ ′′ (Σ k ). Since Σ k ∈ T ( L ), recall from the beginning of theproof, that there is a neighborhood ˜Ω of spt(Σ) and there are associatedsqueezing maps P ′ r ,t : ˜Ω → ˜Ω. By construction, | Φ i | ( a ) and | Φ i | ( a ) areoutside of ˆ N µ . We want to move each of the other | Φ i | ( b l ) outside of ˆ N µ aswell, with a mass control on the deformation. Fix l ∈ { , ..., m − } . By thechoice of ǫ ′′ < ǫ ′ , and the first construction 5.9 in [28] (cf [33, Lemma 7.1]),there is a path { C t } t ∈ [0 , continuous in the mass topology from C = Φ i ( b l )to a cycle C with • spt( C ) ⊂ ˜Ω, • for all t ∈ [0 , | C t | ∈ B F ǫ ′ (Σ k ( l ) ) ∩ B F ǫ ′ (Σ k ( l − ).Thanks to the second item above and Lemma 10, ∀ s ∈ [0 , t ′ ] , ( P ′ r ,s ) ♯ | C | ∈ B F ǫ (Σ k ( l ) ) ∩ B F ǫ (Σ k ( l − ) , || ( P ′ r ,t ′ ) ♯ | C ||| ( N ) ≤ L − κ for κ , t ′ chosen at the beginning of the proof, in particular by (8)( P ′ r ,t ′ ) ♯ | C | / ∈ ˆ N µ . Denote by A l : [0 , → Z n ( N ; F ; Z ) the (reparametrized) concatenation of { C t } t ∈ [0 , and { ( P ′ r ,t ) ♯ C } t ∈ [0 , . Let A − l denote the same path with reverseparametrization. Up to reparametrization, Φ i (cid:12)(cid:12) [ a ,a ] is homotopic in the F -topology to the following concatenation:(Φ i (cid:12)(cid:12) [ b ,b ] + A ) + ( A − + Φ i (cid:12)(cid:12) [ b ,b ] + A ) + ... + ( A − m − Φ i (cid:12)(cid:12) [ b m − ,b m ] ) . Each subsum in parentheses is a path p l : [0 , → Z n ( N ; F ; Z ) ( l =1 , ..., m ) so that the (varifold) image of | p l | is included in a ball of theform B F ǫ (Σ k ), whose endpoints satisfy | p l (0) | , | p l (1) | / ∈ ˆ N µ . We can apply INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 19 the first, second and third constructions 5.9, 5.11, 5.13 in [28] (with thesqueezing maps P ′ r ,t replacing their maps P t ) to p l and get a path q l whichis F -continuous and homotopic to p l in the flat topology, with the followingproperties: • the endpoints are the same p l ( j ) = q l ( j ) ( j = 0 ,
1) and are not inˆ N µ , • max t ∈ [0 , M ( q l ( t )) ≤ max { M ( p l (0)) , M ( p l (1)) } + 1 /i ≤ max t ∈ [0 , M (Φ i ( t )) + 1 /i, • there exists ˆ ǫ = ˆ ǫ ( µ ) > i is large enough,inf { F (Σ , | q l | ([0 , ∈ T ( L ) } ≥ ˆ ǫ. The last item follows from arguments very similar to Claims 1 and 2 in theproof of [28, Deformation Theorem C], and item (5) of Lemma 10. Finallywe concatenate q ,..., q m − to get Ψ i (cid:12)(cid:12) [ a ,a ] and we proceed similarly for theother intervals [ a , a ],...,[ a p − , a p ] to get Ψ i . (cid:3) Existence of saddle point minimal hypersurfaces.
Equipped withthe previous deformation theorems, we prove the existence of saddle pointsin non-bumpy metrics. Remember that if a degenerate stable minimal hy-persurface satisfies Condition [M] then it is of type I, II, or III (AppendixA). The following theorem is an extension of Proposition 6.
Theorem 12.
Let ( N n +1 , g ) be a compact manifold with boundary, with ≤ n ≤ , such that ∂N is locally area minimizing inside N . Then thereis a saddle point minimal hypersurface Γ inside the interior Int( N ) , whoseindex is at most one and whose n -volume is bounded by W + 1 where W isthe width of ( N, g ) .Proof. We can assume that any minimal hypersurface of n -volume boundedby W and index at most one satisfy Condition [M] (see Appendix A).Let W be the width of N and let { Φ i } i ∈ N be a pulled-tight sequence ofsweepouts so that L ( { Φ i } i ∈ N ) = W. Since W is larger than the n -volume of any component of ∂N , by Condition[M] each component of ∂N is either strictly stable or is degenerate stableand is of Type II. We can also suppose that [ ⋆ ] W is satisfied (see beforeRemark 2.1) otherwise there is already a saddle point Γ of index at mostone and n -volume bounded by W + 1. We first apply Deformation Theo-rem D. Then by using the monotonicity formula, Lemma 8, Lemma 10 andLemma 22 (4), we check that T ( W ) is at F -positive distance from S ( W ).Hence we can apply Deformation Theorem A with K = ¯ B F ˆ ǫ ( T ( W )) (whereˆ ǫ might be chosen smaller than the one given by Deformation Theorem D). Since by arguments similar to those used in Remark 2.1, the set of varifoldssatisfying the assumptions of Deformation Theorem C is finite, we apply De-formation Theorem C to these varifolds. Let { Ψ i } be the resulting sequenceof sweepouts. By Almgren-Pitts’ theory (see [28, Theorem 3.8] and use thatlim inf i →∞ ( W − max j =0 , M (Ψ i ( j ))) >
0, see also proof of Theorem 24 inAppendix B), an element V of Λ ( { Ψ i } ) has smooth support and mass W .No component of spt( V ) is 2-unstable or degenerate stable of Type I, andthe components of spt( V ) (their double covers if not 2-sided) cannot be allstrictly stable or degenerate stable of Type II. Hence, since we are assuming[ ⋆ ] W , at least one of its components satisfies the following: • either it has Morse index one, • or is stable, 1-sided and its double cover is unstable.In both cases, this minimal hypersurface is a saddle point and necessarilycontained in the interior of N . (cid:3) Zero-infinity dichotomy for manifolds thick at infinity
Local version of Gromov’s result for manifolds thick at infinity.
Consider a complete manifold ( M n +1 , g ). Recall that saddle point minimalhypersurfaces are closed embedded minimal hypersurfaces satisfying a nat-ural saddle point condition. By “compact domain”, we mean a compact( n + 1)-dimensional submanifold of M with smooth boundary.Let B be a compact domain and Σ ⊂ ( M, g ) be a closed embedded min-imal hypersurface which may be empty. Suppose that Σ is locally area-minimizing. We say that B \ Σ has a singular weakly mean convex foliationif there is a bigger compact domain B containing B ∪ Σ and endowed witha metric g ′ coinciding with g on a neighborhood of B ∪ Σ, so that there is afamily of closed subsets of B , { K t } t ∈ [0 , , satisfying: • K = B , for all neighborhood N of Σ, K t ∩ ( B \ N )) = ∅ for t closeenough to 1 and Σ ⊂ K t for all t ∈ [0 , • ∂K t has the regularity of the level set flow, • ∂K t is mean convex in the sense of mean curvature flow, and the nonstrictly mean convex level sets ∂K t are smoothly embedded closedminimal hypersurfaces in ( B , g ′ ),The following theorem is a more precise version of Theorem 4 for manifoldsthick at infinity. Theorem 13.
Let ( M, g ) be an ( n + 1) -dimensional complete manifold with ≤ n ≤ , thick at infinity, and let B ⊂ M be a compact domain. Then(1) either M contains a saddle point minimal hypersurface intersecting B ,(2) or there is an embedded closed locally area minimizing hypersurface Σ B ⊂ ( M, g ) (maybe empty) such that B \ Σ B has a singular weaklymean convex foliation. INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 21
Proof.
We suppose that M is thick at infinity. Let us resume from the proofof Theorem 4. Recall that B is contained in a ball B r ( p ), itself contained in D . We modify the metric g near ∂D to get g D so that ∂D becomes meanconvex for g D . For any subdomain D ′ ⊂ ( D, g D ) with C , weakly meanconvex boundary and minimal hypersurface S ⊂ Int( D ′ ) \ B , we consider themetric completion D of D ′ \ S .We find a solution to a constrained minimization problem, yielding acompact manifold B ′′ depending on D . If there is a saddle point minimalhypersurface Γ such that a thin mean concave neighborhood N Γ of Γ is embedded in B ′′ \ B then the metric completion of B ′′ \ N Γ gives a newmanifold D where we can solve the constrained minimization problem, get amanifold B ′′ . If there is a saddle point minimal hypersurface Γ embedded in B ′′ \ B , we repeat the process and get B ′′ . If for a j , B ′′ j \ B does not containa saddle point minimal hypersurface, then we define B ′′∞ := B ′′ j . Supposethat the sequence of B ′′ j is infinite, the n -volume of B ′′ j is strictly decreasingand we can suppose that we chose the saddle point minimal hypersurfaces insuch a way that lim j →∞ Vol n ( B ′′ j ) is as small as possible, among all choicesof sequence of saddle points Γ , Γ , Γ , ... . Then B ′′ j subsequently converges inthe Gromov-Hausdorff topology to a closed mean convex set B ′′∞ in which B is isometrically embedded, and by minimality of its volume, there is no saddlepoint minimal hypersurface embedded in B ′′∞ \ B . Since the C , boundary ofeach B ′′ j has a minimizing property and its n -volume is at most Vol n ( ∂B ), bycompactness similar to [41], B ′′∞ has a C , mean convex boundary smoothoutside of B .Let V be the union of minimal components of ∂B ′′∞ which are unstable ordegenerate stable of Type III, and let N V be a neighborhood of V foliatedby weakly mean convex hypersurfaces (Appendix A). We can run the levelset flow (Appendix C) to ∂B ′′∞ \ N V and get { A t } t ≥ . If the level set flowgets “stuck” at a 2-sided minimal hypersurface S on one side, there are afew possibilities.Either T t ≥ A t ∩ Int( B ) = ∅ and we are done by taking Σ B = ∅ .Or T t ≥ ˜ A t ∩ Int( B ) = ∅ and if we suppose that S does not satisfy Con-dition [M], some saddle point minimal hypersurfaces close to S intersect B and they have good index and n -volume bounds: if this occurs for any D , D , B ′′∞ for take a sequence of larger and larger D ( r → ∞ ), we get asequence of saddle points Γ ( k ) of controlled index and n -volume. Taking asubsequence limit, we get a minimal hypersurface intersecting B , which isclosed by thickness at infinity. Actually the metric g D on each D coincideswith g on larger and larger regions as D becomes larger so for k large, Γ ( k ) is a saddle point minimal hypersurface for the original metric g and we aredone.In the case T t ≥ A t ∩ Int( B ) = ∅ and S satisfies Condition [M], S iseither strictly stable or degenerate stable of Type I or II (for the metric g D ).If S is of Type I, then the level set flow approaches S from above, and it is possible to prolongate the foliation { A t } t ≥ by hand beyond S , so thataround S the foliation is smooth weakly mean convex. After this foliationgoes beyond S , we can run the level set flow again, and we can repeat thatprocess, extending the foliation whenever the level set flow converges to adegenerate stable minimal hypersurface of Type I. This way, we construct afoliation { ˜ A t } t ≥ , that we can suppose sweeps out a region of B of maximalvolume. We can make sure to not get trapped at a degenerate stable minimalhypersurface of Type I by a compactness argument.The previous construction can only stop if T t ≥ ˜ A t ∩ Int( B ) = ∅ , or if asaddle point minimal hypersurface intersects B , or if the foliation arrives atminimal hypersurfaces that are strictly stable or degenerate stable of TypeII. We will assume the last case in the remaining of the proof. We write B core := T t ≥ ˜ A t . It is a union of an ( n + 1)-dimensional compact manifold B (1) core and a minimal hypersurface B (2) core . Each boundary component of B (1) core is either strictly stable or degenerate stable of Type II (hence locally areaminimizing). Now two cases can happen: • either the ( n +1)-dimensional volume of B core ∩ B is 0, then B (2) core \ ∂B ′′∞ is a closed embedded locally area-minimizing hypersurface Σ D,B ⊂ ( B ′′∞ , g D ) of n -volume at most Vol n ( ∂B ) + 1, so that B \ Σ D,B has aweakly mean convex foliation, • or the ( n +1)-dimensional volume of B core ∩ B is positive so that B (1) core is a non-trivial ( n + 1)-dimensional compact manifold; in that caseTheorem 12 produces a saddle point Γ ⊂ ( B (1) core , g D ) of controlled n -volume and Morse index, which intersects B by volume minimalityof B ′′∞ (see beginning of proof).If we take r → ∞ , larger and larger domains D ⊃ B r ( p ), and if the firstbullet above always occurs for r large, then by thickness at infinity and [41],the diameter of Σ D,B is uniformly bounded so for r large enough Σ B := Σ D,B is a closed embedded locally area minimizing hypersurface for the originalmetric g , and B \ Σ B has a weakly mean convex foliation. This is item (2)of the theorem.If for a sequence of radii r j → ∞ and choice of ( D j , g D j ), the second bulletoccurs, then we take a converging subsequence of saddle points Γ j (see [41]),and since ( M, g ) is thick at infinity, Γ j is actually a saddle point for theoriginal metric g if j is large and it intersects B . This is item (1) of thetheorem. (cid:3) Remark 3.1.
By inspecting the proof of Theorem 13, we see that it alsoholds if ( M, g ) , thick at infinity, has a non-empty boundary and if the com-ponents of ∂M are closed minimal hypersurfaces: for any compact domainwith smooth boundary B ⊂ M , the dichotomy of Theorem 13 holds true.Here some components of ∂M might be included in B . INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 23
Min-max in a manifold generated by a saddle point minimalhypersurface with the level set flow.
Let ( M n +1 , g ) be a complete man-ifold thick at infinity. We assume in this subsection that the metric g satisfiesCondition [M].We will often use { Σ t } t ∈ [0 , or similar notations for 1-sweepouts of a re-gion R of M . Sometimes we will define Σ t as a hypersurface, even thoughrigorously speaking, each Σ t should be a current in Z n,rel ( R, Z ) (see [1,Definition 1.20], [25, 2.2]). For simplicity we will also denote by Vol n (Σ t ) itsmass instead of using M (Σ t ).Let Γ be a saddle point minimal hypersurface in ( M, g ). We will treatthe case where Γ is 2-sided for simplicity, but the case where it is 1-sided iscompletely analogous. Γ has to be either unstable, or degenerate stable ofType III, and so we can find a neighborhood N Γ of Γ and a diffeomorphism φ : Γ × ( − δ , δ ′ ) → N Γ such that φ (Γ × { } ) = Γ, the mean curvatureof φ (Γ × { s } ) is either vanishing or non-zero pointing away from Γ, and φ (Γ × {− δ } ), φ (Γ × { δ ′ } ) have non-zero mean curvature.Let K := M \ N Γ . Let B k be an exhausting sequence of compact do-mains with smooth boundary, containing N Γ . We choose a metric g k on B k verifying:(1) ∂B k is minimal strictly stable with respect to g k ,(2) || g k − g || C ≤ µ k , where the right-hand side is the C distance between g and g k on B k , computed with g , and µ k converges to zero as k → ∞ ,(3) g = g k except in a 1 /k -neighborhood of ∂B k .For each integer k , let K ( k )0 := ( B k ∩ K , g k ). It is a strictly mean con-vex subset of ( B k , g k ). We run the level set flow to K ( k )0 and get a family { K ( k ) t } t ≥ . Denote by X k the metric completion of B k \ T t ≥ K ( k ) t . X k is acompact manifold with closed minimal stable boundary components. Eachof them has an n -volume bound coming from monotonicity properties of thelevel set flow and is locally area minimizing inside X k . Besides the levelset flow ∂K ( k ) t converges to ∂X k smoothly (see Appendix C). Since interiorpoints of X k are identified with points of M , there is a natural map X k → M .As k → ∞ , X k subsequently converges in the Gromov-Hausdorff topologyto a manifold X that is naturally endowed with the metric g . X is a (maybenon-compact) manifold with minimal stable boundary components. Theirtotal n -volume is finite, each of them is compact because ( M, g ) is thick atinfinity, and locally area minimizing inside X . By Condition [M], it meansthat each component of ∂X is either strictly stable or degenerate stable ofType II. We will refer to this construction by saying that X is generated byΓ, K in ( M, g ).Let C ( X k ) denote the result of gluing the compact manifold X k to straighthalf-cylinders ( ∂X k × [0 , ∞ ) , g (cid:12)(cid:12) ∂X ⊕ dt ) along ∂X k . The final metric (stilldenoted by g k ) is Lipschitz continuous around ∂X k in general. Let us define˜ ω p ( X k , g k ) := ω p ( C ( X k ) , g k ) where the p -widths ω p of a possibly non-compact manifold are defined in [42,Definition 8]. Note that if X k has no boundary then ˜ ω p ( X k , g ) = ω p ( X k , g ).It is clear that the first width of ( C ( X k ) , g k ) is finite. Indeed let { Σ t } t ∈ [0 , bean arbitrary 1-sweepout of X k \ K ( k )0 = X \ K with Σ = 0 and Σ = ∂K ,let { S ( k ) t } t ∈ [0 , ∞ ) with S k ) t := ∂X k × { t } ⊂ C ( X k ). Then the three families { Σ t } t ∈ [0 , , { ∂K ( k ) t } t ≥ , { S ( k ) t } t ≥ concatenated gives an explicit 1-sweepoutof any bounded subset of ( C ( X k ) , g k ) and ω ( C ( X k ) , g k ) ≤ max { sup t ∈ [0 , Vol n (Σ t ) , sup t Vol n ( ∂K ( k ) t ) , sup t Vol n ( S ( k ) t ) }≤ max { sup t ∈ [0 , Vol n (Σ t ) , Vol n ( ∂K ) } < ∞ . In [29, proof of Theorem 5.1, Claim 5.6], it is explained that given a compactmanifold Y , given a 1-sweepout Φ : [0 , → Z n,rel ( Y, Z ) with sup t ∈ [0 , M (Φ ( t )) ≤ C , one can construct p -sweepouts Φ p : R P p → Z n,rel ( Y, Z ) for any p , satis-fying sup t ∈ R P p M (Φ p ( t )) ≤ p.C . In particular, we have the following general Fact: if the first width is finite, then all the widths are finite and the p -width is bounded by p times the first width.Using the Fact, it is simple to check that for each p , ˜ ω p ( X k , g k ) is boundedbetween to positive constants independent of k . We choose a subsequenceof B k (that we do not rename) in a way that ˜ ω p ( X k , g k ) converges for each p and we define ˜ ω p ( X, g ) := lim k →∞ ˜ ω p ( X k , g k ) . For each k we also set A (Γ , g k ) := max { Vol n ( ˆ C ); ˆ C component of ∂X k } , where the n -volume of ˆ C is computed with g k . By the monotonicity propertyof the level set flow, the sequence A (Γ , g k ) is bounded. We can assume thatthis sequence converges (by taking a subsequence if necessary) and we define A (Γ , g ) := lim k →∞ A (Γ , g k ) . In what follows, ∂X (resp. X , A (Γ , g )) will roughly play the role of ∂U (resp. U , the n -volume of the largest boundary component of U ) in [42,Section 2]. For simplicity, we will write A , A k , ˜ ω p ( g ), ˜ ω p ( g k ), for A (Γ , g ), A (Γ , g k ), ˜ ω p ( X, g ), ˜ ω p ( X k , g k ) respectively.By [42, Theorem 10], for any fixed p , there is a stationary integral varifold V k with support a closed minimal hypersurfaces of Morse index at most p embedded inside the interior of ( X k , g k ), each component intersecting M \ K by the maximum principle, such that the total mass of V k is ˜ ω p ( g k ). Mul-tiplicities of the 1-sided components of spt( V k ) are even. Making k → ∞ and taking a subsequence, V k converges in the varifold sense to a stationaryintegral varifold V k with support a minimal hypersurface embedded inside INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 25 the interior of X ([41]), and closed because ( M, g ) is thick at infinity. More-over, since g = g k except very close to ∂B k , V k is stationary for the originalmetric g if k is large. In other words, we just proved the following Proposition 14.
For all p there are disjoint connected closed embeddedminimal hypersurface Γ ( p ) i ⊂ (Int( X ) , g ) ( ≤ i ≤ J ) and integers m p,i ( m p,i is even when Γ ( p ) i is 1-sided) satisfying ˜ ω p ( X, g ) = J X i =1 m p,i Vol n (Γ ( p ) i ) . We denote by Vol n ( ., g k ) the n -volume computed with g k . If ¯ D is a com-pact domain of M , for all k it induces a compact subset ¯ D ′ in X k by pullingback with the natural map X k → M , and since X k ⊂ C ( X k ), ¯ D ′ in turn in-duces a compact subset of C ( X k ) that we call i k ( ¯ D ). For a region R ⊂ C ( X k )with rectifiable boundary, we will denote by Z n,rel ( R ; Z ) the space of rela-tive cycles mod 2 in the closure of R (see [1, Definition 1.20], [25, 2.2]).Before continuing, let us remark the following. Lemma 15.
For all ¯ ǫ > , there is a compact domain ¯ D ⊂ M so that forany k large enough, the region X k \ i k ( ¯ D ) ⊂ C ( X k ) has a -sweepout { S t } t ∈ [0 , where S t ∈ Z n,rel ( X k \ i k ( ¯ D ); Z ) for each t ∈ [0 , , satisfying sup t ∈ [0 , { Vol n ( S t , g k ) } ≤ A + ¯ ǫ. Proof.
Recall that N Γ is the neighborhood of Γ introduced at the beginningof this subsection.For any k , let Y k be the compact domain image of X k under the naturalmap X k → M . By definition of A and the properties of g k , for ¯ k largeenough A ¯ k ≤ A + ¯ ǫ/
2. Then the boundary components of Y ¯ k are closed2-sided and each of them has n -volume at most A ¯ k . For k large enough, B k contains Y ¯ k in its interior. Let ˆ C be any component of ∂Y ¯ k , that we consideras a cycle in Z n ( B k ; Z ). Consider the following minimization problem withconstraint (we used a similar minimization problem in the proof of Theorem4): minimize Vol n ( ˆ C , g k ) among cycles ˆ C such that • there is a path { ˆ C t } t ∈ [0 , continuous in the F -topology with ˆ C = ˆ C , • the support of each ˆ C t is in B k \ N Γ , • Vol n ( ˆ C t , g k ) ≤ A + ¯ ǫ for all t ∈ [0 , C and a path { ˆ C t } t ∈ [0 , ) exists by compactnessin the flat topology, and interpolation results [28, Proposition A.2]. Considerthe image of { ˆ C t } t ∈ [0 , by the Almgren map, A ( { ˆ C t } ) ∈ I n +1 ( B k ; Z ) (seeAppendix B). We call it R ( ˆ C ) for simplicity and to avoid confusion with thenotation A = A (Γ , g ). Now we can repeat that construction for any component of ∂Y ¯ k . Consider Y ¯ k as an element of I n +1 ( B k ; Z ) and let Z k := Y ¯ k + X ˆ C component of ∂Y k R ( ˆ C ) ∈ I n +1 ( B k ; Z ) . The support of Z k is non-empty because it contains N Γ and is a compact( n + 1)-dimensional region with weakly mean convex boundary for g k ( ∂Z k isnot necessarily smooth but is locally the intersection of domains with smoothweakly mean convex boundary). Hence Z k serves as a barrier domain forthe level set flow, which means that if Y k is the image of X k by the naturalmap X k → M , then by a slight abuse of notations Y k ⊂ Z k . Set ¯ D := Y ¯ k . By concatenating the pathes { ˆ C t } one after the other (for ˆ C component of ∂Y ¯ k = ¯ D ) and restricting these cycles to Y k \ ¯ D , we construct a 1-sweepout { S (1) t } of Y k \ ¯ D such that sup t Vol n ( S (1) t , g k ) ≤ A + ¯ ǫ , which pulls back to asweepout called { S t } of X k \ i k ( ¯ D ) ⊂ C ( X k ), where i k ( ¯ D ) ⊂ X k ⊂ C ( X k ) hasbeen defined just before this lemma and where S t ∈ Z n,rel ( X k \ i k ( ¯ D ); Z ) foreach t ∈ [0 , t ∈ [0 , { Vol n ( S t , g k ) } ≤ A + ¯ ǫ so the lemma is proved. (cid:3) We want to show the analogue of [42, Theorem 9] for ˜ ω p ( g ): Proposition 16. ∀ p ≥ ω p +1 ( g ) − ˜ ω p ( g ) ≥ A , lim p →∞ ˜ ω p ( g ) p = A . Proof.
By Theorem 9 of [42], for all p ≥ ω p +1 ( g k ) − ˜ ω p ( g k ) ≥ A k and˜ ω p ( g k ) ≥ p. A k , hence passing to the limit when k → ∞ ,(9) ˜ ω p +1 ( g ) − ˜ ω p ( g ) ≥ A , ˜ ω p ( g ) ≥ p. A . Let ¯ ǫ > k be large enough so that Lemma 15 is satisfiedand A k ≤ A + ¯ ǫ . Let D be a compact region of C ( X k ) containing X k . Theregion D \ X k has a 1-sweepout { T t } t ∈ [0 , with(10) sup t Vol n ( T t ) ≤ A + ¯ ǫ. To explain that, let Σ , ..., Σ m be the components of ∂X k , and let L be alarge number. Consider the function f L : ∂X k × [0 , L ] → R f L ( x, t ) := ( j − L + t if ( x, t ) ∈ Σ j × [0 , L ] . INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 27
Then the level sets of f L gives a 1-sweepout of D \ X k as desired if L is largeenough so that X k ∪ ∂X k × [0 , L ] contains D , i.e. we consider T t := f − L ( t ) ∩ ( D \ X k ) ∈ Z n,rel ( D \ X k ; Z ) . Now by concatenating { S t } t from Lemma 15 with { T t } t ∈ [0 ,L ] , we get a1-sweepout { U s } s ∈ [0 ,L +1] of the disjoint union of X k \ i k ( ¯ D ) ⊔ D \ X k satisfyingsup t Vol n ( U t ) ≤ A + ¯ ǫ. Note that U s is not in general a cycle in Z n ( C ( X k ); Z ) because the relativecycles S t , T t may have boundaries, so { U s } is not a 1-sweepout of the union X k \ i k ( ¯ D ) ∪ D \ X k considered as a subset of C ( X k ). Next we will see how toform a genuine sweepout made of cycles out of { U s } .Again by the proof of [29, Theorem 5.1, Claim 5.6], we can construct from { U s } for each p ≥ p -sweepouts Φ p (with domain R P p ) of the disjointunion X k \ i k ( ¯ D ) ⊔ D \ X k such thatsup x ∈ R P p M (Φ p ( x ) , g k ) ≤ p. ( A + ¯ ǫ ) . On the other hand, since a p -sweepout of ¯ D ⊂ M lifts to a p -sweepout of i k ( ¯ D ) ⊂ C ( X k ), [29, Theorem 5.1], there is a sequence of sweepouts Ψ p of i k ( ¯ D ) (with domain R P p ) so that for a constant C = C ( ¯ D, g ) (independentof k ), ∀ p ≥ x ∈ R P p M (Ψ p ( x ) , g k ) ≤ C p n +1 . As explained in the proof of Theorem 9 in [42], we can glue the p -sweepoutsΦ p and Ψ p parametrized by R P p using [25] and get a new p -sweepout ˆΦ p of D ⊂ C ( X k ) with domain R P p such that:sup x ∈ R P p M ( ˆΦ p ( x ) , g k ) ≤ p. ( A + ¯ ǫ ) + C p n +1 + Vol n ( ∂ ( i k ( ¯ D )) , g k )+ Vol n ( ∂ ( X k \ i k ( ¯ D )) , g k ) + Vol n ( ∂X k , g k ) . As ¯ D is fixed and ∂X k has n -volume less than twice that of Γ by monotonic-ity of the level set flow, the last three n-volumes above are bounded by aconstant C independent of k .Taking D arbitrarily large, we get for all k large ˜ ω p ( g k ) ≤ p. ( A + ¯ ǫ ) + C p n +1 + C and thus˜ ω p ( g ) ≤ p. ( A + ¯ ǫ ) + C p n +1 + C , which implies that lim sup p →∞ ˜ ω p ( g ) p ≤ A + ¯ ǫ . Since ¯ ǫ was arbitrarily small,we conclude with (9) that the proposition is true. (cid:3) Finally we have:
Lemma 17.
Suppose that any closed embedded minimal hypersurface in X intersecting Γ is a saddle point minimal hypersurface. Then for all closedembedded minimal hypersurface Σ ⊂ Int( X ) intersecting Γ , Vol n (Σ) > A (Γ , g ) if Σ is 2-sided , n (Σ) > A (Γ , g ) if Σ is 1-sided . Proof.
Let us check the lemma when Σ is 2-sided, the other case being simi-lar. It is a saddle point by assumption, so there is a mean concave neighbor-hood N Σ of Σ foliated by hypersurfaces with mean curvature vector pointingaway from Σ when non-zero. The boundary ∂N Σ has two connected compo-nents Σ , Σ whose n -volume are both strictly less than Vol n (Σ). For k large, N Σ ⊂ Int( X k ) since g k = g on larger and larger balls. For k large, minimalhypersurfaces in ( X k , g k ) with n -volume at most max { Vol n (Σ ) , Vol n (Σ ) } and intersecting Γ are saddle point minimal hypersurfaces: to see this sup-pose there is a sequence of minimal hypersurfaces S k i ⊂ ( X k i , g k i ) eitherstrictly stable or degenerate stable of Type I or II, intersecting Γ and of n -volume bounded uniformly. Then a subsequence converges, which is closedby thickness at infinity and it means that for k i large, S k i is minimal for g ,intersects Γ but is not a saddle point: contradiction. Now since moreoverminimal hypersurfaces in (Int( X k ) , g k ) not intersecting Γ are not locally areaminimizing but degenerate stable of Type I, by arguments in the proof of[42, Lemma13] applied to ( X k , g k ),max { Vol n (Σ ) , Vol n (Σ ) } > A k . Since A k converges to A , we get the lemma. (cid:3) Zero-infinity dichotomy for the space of cycles in manifoldsthick at infinity.
We now state our main theorem, which can be thoughtof as an extension of Yau’s conjecture to non-compact manifolds thick atinfinity:
Theorem 18 (Zero-infinity dichotomy) . Let ( M, g ) be an ( n +1) -dimensionalcomplete manifold with ≤ n ≤ , thick at infinity. Then the following di-chotomy holds true:(1) either ( M, g ) contains infinitely many saddle point minimal hyper-surfaces,(2) or there is none; in that case for any compact domain B , there is anembedded closed area minimizing hypersurface Σ B ⊂ ( M, g ) (maybeempty) such that B \ Σ B has a singular weakly mean convex foliation. By inspecting the proof of the above theorem, we can check that it alsoholds more generally for manifolds M n +1 (2 ≤ n ≤
6) thick at infinity withminimal boundary, such that each component of ∂M is closed. In particu-lar, if M is compact with minimal boundary, then the following dichotomyholds: either there are infinitely many saddle point minimal hypersurfacesin the interior of M , or there is a closed embedded area minimizing minimal INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 29 hypersurface Σ ⊂ M such that M \ Σ has a singular weakly mean convexfoliation.As briefly mentionned in the introduction, there is an interpretation ofthe zero-infinity dichotomy in Morse theoric terms. Let Z n ( M, Z ) be thespace of integral cycles with bounded support in M , endowed with the flattopology. It is a space of generalized closed hypersurfaces and the massfunctional M extends the notion of n -volume for smooth hypersurfaces.The dichotomy then says that either Z n ( M, Z ) contains infinitely many“non-trivial critical points” of M whose supports are smooth minimal hy-persurfaces, or Z n ( M, Z ) is locally simple: the only “critical points” aresupported on smooth stable minimal hypersurfaces and for each boundedregion B of M , the set of elements of Z n ( M, Z ) with support inside B canbe contracted to M -minimizing elements of Z n ( M, Z ) by a retraction flowwhich is M -nonincreasing and continuous in the flat topology.The proof of Theorem 18 will follow from Theorem 13 and the following: Theorem 19.
Let ( M, g ) be an ( n + 1) -dimensional complete manifold with ≤ n ≤ , thick at infinity. If there exists a saddle point minimal hypersur-face, then there exists infinitely many.Proof. We suppose that M is thick at infinity. Unless specified, minimalhypersurfaces are closed embedded. We assume that the metric g satisfiesCondition [M], otherwise there are already infinitely many saddle points(Appendix A).Let Γ be a saddle point minimal hypersurface in ( M, g ). We are now in thesituation treated by Subsection 3.2. We will explain the case where Γ is 2-sided for simplicity, but the case where it is 1-sided is completely analogous.Γ has to be either unstable, or degenerate stable of Type III, and so we canfind a neighborhood N Γ of Γ and a diffeomorphism φ : Γ × ( − δ , δ ′ ) → N Γ such that φ (Γ ×{ } ) = Γ, the mean curvature of φ (Γ ×{ s } ) is either vanishingor non-zero pointing away from Γ, and φ (Γ × {− δ } ), φ (Γ × { δ ′ } ) have non-zero mean curvature.Let K := M \ N Γ . Let X be generated by Γ, K in ( M, g ) as explained inSubsection 3.2. Recall that X is the limit of compact manifolds X k .Suppose first that there is a closed minimal hypersurface S embeddedinside the interior of X , intersecting Γ, which is either strictly stable or de-generate stable of Type I or II. If S is not non-separating degenerate stableof Type I , let Y denotes the metric completion of X \ S . Otherwise S is 2-sided, and if X ′ is the metric completion of X \ S , two boundary components T , T of same n -volume come from S . One (say T ) is locally area minimiz-ing inside X ′ , the other component T is not. We minimize the n -volume ofa hypersurface in X ′ close to T in its homology class and get a 2-sided lo-cally area minimizing closed minimal hypersurface W that separates T from T : let Y be defined as the metric completion of the component of X ′ \ W containing T . In any case, Y has locally area minimizing boundary, whosecomponents are closed. Let Γ ∗ ⊂ Y denote the preimage of Γ under the natural map Y → M . We apply Theorem 13 to a domain B ⊂ Y containingΓ ∗ (see Remark 3.1), and we claim that only case (1) can happen: if case(2) was true, there would be a locally area minimizing minimal hypersurfaceΣ B and a weakly mean convex foliation { A t } t ≥ of B \ Σ B . Since ∂B ∩ ∂Y islocally area minimizing, there is a first time T when A T touches Γ ∗ , whichof course contradicts the maximum principle. Thus we deduce that thereis a saddle point Γ ⊂ Y , different from Γ. We can reapply this discussionto the new saddle point Γ , get a manifold X (1) with compact locally areaminimizing boundary, and if X (1) contains in its interior a closed minimalhypersurface S intersecting Γ , which is either strictly stable or degeneratestable of Type I or II, we get a new saddle point Γ different from Γ and Γ .If at each step, X ( j ) contains in its interior a minimal hypersurface eitherstrictly stable or degenerate stable of Type I or II, then we produce an infi-nite sequence of distinct saddle points Γ j , and the conclusion of the theoremis verified.For these reasons, we now only need to assume that if X , Γ are as above,minimal hypersurfaces embedded in the interior of X intersecting Γ areeither unstable or degenerate stable of Type III, that is, they are all saddlepoints. Note that this condition implies the(11)Frankel property inside X for minimal hypersurfaces intersecting Γ:any two closed embedded minimal hypersurfaces in Int( X )intersecting Γ have to intersect each other.To check this, first notice by the maximum principle applied to ( X k , g k ) for k large, that any minimal hypersurface in Int( X ) disjoint from Γ is of the form φ (Γ × { s } ) for some s ∈ ( − δ , δ ′ ) \{ } , so it is degenerate stable of Type I.Secondly, if there were two disjoint saddle point minimal hypersurfaces S , S in Int( X ) (intersecting Γ), by a minimization argument we would get aminimal hypersurface S in Int( X ) which is • either strictly stable or degenerate stable of Type II, and in thesecases S intersects Γ, contradicting our assumption on X , • or degenerate stable of Type I, in that case S has to be S or S , acontradiction.Before continuing, we explain why all connected components of the min-imal hypersurfaces we will construct from now on will always intersect Γ.If Γ is unstable then we could have chosen φ so that all the hypersurfaces φ (Γ × { s } ) with s = 0 are non-minimal with mean curvature vector pointingaway from Γ. In that case by the maximum principle any minimal hypersur-face in (Int( X ) , g ) intersects Γ so we are done. However if Γ is degeneratestable of Type III, it might happen that some φ (Γ × { s } ) distinct from Γare also minimal. By Lemma 23 there is a sequence of metrics h ( q ) converg-ing to g so that Γ is still a saddle point and any φ (Γ × { s } ) with s = 0 isnon-minimal. Hence any minimal hypersurface in (Int( X ) , h ( q ) ) intersects Γ INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 31 by the maximum principle. Applying min-max theory to (
X, h ( q ) ) will pro-duce for each integer p a minimal hypersurface with integer multiplicitiesin Int( X ), intersecting Γ, of Morse index and n -volume bounded indepen-dently of q . By [41], we take a subsequence limit as q → ∞ and get aminimal hypersurface with similar properties in (Int( X ) , g ): in particulareach component intersects Γ. Thus by (11), the usual Frankel property,even though it may not be satisfied for all minimal hypersurfaces in Int( X ),will be satisfied for all the minimal hypersurfaces we will consider until theend of this proof (since they are constructed by min-max). In what follows,we will implicitly assume the use of such a limiting procedure involving h ( q ) .For the end of the proof, the strategy to produce infinitely many saddlepoints is to use arguments of the solution of Yau’s conjecture [29] [42]. Inthe simplest case where X is actually closed compact then by the Frankelproperty in X satisfied by minimal hypersurfaces intersecting Γ, [29] impliesthe existence of infinitely many minimal hypersurfaces in X which are saddlepoint minimal hypersurfaces by assumption on X .In Subsection 3.2, we defined some numbers ˜ ω p ( X, g ) ( p ≥
1) and A (Γ , g ).Suppose that X is non-compact without boundary and A (Γ , g ) = 0. ByProposition 14 and by (11), the widths ˜ ω p ( X, g ) are all finite, for each p thereis a connected closed embedded minimal hypersurface Γ ( p ) ⊂ (Int( X ) , g ) anda positive integer m p (which is even when Γ ( p ) is 1-sided) satisfying(12) ˜ ω p ( X, g ) = m p Vol n (Γ ( p ) ) . Moreover we have the following asymptotics (Propostion 16):lim p →∞ ˜ ω p ( X, g ) /p = 0 . We claim that this implies the existence of infinitely many saddle pointminimal hypersurfaces in Int( X ). Suppose by contradiction that there areonly finitely many minimal hypersurfaces Σ , ..., Σ L ⊂ Int( X ) intersectingΓ. Then ˜ ω p ( X, g ) is a strictly increasing sequence. To explain this, considerthe compact manifolds ( X k , g k ) defined in Subsection 3.2; by thickness atinfinity, for any p and for k large the min-max minimal hypersurface in( X k , g k ) whose n -volume with multiplicity is ˜ ω p ( X k , g k ) is actually a minimalhypersurface for the original metric g (see paragraph right before Proposition14). Consequently we can find a subsequence { X k i } of { X k } , so that bothsequences { ˜ ω p ( X k i , g k i ) } and { ˜ ω p +1 ( X k i , g k i ) } stabilize to a constant, i.e. ∀ i ≥ ω p ( X k i , g k i ) = ˜ ω p ( X, g ) , ˜ ω p +1 ( X k i , g k i ) = ˜ ω p +1 ( X, g ) . So if ˜ ω p ( X, g ) = ˜ ω p +1 ( X, g ), then in particular ˜ ω p ( X k , g k ) = ˜ ω p +1 ( X k , g k )but this is not possible by [42, Theorem 9 (1)] and since ∂X k = ∅ ( X isnon-compact). The counting argument of [29, Section 7] is then enough toget the existence of infinitely many minimal hypersurfaces (contradictingour assumption that there were only finitely many). The theorem is thenproved in that case. Suppose finally that X is non-compact (with empty or non-empty bound-ary) and A (Γ , g ) >
0. We appeal to the method of [42] as follows. InSubsection 3.2, we saw that ˜ ω p ( X, g ) are finite numbers, and by Proposition14 combined with (11) for all p there is a connected closed embedded min-imal hypersurface Γ ( p ) ⊂ Int( X ) and a positive integer m p (which is evenwhen Γ ( p ) is 1-sided) satisfying˜ ω p ( X, g ) = m p Vol n (Γ ( p ) ) . Furthermore, by Proposition 16, for all p ˜ ω p +1 ( X, g ) − ˜ ω p ( X, g ) ≥ A (Γ , g ) , lim p →∞ ˜ ω p ( X, g ) /p = A (Γ , g ) . Additionally, by Lemma 17 for any closed embedded minimal hypersurfaceΣ ⊂ Int( X ), Vol n (Σ) > A (Γ , g ) if Σ is 2-sided , n (Σ) > A (Γ , g ) if Σ is 1-sided . Consequently the previous identities and estimates combined with the arith-metic lemma [42, Lemma 14] imply that there are infinitely many closed em-bedded minimal hypersurfaces inside Int( X ) intersecting Γ, which we recallhave to be saddle points by assumption on X . This finishes the proof of thetheorem. (cid:3) Proof of Theorem 18.
Theorem 18 readily ensues from Theorem 13 and The-orem 19. The only non trivial point is that in item (2), for any compactdomain B , the minimal hypersurface Σ B is globally area minimizing in its Z -homology class in ( M, g ) (instead of just locally area minimizing as initem (2) of Theorem 13). Assume that there are no saddle points, let B be a bounded domain and Σ B the associated locally area minimizing mini-mal hypersurface given by Theorem 13 (2). Suppose that there is anotherclosed embedded hypersurface Γ with Vol n (Γ) ≤ Vol n (Σ B ) and such that ∂D = Γ ∪ Σ B for some compact domain D . Let X be a compact domaincontaining D , by Theorem 13 (2), there are a locally area minimizing hy-persurface Σ X ⊂ ( M, g ), a bigger region Y ⊃ X with a metric g Y coincidingwith g on X , and a collection of weakly mean convex closed sets { K t } t ≥ such that K t foliates X \ Σ X . We can minimize the n -volume of Γ inside K (which is mean convex) and obtain a locally area minimizing minimalhypersurface Γ ′ ⊂ ( K , g Y ), of n -volume at most Vol n (Σ B ). The non strictlymean convex K t have smooth minimal boundary ∂K t which have to be de-generate stable of Type I. Hence Σ B ⊂ Σ X and Γ ′ ⊂ Σ X . But since thesetwo hypersurfaces are Z -homologous inside K , Σ B = Γ ′ . Moreover a pos-teriori, we now know that Γ was actually already locally area minimizing,hence Γ ⊂ Σ X and so again Γ = Σ B . This proves that Σ B ⊂ ( M, g ) is theunique area minimizer in its Z -homology class inside ( M, g ). INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 33 (cid:3) Yau’s conjecture for finite volume hyperbolic -manifolds Combining methods from previous sections and [9, 10], we prove thatYau’s conjecture holds true for finite volume hyperbolic 3-manifolds.
Theorem 20.
In any finite volume hyperbolic -manifold M , there are infin-itely many saddle point minimal surfaces. In particular, there are infinitelymany closed embedded minimal surfaces.Proof. We assume that g hyp satisfies Condition [M], otherwise the conclusionof the theorem is already true. Let p ∈ M and denote by B r ( p ) the geodesicball of radius r centered at p in ( M, g hyp ).If (
M, g hyp ) is not compact then outside of a compact subset, M is madeof hyperbolic cusps C , ..., C K , each of them is naturally foliated by mean-concave tori { T ( k ) t } t ≥ ( k = 1 , ..., K ). Let us cut M along a collection of tori { T (1) t , ..., T ( K ) t K } to get M ′ , and deform g hyp a bit around these tori to obtaina metric g ′ with respect to which these tori become minimal stable, andsuch that { T ( k ) t } t ∈ [0 ,t k ) remain strictly mean concave (mean curvature vectorpointing towards T t k ). If t k are chosen large enough and the deformationssmall enough then by [10], for all A >
0, there is R = R ( A ) independent of { t k } k ∈{ ,...,K } so that any closed embedded minimal surface in ( M ′ , g ′ ) withMorse index bounded by A is contained in B R ( p ) or is a component of ∂M m .Hence by choosing t k larger and larger, we construct a sequence of compactmanifolds ( M m , g m ) approximating the hyperbolic manifold ( M, g hyp ) withthe following properties: • M m ⊂ M , the metrics g m and g hyp coincide on ( B m ( p ) , g hyp ), • for all ǫ >
0, there is a radius r = r ( ǫ ) so that for all m large,( M m \ B r ( p ) , g m ) has a 1-sweepout { Σ t } t ∈ [0 , with sup t ∈ [0 , Vol n (Σ t ) ≤ ǫ , • ∂M m is non-empty, is a strictly stable minimal surface and has areaconverging to zero, • for all A >
0, there is R = R ( A ) so that any closed embeddedminimal surface in ( M m , g m ) not equal to a component of ∂M m , andwith Morse index bounded by A is contained in B R ( p ), • the widths of ( M m , g m ) are uniformly bounded independently of m .By these properties, applying Theorem 12 to each ( M m , g m ) and using thefourth bullet, we get a saddle point minimal surface Γ ⊂ ( M, g hyp ).Let us explain how to adapt the proof of Theorem 19. As in the proofof Theorem 19, there is a thin strictly mean concave neighborhood N Γ ofΓ ⊂ ( M, g hyp ) which is foliated by surfaces with mean curvature pointingaway from Γ when non-zero. For each m large, define K m := M m \ N Γ . Let X m be generated by Γ, K m in ( M m , g m ). Then X m is compact and has locallyarea minimizing boundary. By the properties of g m and of hyperbolic cusps,( X m , g m ) converges (say in the Gromov-Hausdorff distance) to a manifold X with compact boundary endowed with a hyperbolic metric still denoted by g hyp . Let ( C ( X ) , h ) be the result of gluing ( X, g hyp ) to a straight half-cylinder( ∂X × [0 , ∞ ) , g hyp (cid:12)(cid:12) ∂X ⊕ dt ). For each positive integer p , let˜ ω p ( X, g hyp ) = ω p ( C ( X ) , h )(see Definition 8 in [42]) and define A (Γ , g hyp ) = max { Vol n ( C ); C is a component of ∂X } . Similarly we introduce for all m large and all p , ˜ ω p ( X m , g m ) and A (Γ , g m ).It is not hard to see that(13) lim m →∞ A (Γ , g m ) = A (Γ , g hyp ) . Besides we have ∀ p, lim m →∞ ˜ ω p ( X m , g m ) = ˜ ω p ( X, g hyp ) . The inequality ≥ follows from definitions, whereas ≤ comes from the secondbullet in the list of properties of g m , the Fact of Subsection 3.2 and thepossibility to glue two p -sweepouts together with good bound on the mass(see end of proof of Theorem 16).If there is a closed minimal surface S in Int( X ) intersecting Γ, eitherstrictly stable or degenerate stable of Type I or II, then by the same argu-ments as in the proof of Theorem 19 applied to ( X m , g m ), for m large we getanother saddle point minimal surface Γ ⊂ ( X, g hyp ) and we can continue.Either we get infinitely many saddle points or the process stops.Hence we can assume that actually any minimal surface intersecting Γ is asaddle point. As in the proof of Theorem 19, by an approximation argument,minimal surfaces produced by min-max can be supposed to intersect Γ.From here, we can apply almost verbatim the end of proof of Theorem 19to (
X, g hyp ) by using ( X m , g m ) for m large. More precisely: if X is compactwithout boundary then we conclude using [29]. If X is non-compact, for all p we produce as in the proof of Proposition 14 a connected minimal surfaceΓ ( p ) m ⊂ (Int( X m ) , g m ) intersecting Γ, of index at most p , satisfying for aninteger k p,m (which is even when Γ ( p ) m is 1-sided):˜ ω p ( X m , g m ) = k p,m Vol n (Γ ( p ) m ) . By the fourth bullet in the list of properties of ( M m , g m ), for each p , Γ ( p ) m is aminimal hypersurface for the original hyperbolic metric if m is large enough.By taking a converging subsequence, there is a connected closed embeddedminimal surface Γ ( p ) ⊂ (Int( X ) , g hyp ) intersecting Γ and a positive integer k p (which is even when Γ ( p ) is 1-sided) so that:(14) ˜ ω p ( X, g hyp ) = k p Vol n (Γ ( p ) ) . Here the use of the fourth bullet in the properties of g m is essential to geta closed limit surface, since ( M, g hyp ) is not thick at infinity in general.
INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 35
Moreover since for all m , ˜ ω p +1 ( X m , g m ) − ˜ ω p ( X m , g m ) ≥ A (Γ , g m ), we getfrom (13) for all p :˜ ω p +1 ( X, g hyp ) − ˜ ω p ( X, g hyp ) ≥ A (Γ , g hyp ) . Since outside a compact set, X is made of finitely many cusps, for any ǫ > R ′ so that any bounded domain of X \ B R ′ ( p ) has a 1-sweepout { Σ t } t ∈ [0 , with sup t Area(Σ t ) ≤ ǫ . Thus by arguments of Subsection 3.2, weget lim p →∞ ˜ ω p ( X, g hyp ) p = A (Γ , g hyp ) . As in the proof of Theorem 19, by (13) we have for any minimal surface in(Int( X ) , g hyp ) intersecting Γ:Vol n (Σ) > A (Γ , g hyp ) if Σ is 2-sided,2 Vol n (Σ) > A (Γ , g hyp ) if Σ is 1-sided.We have the ingredients to conclude. If X is non-compact and has non-empty boundary then [42, Lemma 14] imply the existence of infinitely manyminimal surfaces inside Int( X ) intersecting Γ. They are saddle points byassumption on X and Γ. It remains to treat the case where X is non-compact and has empty boundary. Suppose towards a contradiction that { Γ ( p ) } p ≥ is a finite set. Then ˜ ω p ( X, g hyp ) is strictly increasing in p : indeedif ˜ ω p ( X, g hyp ) = ˜ ω p +1 ( X, g hyp ), by (14) and the finiteness of { Γ ( p ) } p ≥ , forsome m arbitrarily large˜ ω p ( X m , g m ) = ˜ ω p ( X, g hyp ) = ˜ ω p +1 ( X, g hyp ) = ˜ ω p +1 ( X m , g m ) , which is not possible because of ∂X m = ∅ and [42, Theorem 9 (1)]. Finallythe counting argument of [29, Section 7] applies and shows that { Γ ( p ) } p ≥ isactually an infinite set. (cid:3) Density of the union of finite volume minimalhypersurfaces
In our setting the natural topology on the space of complete metrics isthe usual strong C ∞ -topology, or “Whitney C ∞ -topology” (see [14, ChapterII, § b , b , ... be a sequence of openballs forming a locally finite covering M , and for each complete metric g ,and i , k , ǫ , set O ( g, i, k, ǫ ) := { g ′ ; || ( g ′ − g ) (cid:12)(cid:12) b i || C k ≤ ǫ } . Now consider g be a complete metric, e = ( ǫ , ǫ , ... ) a sequence of positivenumbers, k = ( k , k , ... ) a sequence of integers and set O ( g, e , k ) := ∞ \ i =1 O ( g, i, k i , ǫ i ) . Then by definition the strong C ∞ topology on the space of complete metricsof M has a basis of open sets given by { O ( g, e , k ) } g, e , k . The weak C ∞ -topology where a basis of open sets is given by { O ( g, i, k, ǫ ) } g,i,k,ǫ is lessrelevant for us, because it is “easier” to change the behavior at infinity andto be generic in this topology. The space of complete metrics endowed withthe strong topology is a Baire space.As usual, a closed minimal hypersurface is said to be non-degenerate whenit has no non-trivial Jacobi fields.Recall from the introduction that F thin (resp. T ∞ ) is the family of com-plete metrics on M with a thin foliation at infinity (resp. thick at infinity). F thin and F thin ∩ Int( T ∞ ) are non-empty open subsets for the strong topol-ogy. For instance, consider the following example already mentioned in theintroduction: if N is a closed manifold, there is a metric h on N × R witha thin foliation at infinity while also satisfying condition ⋆ k of [33], hencebeing thick at infinity. Since ⋆ k is an open condition, a neighborhood of h is in F thin ∩ Int( T ∞ ). Note that if M is compact, any metric on M is in F thin ∩ Int( T ∞ ). The following theorem generalizes the density theorem ofIrie, Marques and Neves [22] to these manifolds: Theorem 21.
Let M be a complete ( n + 1) -dimensional manifold with ≤ n ≤ .(1) For any metric g in a C ∞ -dense subset of F thin , the union of completefinite volume embedded minimal hypersurfaces in ( M, g ) is dense.(2) For any metric g ′ in a C ∞ -generic subset of F thin ∩ Int( T ∞ ) , theunion of closed embedded minimal hypersurfaces in ( M, g ′ ) is dense.Proof. Proof of (1):
Suppose that M is endowed with a metric g with athin foliation at infinity. We would like to find a metric h ∈ F thin arbitrarilyclose to g ∈ F thin in the C ∞ -topology, such that the union of complete finitevolume embedded minimal hypersurfaces in ( M, h ) is dense.Since (
M, g ) ∈ F ∞ , for any µ >
0, there is a compact subset C ⊂ M so that any bounded domain of M \ C has a foliation { Σ t } t ∈ [0 , (given forexample by the level sets of the function defining the thin foliation at infinity)such that sup t ∈ [0 , Vol n (Σ t ) ≤ µ. Hence by techniques explained in Subsection 3.2 and Proposition 16, thewidths ω p ( M, g ) are finite and satisfy(15) lim p →∞ ω p ( M, g ) p = 0 . Recall that b , b , ... are open balls forming a locally finite covering of M . Fix a sequence of integers ˆ k = ( k , k , ... ) and a sequence of positivenumbers ˆ ǫ = ( ǫ , ǫ ... ). In what follows, the norms || . || C k are all computedwith respect to the background metric g . By abuse of notations, for any INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 37 symmetric 2-tensor g ′ and constant c , we will write || g ′ || C ˆ k ≤ ˆ ǫc instead of ∀ j, || g ′ (cid:12)(cid:12) b j || C kj ≤ ǫ j c. Let B r q ( x q ) ⊂ ( M, g ) be a sequence of open balls centered at points x q , ofradii 2 r q , forming a base of open sets for the usual topology of M . We willconstruct successive deformations of g , called h = g , h , h ..., convergingto a metric h ∞ satisfying || g − h ∞ || C ˆ k ≤ ˆ ǫ , such that the union of completefinite volume embedded minimal hypersurfaces in ( M, h ∞ ) is dense. Supposethat the metrics h , h , ..., h L − have already been constructed and that(16) || g − h L − || C ˆ k ≤ ˆ ǫ L − X i =1 i . We construct h L as follows.First we can find compact approximations ( M m , g m ) of ( M, h L − ) in thefollowing manner. Let f : M → [0 , ∞ ) be the function defining the thinfoliation at infinity. One can check thatlim t →∞ Vol n ( f − ( t ) , h L − ) = lim t →∞ Vol n ( f − ( t ) , g ) = 0 . We can assume that positive integers are not critical values of f . Con-sider M ⊂ ... ⊂ M m ... the exhaustion of M by compact domains M m := f − ([0 , m ]). We perturb slightly the metric g on M m into g m so that • the metric g m is bumpy, • the boundary ∂M m is a strictly stable minimal hypersurface withrespect to g m , • if we restrict the tensors g m , h L − to M m then || g m − h L − || C ≤ m , • if we restrict the tensors g m , h L − to M m \ N m where N m is a 1 /m -neighborhood of ∂M m , then || g m − h L − || C k ≤ m .Let ˜ ω p ( M m , g m ) := ω p ( C ( M m ) , g m ) where C ( M m ) is the result of gluing to M m a straight half-cylinder ( ∂M m × [0 , ∞ ) , g m (cid:12)(cid:12) ∂M m ⊕ dt ) along ∂M m (as inSubsection 3.2).Let q L be the first integer for which B r qL ( x q L ) does not intersect anycomplete finite volume embedded minimal hypersurface in ( M, h L − ). If q L does not exists, we are done by taking h L = h ∞ . Otherwise let s L be anonnegative symmetric 2-tensor with support B r qL ( x q L ) and equal to h L − inside B r qL ( x q L ). There is a µ L > ∀ t ∈ [0 , µ L ] , || ts L || C ˆ k ≤ ˆ ǫ L . Consider the deformation of g m given by g m ( t ) = g m + ts L , t ∈ [0 , µ L ] . For each fixed p , by the second bullet, Vol n ( ∂M m , g m ) goes to zero so byarguments in proof of Theorem 20,lim m →∞ ˜ ω p ( M m , g m ) = ω p ( M, h L − ) , lim m →∞ ˜ ω p ( M m , g m + µ L s L ) = ω p ( M, h L − + µ L s L ) . (18) Claim:
There is an integer ¯ p , such that for all m large enough,˜ ω ¯ p ( M m , g m + µ L s L ) > ˜ ω ¯ p ( M m , g m ) . We postpone its proof. Fix ¯ p as in the Claim. From [42, Theorem 10]that for all t ∈ [0 , µ L ], there are connected closed minimal hypersurfacesΓ , ..., Γ P embedded in Int( M m ) and positive integers q , ..., q P so that˜ ω ¯ p ( M m , g m + ts L ) = P X i =1 q i Vol n (Γ i , g m + ts L ) . The hypersurfaces Γ i can be chosen to have index at most ¯ p [28]. Since g m is bumpy the set of numbers { P X i =1 q ′ i Vol n (Γ ′ i , g m ); q ′ i integers , Γ ′ i ⊂ ( M m , g m ) closed minimal hypersurfaces } is countable. Moreover, since ˜ ω ¯ p ( M m , g m + ts L ) is continuous in t , for all m large, there exists t ′ m ∈ (0 , µ L ) for which there is a minimal hypersurfaceΓ ( m ) ⊂ ( M m , g m + t ′ m s L ) intersecting the support of s L , B r qL ( x q L ), of indexat most ¯ p . Taking a subsequence limit as m → ∞ ([41]), we get a t ′∞ ∈ [0 , µ L ]and a complete finite volume embedded minimal hypersurface in ( M, h L − + t ′∞ s L ) intersecting B r qL ( x q L ). Set h L = h L − + t ′∞ s L . By (17), we have || h L − h L − || C ˆ k ≤ ˆ ǫ L so indeed || g − h L || C ˆ k ≤ ˆ ǫ P Li =1 12 i . Wecontinue this construction and get by completeness a limit metric h ∞ with || g − h ∞ || C ˆ k ≤ ˆ ǫ , for which the union of complete finite volume embeddedminimal hypersurfaces in ( M, h ∞ ) is dense. Note that it was very helpful touse [42, Theorem 10] and construct closed minimal hypersurfaces in orderto take advantage of the bumpy metric theorems [43, 45].Let us prove the Claim (which plays here the role of the Weyl Law [25] in[22]). Because of (18), we only need to show that for a certain ¯ p , ω ¯ p ( M, h L − + µ L s L ) > ω ¯ p ( M, h L − ) . The large inequality is always true since s L is nonnegative, the point is toshow that these two terms are not equal. Note that as a consequence of (15)and (16), (17), there is a subsequence { p k } and a sequence { δ k } such that(19) ∀ k ω p k +1 ( M, h L − ) − ω p k ( M, h L − ) = δ k , lim k →∞ δ k = 0 . Suppose towards a contradiction that for all k we have ω p k +1 ( M, h L − + µ L s L ) = ω p k +1 ( M, h L − ) , INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 39 ω p k ( M, h L − + µ L s L ) = ω p k ( M, h L − ) . Let ¯ δ > D ⊂ M be a compactdomain containing B r qL ( x r qL ). Consider a ( p k + 1)-sweepout Φ : X →Z n,rel ( D ; Z ) so thatsup x ∈ X M (Φ( x ) , h L − + µ L s L ) ≤ ω p k +1 ( M, h L − + µ L s L ) + ¯ δ where M ( .g ′ ) denotes the mass computed with a metric g ′ . For clarity let uswrite B r q ( x r q ) instead of B r qL ( x r qL ). Now the key remark is that Φ restrictedto X := { x ∈ X ; M (Φ( x ) x B r q ( x r q ) , h L − + µ L s L ) ≥ ω ( B r q ( x r q ) , h L − + µ L s L ) / } is a p k -sweepout of D . The proof is a Lusternik-Schnirelmann type argumentused in [15, 16] (see also [19, Section 3] [29, Section 8]). Indeed suppose thatit is not, remark that Φ restricted to X := X \ X is clearly not a 1-sweepout(a 1-sweepout of M has to be a 1-sweepout of B r q ( x r q ) after restricting theimage currents to B r q ( x r q )). Consider λ := Φ ∗ (¯ λ ) ∈ H ( X, Z ) as in [29,Definition 4.1]. Consider the inclusion maps i a : X a → X ( a = 1 , i ∗ ( λ p k ) = 0 ∈ H p k ( X , Z ), i ∗ ( λ ) = 0 ∈ H ( X , Z ). Consider the exact sequences H p k ( X, X ; Z ) j ∗ → H p k ( X ; Z ) i ∗ → H p k ( X ; Z ) ,H ( X, X ; Z ) j ∗ → H ( X ; Z ) i ∗ → H ( X ; Z ) . Then we can find λ ∈ H p k ( X, X ; Z ), λ ∈ H ( X, X ; Z ) so that j ∗ ( λ ) = λ p k , j ∗ ( λ ) = λ , which implies λ p k +1 = j ∗ ( λ ) ⌣ j ∗ ( λ ) = j ∗ ( λ ⌣ λ )but the last term has to be zero since it belongs to H p k +1 ( X, X ∪ X ; Z ) = H p k +1 ( X, X ; Z ) = 0. This is impossible by definition of ( p k + 1)-sweepouts,hence Φ (cid:12)(cid:12) X is a p k -sweepout. Then since s L = h L − in B r q ( x r q ), if weestimate the mass for the metric h L − :sup x ∈ X M (Φ( x ) , h L − ) ≤ sup x ∈ X M (Φ( x ) , h L − + ǫs L ) − (1 − (1 + µ L ) − n/ ) ω ( B r q ( x r q ) , h L − + µ L s L ) / ≤ ω p k +1 ( M, h L − ) + ¯ δ − (1 − (1 + µ L ) − n/ ) ω ( B r q ( x r q ) , h L − + µ L s L ) / ≤ ω p k ( M, h L − ) + δ k + ¯ δ − (1 − (1 + µ L ) − n/ ) ω ( B r q ( x r q ) , h L − + µ L s L ) / . However, as k goes to infinity, δ k converges to zero while the last term is pos-itive and independent of k , D , ¯ δ : it follows that if we chose k large and then¯ δ smaller than a fraction of the last term, then sup x ∈ X M (Φ( x ) , h L − ) < ω p k ( M, h L − ) for any D large enough. This obviously contradicts the defi-nition of p k -width ([42, Definition 8]). Hence the Claim is verified. Proof of (2):
Let U := F thin ∩ Int( T ∞ ). Reasoning as in [22], it is enoughto show that for a bounded open set U ⊂ M , the space M U of metrics g in U such that there is a non-degenerate closed embedded minimal hypersurface in( M, g ) intersecting U is open and dense inside U . Openness follows from [43].Denseness is proved as follows. Fix a sequence of integers ˆ k = ( k , k , ... ) anda sequence of positive numbers ˆ ǫ = ( ǫ , ǫ , ... ), we follow the same notationsas previously. We pick a metric ¯ g ∈ U , let ¯ s be a nonnegative symmetric2-tensor with support U and equal to ¯ g in an open ball B ⊂ U . For µ > ∀ t ∈ [0 , µ ] , || t. ¯ s || C ˆ k ≤ ˆ ǫ/ g and by choosing ǫ , ǫ ... even smaller ifnecessary, we can suppose that(21) ∀ t ∈ [0 , µ ] , ¯ g + t. ¯ s ∈ U . Consider the compact manifolds ( M m , g m ) introduced previously, but with¯ g (resp. U , ¯ s ) replacing h L − (resp. B r qL ( x q L ), s L ), so that the bumpymetrics g m converge locally to ¯ g . Consider a k large, so that the Claim isvalid. As in the proof of item (1), this implies that there is t ′∞ ∈ [0 , ǫ ] suchthat ( M, ¯ g + t ′∞ ¯ s ) contains a finite volume connected complete embeddedminimal hypersurface Γ intersecting U . Since ¯ g + t ′∞ ¯ s ∈ T ∞ by (21), Γ isclosed. Then we can perturb ¯ g + t ′∞ ¯ s conformally like in [22, Proposition2.3] to get by (20) a metric g ∈ U satisfying || g − ¯ g || C ˆ k ≤ ˆ ǫ and for which Γis a non-degenerate closed embedded minimal hypersurface intersecting U .The denseness of M U in U is checked, which finishes the proof. (cid:3) We end this section with a slightly non rigorous construction of non-compact manifolds of finite volume which do not obey the Weyl law [25],which justify our use of a new argument in the proof of Theorem 21 comparedto [22].
Remark 5.1. By [35] , for any A, ǫ > there is a metric on the -sphere S of volume such that any surface separating S into two regions of volumemore than ǫ has area greater than A . From this, one should be able to con-struct a sequence of spheres S k := ( S , g k ) of volume less than k with firstwidth in the sense of Almgren-Pitts equal to . Then by forming the infiniteconnected sum of S i with thin necks, one gets a non-compact manifold of fi-nite volume but with widths ω p growing linearly, by a Lusternik-Schnirelmanntype argument. In this construction, there is a lot a freedom in the rate ofgrowth of ω p by playing with the sizes of S k . In particular, such a finitevolume manifold generally does not satisfy a Weyl law. INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 41
Appendix A: Degenerate stable minimal hypersurfaces
We collect some simple facts about the structure of neighborhoods ofminimal hypersurfaces.Let (
N, g ) be a compact Riemannian manifold. In this Appendix, minimalhypersurfaces are smooth closed embedded. We say that a 2-sided minimalhypersurface is degenerate if its Jacobi operator has a non-trivial kernel. Ifsuch a hypersurface is degenerate and stable, then the kernel of its Jacobioperator is spanned by a positive eigenfunction. Note that for a 2-sidedminimal hypersurface which is either unstable or non-degenerate stable, itis well-known that the hypersurface has a neighborhood foliated by closedleaves which, when not equal to the minimal hypersurface itself, have non-zero mean curvature vector. A similar result is true for degenerate stableminimal hypersurfaces, as we noted in [42, Lemma 11].
Lemma 22.
Let Γ be a 2-sided degenerate stable minimal hypersurface inthe interior of ( N, g ) and ν a choice of unit normal vector on Γ . Then thereexist a positive number δ and a smooth map w : Γ × ( − δ , δ ) → R with thefollowing properties:(1) for each x ∈ Γ , we have w ( x,
0) = 0 and φ := ∂∂t w ( x, t ) | t =0 is apositive function in the kernel of the Jacobi operator of Γ ,(2) for each t ∈ ( − δ , δ ) , we have R Γ ( w ( ., t ) − tφ ) φ = 0 ,(3) for each t ∈ ( − δ , δ ) , the mean curvature of the hypersurface Γ t := { exp( x, w ( x, t ) ν ( x )); x ∈ Γ } is either positive or negative or identically zero,(4) if Γ t is minimal for a t ∈ ( − δ , δ ) , its Morse index is at most one.Proof. The first three items were proved in [42]. The last item follows fromthe fact that, since the first eigenvalue of the Jacobi operator is simple, ifa sequence of connected minimal hypersurfaces S k converges smoothly to astable minimal hypersurface, then for k large S k has index at most one. (cid:3) If the minimal hypersurface Γ is 1-sided, one can still apply the previouslemma in a double-cover of N where Γ lifts to a 2-sided hypersurface. If Γis a boundary component of N then the lemma is still valid on the interiorside of Γ. Discussion:
Note that Lemma 22 implies the following for a 2-sideddegenerate stable minimal hypersurface Γ (the situation is completely similarfor 1-sided minimal hypersurfaces): by the first variation formula if Γ isdegenerate stable, the n -volume of the hypersurfaces Γ t is a smooth function A : ( − δ , δ ) → R so that the sign of ∂ t A is the sign of the mean curvature of Γ t (after thecorrect continuous choice of unit normal). This function A associated to Γis defined for 2-sided Γ embedded inside Int( N ) ; if Γ is 1-sided, then wecall A the analogue function associated to the double 2-sided cover and if Γ is a boundary component of N , then A is only defined on [0 , δ ). If thefunction A is not strictly monotonous on any intervals of the form ( − δ ,
0) or(0 , δ ) where δ ≤ δ , then there are clearly an infinite sequence of t k ∈ (0 , δ )converging to 0 so that for each k , A restricted to an open interval containing t k achieves a maximum at t k . In other words, the minimal hypersurfaces Γ t k are saddle point minimal hypersurfaces (as defined in the introduction). Ifon the contrary the function A is strictly monotonous on both ( − δ ,
0) and(0 , δ ), then we have three possibilities: • Type I: A is monotonous on ( − δ , δ ), • Type II: A achieves a strict minimum at 0, • Type III: A achieves a strict maximum at 0 (in this last case Γ is asaddle point).Because we are interested in constructing saddle point minimal hypersur-faces, we introduce the following condition: we say thata degenerate stable minimal hypersurface Γ(resp. the metric g ) satisfies Condition [M]if, with the previous notations, the function A associated to Γ (resp. toany degenerate stable minimal hypersurface) is strictly monotonous on both( − δ ,
0) and (0 , δ ) for δ > N, g ).Let Γ be stable degenerate of Type II or III. The next lemma enablesto construct approximations of g for which the hypersurfaces Γ t are non-minimal if t = 0. Lemma 23.
Let ( N, g ) be a compact manifold with boundary and let Γ be adisjoint union of degenerate stable minimal hypersurfaces of Type II or III,embedded either in Int( N ) or in ∂N . Then there is a sequence of metrics h ( q ) converging to g in the C ∞ -topology so that Γ is still minimal for h ( q ) and with the previous notations, for each component Γ ′ of Γ :(1) if Γ ′ is of Type II, with respect to h ( q ) , Γ ′ is strictly stable, moreoverfor all t ∈ ( − δ , ∪ (0 , δ ) the mean curvature vector of Γ ′ t is neverzero and points towards Γ ′ ,(2) if Γ ′ is of Type III, with respect to h ( q ) , Γ ′ is unstable, moreover forall t ∈ ( − δ , ∪ (0 , δ ) , the mean curvature vector of Γ ′ t is never zeroand points away from Γ ′ .Proof. The two bullets have similar proofs. Let us check (2) for instance.Again for brevity we limit ourselves to the case where Γ is 2-sided embeddded
INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 43 in the interior of (
N, g ). Since the deformations are going to be local, we cansuppose Γ connected. Using previous notations, since Γ is of Type III, thereis a diffeomorphism φ from Γ × ( − δ , δ ) to a neighborhood N Γ of Γ so thateach φ (Γ ×{ s } ) is either minimal or has mean curvature vector pointing awayfrom Γ. Consider a sequence of functions f q : ( − δ , δ ) → [1 ,
2) increasing on( − δ , , δ ), equal to 1 at − δ and δ . We also impose that f q converges smoothly to the constant function equal to 1. Each f q inducesvia φ a function on N Γ that we assume can be extended to a smooth functionon M equal to 1 outside of N Γ (one might need to change a bit f q around theendpoints − δ and δ ). Let us still call f q this function defined on M , andconsider the metrics h ( q ) := f q .g . With respect to h ( q ) , Γ is still minimal, andany hypersurface φ (Γ × { s } ) with s = 0 has now non-zero mean curvaturevector pointing away from Γ. That follows from the following general fact:let S be a 2-sided embedded hypersurface in (Int( N ) , g ) endowed with achoice of unit normal ν and let ϕ : M → R be a smooth function. Considerthe conformal change of metric h := exp(2 ϕ ) g . If A g (resp. A h ) denotesthe second fundamental form of S with respect to ν for g (resp. h ), one cancheck that A h ( a, b ) = exp( ϕ )( A g ( a, b ) + g ( a, b ) dϕ ( ν )) . In particular if S has vanishing mean curvature or if its mean curvaturevector −→ H satisfies h−→ H , ν i <
0, and if dϕ ( ν ) > −→ H h of S with respect to h now satisfies in any case h−→ H h , ν i <
0. Tofinish the proof of (2), we notice that if f q : ( − δ , δ ) → [1 ,
2) was chosen tohave a strictly negative second derivative at 0, then with respect to h q , Γ isan unstable minimal hypersurface. (cid:3) Appendix B: Local min-max constructions in theAlmgren-Pitts’ setting
We explain the local 1-parameter min-max theory in the Almgren-Pittssetting: it is essentially a mixture of [37] and [28, Theorem 1.7].We give a review of the basic definitions from Geometric Measure Theoryand some notions of the Almgren and Pitts’ theory used in the paper. Fora complete presentation, we refer the reader to the book of Pitts [37], toSection 2 in [29] and [28, Section 3].Let N be a compact connected Riemannian ( n + 1)-manifold, assumedto be isometrically embedded in R P . We work with the space I k ( N ; Z ) of k -dimensional flat chains with coefficients in Z and with support containedin N , the subspace Z k ( N ; Z ) ⊂ I k ( N ; Z ) whose elements are boundaries,and with the space V k ( N ) of the closure, in the weak topology, of the set of k -dimensional rectifiable varifolds in R P with support in N .An integral current T ∈ I k ( N ; Z ) determines an integral varifold | T | anda Radon measure || T || ([37, Chapter 2, 2.1, (18) (e)]). If V ∈ V k ( N ), denoteby || V || the associated Radon measure on N . Given an ( n + 1)-dimensionalrectifiable set U ⊂ N , if the associated rectifiable current is an integral current in I n +1 ( N ; Z ), it will be written as [ | U | ]. To a rectifiable subset R of N corresponds an integral varifold called | R | . The support of a currentor a measure is denoted by spt. The notation M stands for the mass ofan element in I k ( N ; Z ). On I k ( N ; Z ) there is also the flat norm F whichinduces the so-called flat topology. The space V k ( N ) is endowed with thetopology of the weak convergence of varifolds. The mass of a varifold isdenoted by M . The F -metric was defined in [37] and induces the varifoldweak topology on any subset of V k ( N ) with mass bounded by a constant.Suppose that ∂ N and ∂ N are disjoint closed sets (which can be empty), ∂ N ∪ ∂ N = ∂N , and C (resp. C ) is the cycle in Z n ( N ; Z ) which isdetermined by ∂ N (resp. ∂ N ). Note that C = C = 0 if ∂N = ∅ . LetΦ : [0 , → Z n ( N ; F ; Z ) be a continuous map so that Φ(0) = C , Φ(1) = C . Let Π be the class of all continuous maps Φ ′ : [0 , → Z n ( N ; F ; Z )homotopic to Φ in the flat toplogy, with endpoints fixed, i.e. there is acontinous map H : [0 , × [0 , → Z n ( N ; F ; Z ) so that H ( .,
0) = Φ( . ), H ( .,
1) = Φ ′ ( . ), H ( j, s ) = C j for j = 0 , s ∈ [0 , π ♯ ( C , C ) bethe family of such homotopy classes Π of maps starting at C and ending at C .In [1], Almgren describes how to associate to a continuous map Φ : [0 , →Z n ( N ; F ; Z ) an element of I n +1 ( N, Z ). Let us explain this construction.There is a number µ > T ∈ I n ( N, Z ) has no boundary and F ( T ) ≤ µ , then there is an S T ∈ I n +1 ( N, Z ) such that ∂S T = T and M ( S T ) = F ( T ) = inf { M ( S ′ ); S ′ ∈ I n +1 ( N, Z ) and ∂S ′ = T } . Such an S T is called an F -isoperimetric choice for T . Now let k be largeenough so that for all i = 1 , ..., k , F (Φ( ik ) − Φ( i − k )) ≤ µ . With the pre-vious notation for F -isoperimetric choices, consider the ( n + 1)-dimensionalintegral current(22) k X i =1 S Φ( ik ) − Φ( i − k ) . This current does not depend of k provided it is sufficiently large. Bythe interpolation formula of [1, Section 6], this sum is also invariant byhomotopies. Hence when Φ ∈ Π ∈ π ♯ ( C , C ), the map which associates toΠ the ( n + 1)-dimensional current (22), defined with Φ, is well defined. Wecall this map the Almgren map and we denote it by A : π ♯ ( C , C ) → I n +1 ( N, Z ) . Given π ♯ ( C , C ), consider the function L : π ♯ ( C , C ) → [0 , ∞ ] definedby L (Π) = inf { sup x ∈ [0 , M (Φ( x )); Φ ∈ Π } . A sequence { Φ i } i ⊂ Π is a min-max sequence iflim sup i →∞ ( sup x ∈ [0 , M (Φ( x ))) = L (Π) . INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 45
The image set of { Φ i } i is Λ ( { Φ i } i ) = { V ∈ V n ( N ); ∃{ i j } → ∞ , x i j ∈ [0 , , such that lim j →∞ F ( | Φ i j ( x i j ) | , V ) = 0 } . If { Φ i } i ⊂ Π is a min-max sequence, define the critical set C ( S ) ⊂ V n ( N )of { Φ i } i as C ( { Φ i } i ) = { V ∈ Λ ( { Φ i } i ); || V || ( N ) = L (Π) . A min-max sequence { Φ i } i ⊂ Π such that every element of C ( { Φ i } i ) isstationary is called pulled-tight.We finally define the width W of ( N, g ) to be(23) W := inf A (Π)=[ | N | ] L (Π) , the infimum being taken over all the possible following choices: we start witha partition ∂N = X ∪ X ( X , X are closed), C i is the cycle in Z n ( N, Z )determined by X i ( i = 1 , ∈ π ♯ ( C , C ) satisfies A (Π) = [ | N | ] . This width should not be confused with the first width ω ( N, g ) which isdefined with relative cycles (see [15, 19, 29, 25]).A metric g is said to be bumpy if no smooth immersed closed minimalhypersurface has a non-trivial Jacobi vector field. White showed that bumpymetrics are generic in the Baire sense [43, 45]. The following theorem isessentially a consequence of the index bound of Marques-Neves [28]. Theorem 24.
Let ( N n +1 , g ) be a compact manifold with boundary endowedwith a bumpy metric g , with ≤ n ≤ . Suppose that the boundary ∂N is astrictly stable minimal hypersurface. Then there exists a stationary integralvarifold V whose support is a smooth embedded minimal hypersurface Σ ⊂ N of index bounded by one, such that || V || ( N ) = W. Moreover one of the components of Σ is contained in the interior Int( N ) .Proof. We consider (
N, g ) as isometrically embedded inside ( ˜
N , ˜ g ) such that ∂ ˜ N is strictly mean convex (the mean curvature vector points inwards), and˜ N \ N is foliated by strictly mean convex hypersurfaces, so that any closedminimal hypersurface embedded in ˜ N is embedded in N .Suppose that ∂ N, ∂ N ⊂ N are disjoint closed sets, ∂ N ∪ ∂ N = ∂X ,and C (resp. C ) is the cycle in Z n ( N, Z ) which is determined by ∂ N (resp. ∂ N ).By [34], any homotopy class Π ∈ π ♯ ( C , C ) satisfies L (Π) > max( M ( C ) , M ( C ) } . Hence combining [28, Theorem 1.7] and the arguments of [26, Theorem2.1] (see also [47, Theorem 2.7]), we get V as in the statement but with || V || ( N ) = L (Π), and a connected component of spt( V ) which either is 2-sided and has index one, or is 1-sided. This component has to be embeddedin the interior of N .If we apply the previous discussion to a sequence of homotopy classes Π k such that lim k →∞ L (Π k ) = W , we get a sequence of varifolds V k which sub-sequently converge to V ([41]) as in the statement. A connected componentof spt( V ) is in Int( N ) because by the maximum principle, a sequence ofminimal hypersurfaces in Int( N ) cannot converge in the Gromov-Hausdorfftopology to some components of ∂N , which is strictly stable. (cid:3) Appendix C: Facts about mean curvature flow
We find it convenient to use the level set flow formulation of the meancurvature flow. Under the level set flow introduced by Chen-Giga-Goto [3]and Evans-Spruck [12], any closed set K in R n +1 generates a one-parameterfamily of closed sets { K t } t ≥ with K = K . The closed set K is said to beweakly (resp. strictly) mean convex if K t ⊂ K (resp. K t ( K ) for all t > X := ∂K happens to be smooth, then K is mean convex if and onlyif the mean curvature of X is nonnegative.When K is mean convex, the level set flow is given by the level sets of aLipschitz function u satisfying in the viscosity sense − |∇ u | div( ∇ u |∇ u | ) , and ∂K t = { x ∈ R n +1 ; u ( x ) = t } . In that case, there is a unique Lipschitzfunction u giving a viscosity solution of the level set flow [12, Theorem 7.4],[3] (see also [6]). The flow is non-fattening (the level sets have no interior)[44, Corollary 3.3]. The singular set at each time t > n −
1, in particular at all positive time t , ∂K t is a closedhypersurface smooth outside a subset of dimension n − T > { ∂K t } t ∈ [0 ,T ) form a possibly singular mean convexfoliation of K \ K T [44]. By monotonicity, the n -dimensional Hausdorff mea-sure (that we usually denote by Vol n ) of the set ∂K t is nonincreasing in t [44, Corollary 3.6].The previous facts extend naturally to closed ambient manifolds ( N n +1 , g )[21, 44]. The long term behavior can be different: in general a closed meanconvex non-minimal n -dimensional hypersurface X ⊂ N evolving under thelevel set flow will sweep out an ( n + 1)-dimensional manifold with boundary Y ⊂ N , and ∂Y = X ∪ X ∞ , where X ∞ is a closed embedded stable minimalhypersurface. If 2 ≤ n ≤ X ∞ is smooth, the convergence of X t to X ∞ is smooth and one-sheeted (resp. two-sheeted) for 2-sided (resp. 1-sided)components of X ∞ [44, Section 11]. By the avoidance principle (or maximumprinciple, see [21, Lemma 6.3]), if K is a mean convex closed set generating INIMAL HYPERSURFACES IN MANIFOLDS THICK AT INFINITY 47 { K t } t ≥ then any strictly mean concave closed set W ⊂ K satisfies W ⊂ Int( ∩ t ≥ K t ) . References [1] F. J. Almgren. The homotopy groups of the integral cycle groups.
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