aa r X i v : . [ m a t h . DG ] F e b ON THE MULTIPLICITY ONE CONJECTURE IN MIN-MAX THEORY
XIN ZHOUA
BSTRACT . We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric,the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth,Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves.We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certainrelative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows byapproximating such min-max value using the min-max theory for hypersurfaces with prescribed meancurvature established by the author with Zhu.
0. I
NTRODUCTION
Let ( M n +1 , g ) be a closed orientable Riemannian manifold of dimension ≤ ( n + 1) ≤ . In[2], Almgren proved that the space of mod-2 cycles Z n ( M, Z ) is weakly homotopic the Eilenberg-MacLane space K ( Z ,
1) = RP ∞ ; (see also [29] for a simpler proof). Later, Gromov [15, 16], Guth[18], Marque-Neves [28] introduced the notion of volume spectrum as a nonlinear version of spectrumfor the area functional in Z n ( M, Z ) . In particular, the volume spectrum is a non-decreasing sequenceof positive numbers < ω ( M, g ) ≤ · · · ≤ ω k ( M, g ) ≤ · · · → + ∞ , which is uniquely determined by the metric g in a given closed manifold M .By adapting the celebrated min-max theory developed by Almgren [3], Pitts [31] (for ≤ ( n + 1) ≤ ), and Schoen-Simon [33] (for n + 1 = 7 ), Marques-Neves [28, 27] proved that each ω k ( M, g ) isassociated with an integral varifold V k whose support is a disjoint collection of smooth, connected,closed, embedded, minimal hypersurfaces { Σ k , · · · , Σ kl k } , such that(0.1) ω k ( M, g ) = l k X i =1 m ki · Area(Σ ki ) , where { m k , · · · , m kl k } ⊂ N is a set of positive integers, usually called multiplicities . We refer to[39, 6, 10, 17, 9, 23, 7, 47, 50, 32] for other variants of this theory.Our main theorem states that all these integer multiplicities are identically equal to one for a bumpymetric. A metric g is called bumpy if every closed immersed minimal hypersurface is non-degenerate.White proved that the set of bumpy metrics is generic in Baire sense [42, 44]. Theorem A.
Given a closed manifold M n +1 of dimension ≤ ( n + 1) ≤ with a bumpy metric g ,the min-max minimal hypersurfaces { Σ ki : k ∈ N , i = 1 , · · · , l k } associated with volume spectrumare all two-sided and have multiplicity one and index bounded by k . That is m ki = 1 for all k ∈ N , ≤ i ≤ l k , ω k ( M, g ) = l k X i =1 Area(Σ ki ) , and l k X i =1 index(Σ ki ) ≤ k. Remark . This solves the
Multiplicity One Conjecture of Marques-Neves [29, 1.2]; (see also [27]for an earlier weaker version of this conjecture). We refer to Theorem 5.2 for a more detailed statementof this result. Note that by standard compactness analysis (see [35]), the same conclusion concerningtwo-sidedness and multiplicity one also holds true for a metric with positive Ricci curvature.
Remark . This conjecture was proved earlier for 1-parameter min-max constructions under positiveRicci curvature assumption by Marques-Neves [25], the author [48, 49], and Ketover-Marques-Neves[22]. Later it was fully proved for 1-parameter case by Marques-Neves [27]. Recently, Chodosh-Mantoulidis [5] proved this conjecture in dimension three ( n + 1) = 3 for the Allen-Cahn setting; (see[12] for earlier works along this direction); they also proved that the total index is exactly k for their k -min-max solutions when ( n + 1) = 3 . After our results were poseted, Marques-Neves finished theirprogram and also proved the same optimal index estimates for ≤ ( n + 1) ≤ [29, Addendum].One motivation of this conjecture is to prove the Yau’s conjecture [46] on existence of infinitelymany closed minimal surfaces in three manifolds. Combining with the growth estimates of { ω k ( M, g ) } by Marques-Neves [28, Theorem 5.1 and 8.1] and the Frankel Theorem [11], we have Theorem B.
Let M n +1 be a closed manifold of dimension ≤ ( n + 1) ≤ .(a) For each bumpy metric g , there exists infinitely many smooth, connected, closed, embedded,minimal hypersurfaces.(b) If a metric g has positive Ricci curvature, then there exists a sequence of smooth, connected,closed, embedded, minimal hypersurfaces { Σ k } k ∈ N , such that Area(Σ k ) ∼ k n +1 , as k → ∞ . Remark . Result (a) was already known even without the bumpy assumption by combining Marques-Neves [28] and Song [40]. For a set of generic metrics, Irie-Marques-Neves [21] and Marques-Neves-Song [30] proved denseness and equi-distribution for the space of closed embedded minimal hypersur-faces, using the Weyl Law for volume spectrum by Liokumovich-Marques-Neves [24]. Their genericset in principle could be much smaller than the set of bumpy metrics.Result (b) was also obtained by Chodosh-Mantoulidis [5] in dimension three ( n + 1) = 3 .As a direct corollary of the compactness theory (see [35]), our multiplicity one result also gives asolution to the Weighted Morse Index Bound Conjecture by Marques-Neves.
Theorem C.
Let M n +1 be a closed manifold of dimension ≤ ( n + 1) ≤ with an arbitrary metric g . In (0.1), we have X Σ ki : orientable m ki · index(Σ ki ) + X Σ ki : nonorientable m ki · index(Σ ki ) ≤ k. Sketch of the proof.
The key idea of our proof is to approximate the
Area -functional by theweighted A h -functional used in the prescribing mean curvature (PMC) min-max theory developed bythe author with Zhu [52]. Note that the A h -functional is only defined for boundaries of Caccioppolisets; see (1.1). A smooth critical point of A h is a hypersurface whose mean curvature is prescribed bythe restriction of h to itself. There are two crucial parts in the proof. In the first part, we consider min-max construction of minimal hypersurfaces using sweepouts of boundaries of Caccioppoli sets. Weobserve that in a bumpy metric if one approximates Area by a sequence {A ǫ k h } k ∈ N where { ǫ k } k ∈ N → , and if h : M → R is carefully chosen, then the limit min-max minimal hypersurfaces (of min-maxPMC hypersurfaces associated with A ǫ k h ) are all two-sided and have multiplicity one; see Theorem ULTIPLICITY ONE CONJECTURE 3 ω k ( M, g ) can be realizedby the area of some minimal hypersurfaces coming from min-max constructions using sweepouts ofboundaries. We now elaborate the detailed ideas.To implement the idea in the first part, we generalize the PMC min-max theory in [52] to multi-parameter families using continuous sweepouts. Since the space of Caccioppoli sets C ( M ) is con-tractible, there is no nontrivial free homotopy class to do min-max, so we have to consider relativehomotopy class. Heuristically, given a k -dimensional parameter space X , a subset Z ⊂ X , and acontinuous map Φ : X → C ( M ) , we can consider its relative ( X, Z ) -homotopy class Π = Π(Φ ) consisting of all maps Φ : X → C ( M ) that are homotopic to Φ and such that Φ | Z ≡ Φ | Z . Ifthe min-max value L h = inf { max x ∈ X A h (Φ( x )) : Φ ∈ Π } satisfies the nontriviality condition L h > max x ∈ Z A h (Φ ( x )) with respect to the A h -functional, and if h is chosen in a dense subset S ( g ) ⊂ C ∞ ( M ) (depending on the metric g , see [52, Proposition 0.2]), we prove the existence ofa smooth closed hypersurface Σ h of prescribed mean curvature h ; moreover, it is represented as theboundary Σ h = ∂ Ω h for some Caccioppoli set Ω h and A h (Ω h ) = L h ; hence Σ h is two-sided and havemultiplicity one. Σ h is usually called a min-max PMC hypersurface. We also established Morse indexupper bounds following Marques-Neves [27]. That is, we prove that the Morse index of Σ h is boundedfrom above by k (the dimension of parameter space).Given a relative homotopy class Π as above, consider the min-max construction for the Area -functional and let L = inf { max x ∈ X Area( ∂ Φ( x )) : Φ ∈ Π } . If the nontriviality condition L > max x ∈ Z Area( ∂ Φ ( x )) is satisfied, we can approximate L by L ǫh for a fixed h ∈ S ( g ) (to be chosenlater) and small enough ǫ > . We know that ǫ · h also belongs to the dense subset S ( g ) . Denote Σ ǫ as the min-max PMC hypersurface associated with L ǫh . As the family { Σ ǫ : ǫ > } have uniformlybounded area and Morse index, we can pick a subsequence { Σ k = Σ ǫ k : ǫ k → } that converges asvarifolds and also locally smooth and graphically away from finitely many points to some limit minimalhypersurface Σ ∞ with integer multiplicity such that Area(Σ ∞ ) = L . The limit can be extended to aclosed embedded minimal hypersurface Σ ∞ across the bad points, and Σ ∞ also has the same Morseindex upper bound. Hence Σ ∞ is a min-max minimal hypersurface associated with L . As a standardprocess, if the multiplicity is greater than one, or if a component is one-sided, one can obtain solutionsof the Jacobi operator L Σ ∞ of Σ ∞ by taking the limit of the renormalizations of the heights betweenthe top and bottom sheets of Σ k . In particular, there are two possibilities for the limit depending on theorientations of the top and bottom sheets. For simplicity, let us assume that Σ ∞ is connected and two-sided. An easier case happens when the top and bottom sheets have the same orientation, and hencethe limit is a nontrivial nonnegative solution ϕ of the Jacobi equation L Σ ∞ ϕ = 0 which cannot happenin a bumpy metric. When the top and bottom sheets have opposite orientations, the limit is either anontrivial nonnegative solution to the Jacobi equation, or is a solution ϕ of the following equation L Σ ∞ ϕ = 2 h | Σ ∞ , such that ϕ does not change sign . The key observation is that one can find a h ∈ S ( g ) so that the unique solution (as Σ ∞ is non-degenerate) of L Σ ∞ ϕ = 2 h | Σ ∞ must change sign, and hence Σ ∞ must have multiplicity one; (seeLemma 4.2). Indeed, the set of minimal hypersurfaces with bounded area and Morse index in a bumpymetric is finite by the standard compactness results [35]. On each such Σ , we can construct a h Σ ∈ C ∞ (Σ) such that the unique solution f Σ of L Σ f Σ = 2 h Σ must change sign, and we can further makethe support of all such h Σ pairwise disjoint. Since S ( g ) is open and dense, we can pick a h ∈ S ( g ) thatapproximates all h Σ on Σ as close as we want. Then the solution of L Σ ϕ = 2 h | Σ must also changesign. Up to here, we have elucidated how to construct two-sided min-max minimal hypersurfaces withmultiplicity one for sweepouts of boundaries of Caccioppoli sets. XIN ZHOU
Lastly we apply the above multiplicity one result to the volume spectrum. Though the volumespectrum ω k ( M, g ) is defined using cohomological relations, Marques-Neves proved in [27], usingtheir Morse index estimates, that in a bumpy metric ω k ( M, g ) is realized by the min-max value L (Π) for certain free homotopy class Π of maps Φ : X → Z n ( M n +1 , Z ) , where X is some fixed k -dimensional parameter space and Z n ( M n +1 , Z ) is the space of mod-2 cycles. It was observed byMarques-Neves [29] that the space of Caccioppoli sets C ( M ) forms a double cover of Z n ( M n +1 , Z ) via the boundary map ∂ : C ( M ) → Z n ( M n +1 , Z ) . Therefore, by lifting to the double cover, foreach Φ ∈ Π , we can produce a map ˜Φ : ˜ X → C ( M ) , where π : ˜ X → X is a double cover, suchthat ∂ ˜Φ( x ) = Φ( π ( x )) . To produce a nontrivial relative homotopy class, we pick a map Φ ∈ Π suchthat max x ∈ X Area(Φ ( x )) is very close to L (Π) = ω k ( M, g ) . Let Z ⊂ X to be the subset whereeach Φ ( x ) , x ∈ Z , is ǫ -distance away from the set of smooth closed embedded minimal hypersurface Σ with Area(Σ) ≤ L and index(Σ) ≤ k . Note that this set of minimal hypersurfaces is finite in abumpy metric, hence for ǫ small enough the complement Y = X \ Z ⊂ X is topologically trivial inthe sense that Y does not detect the generator of the cohomological ring of Z n ( M n +1 , Z ) . Thereforethe pre-image ˜ Y = π − ( Y ) ⊂ ˜ X is homeomorphic to two disjoint identical copies of Y , denotedas Y + and Y − . On the other hand, since no element in Φ ( Z ) is regular, by Pitts’s combinatorialargument, one can homotopically deform Φ | Z so that max x ∈ Z Area(Φ ( x )) < L . Now consider therelative ( ˜ X, ˜ Z ) -homotopy class ˜Π generated by the map ˜Φ : ˜ X → C ( M ) . One key observation isthat the min-max value L ( ˜Π) ≥ L (Π) > max x ∈ Z Area(Φ ( x )) . To see this, given any homotopicdeformation ˜Ψ : ˜ X → C ( M ) of ˜Φ relative to ( ˜Φ ) | ˜ Z , if max x ∈ Y + Area( ∂ ˜Ψ( x )) < L (Π) , then wecan pass it to quotient and obtain a continuous map Ψ : X → Z ( M, Z ) as Y + and Y − are disjoint and ˜Ψ | ˜ Z ≡ ( ˜Φ ) | ˜ Z , so that max x ∈ X Area(Ψ( x )) < L (Π) , but this is a contradiction as Ψ is homotopic to Φ . Therefore, ˜Π is a nontrivial relative homotopy class in C ( M ) , and its associated min-max minimalhypersurfaces are two-sided and have multiplicity one. Finally, as the metric is bumpy, the min-maxvalue L ( ˜Π) of ˜Π is equal to L (Π) when max x ∈ X Area(Φ ( x )) is close enough to L (Π) = ω k ( M, g ) .Hence we have explained how to construct two-sided min-max minimal hypersurfaces of multiplicityone whose areas realize the volume spectrum.0.2. Outline of the paper.
In Section 1, we establish the multi-parameter version of min-max theoryfor prescribing mean curvature hypersurfaces using continuous sweepouts. In Section 2, we prove sev-eral compactness results for prescribing mean curvature hypersurfaces with uniform area and Morseindex upper bounds. In Section 3, we prove the Morse index upper bound for prescribing mean cur-vature hypersurfaces produced by our min-max theory. In Section 4, we prove that min-max minimalhypersurfaces associated with families of boundaries have multiplicity one in a bumpy metric. Finally,in Section 5, we prove the multiplicity one conjecture for volume spectrum.
Acknowledgements.
I would like to thank my Ph. D. advisor Richard Schoen for innumerable advicesand long-term encouragement and support. I want to thank Brian White for showing me an unpublishednotes and for an enlightening conversation which inspired the key idea of this paper. I also want tothank Fernando Marques and Andre Neves, from whom I learned so many things that are used inthis work, and also for their comments. Finally, thanks to Jonathan Zhu for the collaboration on theprescribing mean curvature min-max theory which is essentially used here, and to Zhichao Wang forcarefully reading the draft and useful comments to improve the presentation. This work was donewhen I visited the Institute for Advanced Study, and I would like to thank IAS for their support andhospitality. The work was partially supported by NSF grant DMS-1811293.
ULTIPLICITY ONE CONJECTURE 5
1. M
ULTI - PARAMETER MIN - MAX THEORY FOR PRESCRIBING MEAN CURVATUREHYPERSURFACES
Here we present an adaption to multi-parameter families of the min-max theory for hypersurfaceswith prescribed mean curvature (abbreviated as PMC) established by the author with Zhu [51, 52].Let S = S ( g ) (depending on the metric g ) be the open and dense subset of C ∞ ( M ) chosen as in[52, Proposition 0.2]. More precisely, S ( g ) consists of all Morse functions h such that the zero set Σ = { h = 0 } is a smooth closed embedded hypersurface, and the mean curvature of Σ vanishesto at most finite order. A hypersurface is almost embedded (sometime also called strongly Alexandrovembedded ) if it locally decomposes into smooth embedded sheets that touch but do not cross. By [52,Theorem 3.11], any almost embedded hypersurface of prescribed mean curvature h ∈ S has touchingset ( n − -rectifiable, and no component is minimal. Notations.
We collect some notions. We refer to [36] and [31, § ( M n +1 , g ) denote a closed, oriented, smooth Riemannian manifold of dimension ≤ ( n +1) ≤ . Assume that ( M, g ) is embedded in some R L , L ∈ N . B r ( p ) denotes the geodesic ball of ( M, g ) . We denote by H k the k -dimensional Hausdorff measure; I k ( M ) (or I k ( M, Z ) ) the spaceof k -dimensional integral (or mod 2) currents in R L with support in M ; Z k ( M ) (or Z k ( M, Z ) ) thespace of integral (or mod 2) currents T ∈ I k ( M ) with ∂T = 0 ; V k ( M ) the closure, in the weaktopology, of the space of k -dimensional rectifiable varifolds in R L with support in M ; G k ( M ) theGrassmannian bundle of un-oriented k -planes over M ; F and M respectively the flat norm [36, § I k ( M ) ; F the varifold F -metric on V k ( M ) and currents F -metric on I k ( M ) or I k ( M, Z ) , [31, 2.1(19)(20)]; C ( M ) or C ( U ) the space of sets Ω ⊂ M or Ω ⊂ U ⊂ M withfinite perimeter (Caccioppoli sets), [36, § § X ( M ) or X ( U ) the space of smooth vectorfields in M or supported in U . ∂ Ω denotes the (reduced)-boundary of [[Ω]] as an integral current, and ν ∂ Ω denotes the outward pointing unit normal of ∂ Ω , [36, 14.2].We also utilize the following definitions:(a) Given T ∈ I k ( M ) , | T | and k T k denote respectively the integral varifold and Radon measure in M associated with T ;(b) Given c > , a varifold V ∈ V k ( M ) is said to have c -bounded first variation in an open subset U ⊂ M , if | δV ( X ) | ≤ c Z M | X | dµ V , for any X ∈ X ( U ); here the first variation of V along X is δV ( X ) = R G k ( M ) div S X ( x ) dV ( x, S ) , [36, § Σ in M , or a set Ω ∈ C ( M ) with fi-nite perimeter, [[Σ]] , [[Ω]] denote the corresponding integral currents with the natural orientation,and [Σ] denotes the corresponding integer-multiplicity varifold.As noted by Marques-Neves [29, Section 5], C ( M ) is identified with I n +1 ( M, Z ) . In particular,the flat F -norm and the mass M -norm are the same on C ( M ) . Given Ω , Ω ∈ C ( M ) , the F -distancebetween them is: F (Ω , Ω ) = F (Ω − Ω ) + F ( | ∂ Ω | , | ∂ Ω | ) . Given Ω ∈ C ( M ) , we will denote B F ǫ (Ω) = { Ω ′ ∈ C ( M ) : F (Ω ′ , Ω) ≤ ǫ } . XIN ZHOU
We are interested in the following weighted area functional defined on C ( M ) . Given h : M → R ,define the A h -functional on C ( M ) as(1.1) A h (Ω) = H n ( ∂ Ω) − Z Ω h d H n +1 . The first variation formula for A h along X ∈ X ( M ) is (see [36, 16.2])(1.2) δ A h | Ω ( X ) = Z ∂ Ω div ∂ Ω Xdµ ∂ Ω − Z ∂ Ω h h X, ν i dµ ∂ Ω , where ν = ν ∂ Ω is the outward unit normal on ∂ Ω .When the boundary ∂ Ω = Σ is a smooth immersed hypersurface, we have div Σ X = H h X, ν i , where H is the mean curvature of Σ with respect to ν ; if Ω is a critical point of A h , then (1.2) directlyimplies that Σ = ∂ Ω must have mean curvature H = h | Σ . In this case, we can calculate the secondvariation formula for A h along normal vector fields X ∈ X ( M ) such that X = ϕν along ∂ Ω = Σ where ϕ ∈ C ∞ (Σ) , [4, Proposition 2.5],(1.3) δ A h | Ω ( X, X ) = II Σ ( ϕ, ϕ ) = Z Σ (cid:0) |∇ ϕ | − (cid:0) Ric M ( ν, ν ) + | A Σ | + ∂ ν h (cid:1) ϕ (cid:1) dµ Σ . In the above formula, ∇ ϕ is the gradient of ϕ on Σ ; Ric M is the Ricci curvature of M ; A Σ is thesecond fundamental form of Σ .1.1. Min-max construction for ( X, Z ) -homotopy class. In this part, we describe the setup for min-max theory for PMC hypersurfaces associated with multiple parameter families in C ( M ) .Let X k be a cubical complex of dimension k ∈ N in some I m = [0 , m and Z ⊂ X be a cubicalsubcomplex.Let Φ : X → ( C ( M ) , F ) be a continuous map (with respect to the F -topology on C ( M ) ). We let Π be the set of all sequences of continuous (in F -topology) maps { Φ i : X → C ( M ) } i ∈ N such that :(1) each Φ i is homotopic to Φ in the flat topology on C ( M ) , and(2) there exist homotopy maps { Ψ i : [0 , × X → C ( M ) } i ∈ N which are continuous in the flattopology, Ψ i (0 , · ) = Φ i , Ψ i (1 , · ) = Φ , and satisfy(1.4) lim sup i →∞ sup { F (Ψ i ( t, x ) , Φ ( x )) : t ∈ [0 , , x ∈ Z } = 0 . Note that a sequence { Φ i } i ∈ N with Φ i = Φ for all i ∈ N belongs to Π . Definition 1.1.
Given a pair ( X, Z ) and Φ as above, { Φ i } i ∈ N is called a ( X, Z ) -homotopy sequenceof mappings into C ( M ) , and Π is called the ( X, Z ) -homotopy class of Φ . Remark . Π can be viewed as the relative homotopy class for Φ in ( C ( M ) , Φ | Z ) . However, wecannot fix the values Φ i | Z to be exactly Φ | Z . In fact, in the later discretization/interpolation process,we will allow Φ i | Z to deviate slightly from Φ | Z ; but the deviations will converge to zero as i → ∞ . Definition 1.3.
The h -width of Π is defined by: L h = L h (Π) = inf { Φ i }∈ Π lim sup i →∞ sup x ∈ X {A h (Φ i ( x )) } . ULTIPLICITY ONE CONJECTURE 7
Definition 1.4.
A sequence { Φ i } i ∈ N ∈ Π is called a min-max sequence if L h (Φ i ) := sup x ∈ X A h (Φ i ( x )) satisfies L h ( { Φ i } ) := lim sup i →∞ L h (Φ i ) = L h (Π) . Lemma 1.5.
Given Φ and Π , there exists a min-max sequence.Proof. Take a sequence {{ Φ αi } i ∈ N } α ∈ N in Π , such that lim α →∞ L h ( { Φ αi } i ∈ N ) = L h (Π) . Now we pick up a new sequence by a diagonalization process. Take a sequence ǫ α → . For each α ,we pick i α ∈ N , such that sup t ∈ [0 , ,x ∈ Z F (Ψ αi α ( t, x ) , Φ ( x )) < ǫ α , and L h ( { Φ αi } ) − ǫ α ≤ sup x ∈ X A h (Φ αi α ( x )) ≤ L h ( { Φ αi } ) + ǫ α , where Ψ αi α is the homotopy between Φ αi α and Φ in the flat topology. Hence the sequence { Φ αi α } α ∈ N belongs to Π and is a min-max sequence. (cid:3) Definition 1.6.
The image set of { Φ i } i ∈ N is defined by K ( { Φ i } ) = { V = lim j →∞ | ∂ Φ i j ( x j ) | as varifolds : x j ∈ X } . If { Φ i } i ∈ N is a min-max sequence in Π , the critical set of { Φ i } is defined by C ( { Φ i } ) = { V = lim j →∞ | ∂ Φ i j ( x j ) | as varifolds : with lim j →∞ A h (Φ i j ( x j )) = L h (Π) } . Now we are ready to state the continuous version of min-max theory for PMC hypersurfaces asso-ciated with a ( X, Z ) -homotopy class. It is a generalization of [52, Theorem 4.8 and Proposition 7.3],and the proof is given in Section 1.4. Theorem 1.7 (Min-max theorem) . Let ( M n +1 , g ) be a closed Riemannian manifold of dimension ≤ ( n + 1) ≤ , and h ∈ S ( g ) which satisfies R M h ≥ . Given a map Φ : X → ( C ( M ) , F ) continuous in the F -topology and the associated ( X, Z ) -homotopy class Π , suppose (1.5) L h (Π) > max x ∈ Z A h (Φ ( x )) . Let { Φ i } i ∈ N ∈ Π be a min-max sequence for Π . Then there exists V ∈ C ( { Φ i } ) induced by anontrivial, smooth, closed, almost embedded hypersurface Σ n ⊂ M of prescribed mean curvature h with multiplicity one.Moreover, V = lim j →∞ | ∂ Φ i j ( x j ) | for some { i j } ⊂ { i } , { x j } ⊂ X \ Z , with lim j →∞ A h (Φ i j ( x j )) = L h (Π) , and Φ i j ( x j ) converges in the F -topology to some Ω ∈ C ( M ) such that Σ = ∂ Ω where its meancurvature with respect to the unit outer normal is h , and A h (Ω) = L h (Π) . XIN ZHOU
Pull-tight.
Now we describe the pull-tight process in [52, Section 5]. Let c = sup M | h | , and L c = 2 L h + c Vol( M ) . Denote A c ∞ = { V ∈ V n ( M ) : k V k ( M ) ≤ L c , V has c -bounded first variation, or V ∈ | ∂ Φ | ( Z ) } . We can follow [51, Section 4] or [52, Section 5] to construct a continuous map: H : [0 , × ( C ( M ) , F ) ∩ { M ( ∂ Ω) ≤ L c } → ( C ( M ) , F ) ∩ { M ( ∂ Ω) ≤ L c } such that:(i) H (0 , Ω) = Ω for all Ω ;(ii) H ( t, Ω) = Ω if | ∂ Ω | ∈ A c ∞ ;(iii) if | ∂ Ω | / ∈ A c ∞ , A h ( H (1 , Ω)) − A h (Ω) ≤ − L ( F ( | ∂ Ω | , A c ∞ )) < here L : [0 , ∞ ) → [0 , ∞ ) is a continuous function with L (0) = 0 , L ( t ) > when t > ;(iv) for every ǫ > , there exists δ > such that x ∈ Z, F (Ω , Φ ( x )) < δ = ⇒ F ( H ( t, Ω) , Φ ( x )) < ǫ, for all t ∈ [0 , this is a direct consequence of (ii) since | ∂ Φ | ( Z ) ⊂ A c ∞ .Note that to construct H , the only modification of [52, § | ∂ Φ | ( Z ) into the definition of A c ∞ as we want to fix the values assumed on Z in the tightening process; all other steps in [52, § § H ( t, Ω) := (cid:0) Ψ | ∂ Ω | ( t ) (cid:1) (Ω) . Lemma 1.8.
Given a min-max sequence { Φ ∗ i } i ∈ N ∈ Π , we define Φ i ( x ) = H (1 , Φ ∗ i ( x )) for every x ∈ X . Then { Φ i } i ∈ N is also a min-max sequence in Π . Moreover, C ( { Φ i } ) ⊂ C ( { Φ ∗ i } ) and everyelement of C ( { Φ i } ) either has c -bounded first variation, or belongs to | ∂ Φ | ( Z ) .Proof. By continuity of H , we know that Φ i is homotopic to Φ ∗ i in the flat topology. By (iv), { Ψ i ( t, x ) = H ( t, Φ ∗ i ( x )) } satisfies (1.4), and hence { Φ i } ∈ Π . By (ii)(iii), A h (Φ i ( x )) ≤ A h (Φ ∗ i ( x )) for every x ∈ X , so { Φ i } is also a min-max sequence. Finally, given any V ∈ C ( { Φ i } ) , then V = lim j →∞ | ∂ Φ i j ( x j ) | where lim j →∞ A h (Φ i j ( x j )) = L h . Denote V ∗ = lim j →∞ | ∂ Φ ∗ i j ( x j ) | . By(iii), lim j →∞ F ( | ∂ Φ ∗ i j ( x j ) | , A c ∞ ) = 0 (as lim j →∞ A h (Φ i j ( x j )) = lim j →∞ A h (Φ ∗ i j ( x j )) = L h ), so V ∗ ∈ A c ∞ . On the other hand, V = lim j →∞ | ∂H (1 , Φ ∗ i j ( x j )) | = H (1 , lim j →∞ | ∂ Φ ∗ i j ( x j ) | ) = H (1 , V ∗ ) = V ∗ . (Note that H is also well defined as a continuous map H : [0 , × { V ∈ V n ( M ) , k V k ( M ) ≤ L c } →{ V ∈ V n ( M ) , k V k ( M ) ≤ L c } .) Hence C ( { Φ i } ) ⊂ C ( { Φ ∗ i } ) and the proof is finished. (cid:3) Definition 1.9.
Let c = sup M | h | . Any min-max sequence { Φ i } i ∈ N ∈ Π such that every element of C ( { Φ i } ) has c -bounded first variation or belongs to | ∂ Φ | ( Z ) is called pulled-tight .1.3. Discretization and interpolation results.
We record several discretization and interpolation re-sults developed by Marques-Neves [26, 28]. Though these results were proven for sweepouts in Z n ( M, Z ) or Z n ( M, Z ) , they work well for sweepouts in C ( M ) . We will point out necessary modi-fications.We refer to Appendix A for the notion of cubic complex structure on X . We refer to [52, Section 4]for the notion of discrete sweepouts. Though all definitions therein were made when X = [0 , , thereis no change for discrete sweepouts on X .Recall that given a map φ : X ( k ) → C ( M ) , the fineness of φ is defined as f ( φ ) = sup {F ( φ ( x ) − φ ( y )) + M ( ∂φ ( x ) − ∂φ ( y )) : x, y are adjacent vertices in X ( k ) } . ULTIPLICITY ONE CONJECTURE 9
Definition 1.10 (c.f. § . Given a continuous (in the flat topology) map
Φ : X → C ( M ) , wesay that Φ has no concentration of mass if lim r → sup {k ∂ Φ( x ) k ( B r ( p )) , p ∈ M, x ∈ X } = 0 . The purpose of the next theorem is to construct discrete maps out of a continuous map in flattopology.
Theorem 1.11.
Let
Φ : X → C ( M ) be a continuous map in the flat topology that has no concentrationof mass, and sup x ∈ X M ( ∂ Φ( x )) < + ∞ . Assume that Φ | Z is continuous under the F -topology. Thenthere exist a sequence of maps φ i : X ( k i ) → C ( M ) , and a sequence of homotopy maps: ψ i : I ( k i ) × X ( k i ) → C ( M ) , with k i < k i +1 , ψ i (0 , · ) = φ i − ◦ n ( k i , k i − ) , ψ i (1 , · ) = φ i , and a sequence of numbers { δ i } i ∈ N → such that(i) the fineness f ( ψ i ) < δ i ;(ii) sup {F ( ψ i ( t, x ) − Φ( x )) : t ∈ I ( k i ) , x ∈ X ( k i ) } ≤ δ i ; (iii) for some sequence l i → ∞ , with l i < k i M ( ∂ψ i ( t, x )) ≤ sup { M ( ∂ Φ( y )) : x, y ∈ α, for some α ∈ X ( l i ) } + δ i ; and this directly implies that sup { M ( ∂φ i ( x )) : x ∈ X ( k ) } ≤ sup { M ( ∂ Φ( x )) : x ∈ X } + δ i . As Φ | Z is continuous in F -topology, we have from (iii) that for all t ∈ I ( k i ) and x ∈ Z ( k i ) M ( ∂ψ i ( t, x )) ≤ M ( ∂ Φ( x )) + η i with η i → as i → ∞ . Applying [26, Lemma 4.1] with S = Φ( Z ) , we get by (ii) that(iv) sup { F ( ψ i ( t, x ) , Φ( x )) : t ∈ I ( k i ) , x ∈ Z ( k i ) } → , as i → ∞ . Now given h ∈ C ∞ ( M ) , denoting c = sup M | h | , then we have from (ii)(iii) that(v) A h ( φ i ( x )) ≤ sup {A h (Φ( y )) : α ∈ X ( l i ) , x, y ∈ α } + (1 + c ) δ i ; and hence sup {A h ( φ i ( x )) : x ∈ X ( k i ) } ≤ sup {A h (Φ( x )) : x ∈ X } + (1 + c ) δ i . Proof. [26, Theorem 13.1] and [28, Theorem 3.9] proved this result when C ( M ) is replaced by Z n ( M ) and Z n ( M, Z ) respectively. The adaption to C ( M ) was done in [49, Theorem 5.1] when X = [0 , and it is the same for general X . (cid:3) The purpose of the next theorem is to construct a continuous map in the F -topology out of a discretemap with small fineness. Theorem 1.12.
There exist some positive constants C = C ( M, m ) and δ = δ ( M, m ) so that if Y is a cubical subcomplex of I ( m, k ) and φ : Y → C ( M ) has f ( φ ) < δ , then there exists a map Φ : Y → C ( M ) continuous in the F -topology and satisfying(i) Φ( x ) = φ ( x ) for all x ∈ Y ;(ii) if α is some j -cell in Y , then Φ restricted to α depends only on the values of φ restricted onthe vertices of α ;(iii) sup { F (Φ( x ) , Φ( y )) : x, y lie in a common cell of Y } ≤ C f ( φ ) . Proof. [28, Theorem 3.10] proved this result when C ( M ) is replaced by Z n ( M, Z ) . We can use thedouble cover ∂ : C ( M ) → Z n ( M, Z ) (see [29, Section 5]) to lift the extension from Z n ( M, Z ) to C ( M ) .Let C = C ( M, m ) and δ = δ ( M ) be given in [28, Theorem 3.10]. Denote ˜ φ = ∂ ◦ φ : Y →Z n ( M, Z ) as the projection of φ into Z n ( M, Z ) . Then f ( ˜ φ ) < δ , so by [28, Theorem 3.10], thereexists a map: ˜Φ : Y → Z n ( M, M , Z ) continuous in the M -topology and satisfying(a) ˜Φ( x ) = ˜ φ ( x ) for all x ∈ Y ;b) if α is some j -cell in Y , then ˜Φ restricted to α depends only on the values of ˜ φ restricted onthe vertices of α ;(c) sup { M ( ˜Φ( x ) , ˜Φ( y )) : x, y lie in a common cell of Y } ≤ C f ( φ ) . By [29, Claim 5.2], ˜Φ can be uniquely lifted to a continuous map Φ : Y → C ( M ) such that ∂ ◦ Φ = ˜Φ and Φ( x ) = φ ( x ) for all x ∈ Y . In fact, given a j -cell α and a fixed vertex x ∈ α ,there is a unique lift Φ : α → C ( M ) such that Φ( x ) = φ ( x ) . By the construction in [29, Claim5.2], F (Φ( x ) , Φ( x )) = F ( ˜Φ( x ) , ˜Φ( x )) ≤ C f ( φ ) for every x ∈ α , so we know by the ConstancyTheorem that Φ( x ) = φ ( x ) for each vertex x ∈ α when δ is small enough. Thus Φ can be obtainedby lifting ˜Φ in each cell of Y .Since ∂ Φ( x ) and ˜Φ( x ) represent the same varifold, Φ is continuous in the F -topology. So we haveproved (i)(ii).For (iii), we have F (Φ( x ) , Φ( y )) = F (Φ( x ) , Φ( y )) + F ( | ∂ Φ( x ) | , | ∂ Φ( y ) | ) ≤ C f ( φ ) . (cid:3) Remark . Note that in general the mass of ∂ Φ( x ) − ∂ Φ( y ) as element in Z n ( M ) may not be equalto that of ˜Φ( x ) − ˜Φ( y ) , so we may not be able to prove the M -continuity for Φ .Following [28, 3.10], we call the map Φ given in Theorem 1.12 the Almgren extension of φ . Wewill record a few properties concerning the homotopy equivalence of Almgren’s extensions. ULTIPLICITY ONE CONJECTURE 11
Before stating the next result, we first recall the notion of homotopic equivalence between discretesweepouts. Let Y be a cubical subcomplex of I ( m, k ) . Given two discrete maps φ i : Y ( l i ) → C ( M ) ,we say φ is homotopic to φ with fineness less than η , if there exist l ∈ N , l > l , l and a map ψ : I (1 , k + l ) × Y ( l ) → C ( M ) with fineness f ( ψ ) < η and such that ψ ([ i − , y ) = φ i ( n ( k + l, k + l i )( y )) , i = 1 , , y ∈ Y ( l ) . The following result is analogous to [28, Proposition 3.11]. We provide a lightly different proof.
Proposition 1.14.
With φ , φ as above, if η < δ ( M, m ) in Theorem 1.12, then the Almgren exten-sions Φ , Φ : Y → C ( M ) of φ , φ , respectively, are homotopic to each other in the F -topology.Proof. By Theorem 1.12, the Almgren extension
Ψ : I × Y → C ( M ) of ψ is continuous in F -topology and is a homotopy between the Almgren extensions Φ ′ , Φ ′ of φ ′ , φ ′ : Y ( l ) → C ( M ) (given by φ ′ i ( y ) = ψ ([ i − , y ) ). Note that Φ ′ i is just a reparametrization of the Almgren extension Φ i of φ i for i = 1 , respectively, so Φ i is homotopic to Φ ′ i in the F -topology. Now let us describe thereparametrization map. Given an arbitrary cell α and k ∈ N , we take α c to be the center cell of α ( k ) .We can define a map n α,k : α → α such that it maps α c to α linearly, and for each x ∈ α \ α c , if wedenote by x c the nearest point projection of x to ∂α c then n α,k maps x to n α,k ( x c ) . This map dilates α c to α and compresses α \ α c to the boundary ∂α , and it is homotopic to the identity map. With thisnotion Φ ′ i | α = Φ i | α ◦ n α,l − l i on each cell α ∈ Y ( l i ) . Hence we finish the proof. (cid:3) The following result is the counterpart of [28, Corollary 3.12].
Proposition 1.15.
Let { φ i } i ∈ N and { ψ i } i ∈ N be given by Theorem 1.11 applied to some Φ therein.Assume that Φ is continuous in the F -topology on X . Then the Almgren extension Φ i is homotopic to Φ in the F -topology for sufficiently large i .In particular, for i large enough, there exist homotopy maps Ψ i : [0 , × X → C ( M ) continuousin the F -topology, Ψ i (0 , · ) = Φ i , Ψ i (1 , · ) = Φ , and lim sup i →∞ sup t ∈ [0 , ,x ∈ X F (Ψ i ( t, x ) , Φ( x )) → . Therefore for given h ∈ C ∞ ( M ) , we have lim sup i →∞ sup x ∈ X A h (Φ i ( x )) ≤ sup x ∈ X A h (Φ( x )) . Proof.
For i large enough such that δ i < δ in Theorem 1.12, we let ¯Ψ i : I × X → C ( M ) be theAlmgren extensions of ψ i . By Theorem 1.11(iv) (with Z = X ) and Theorem 1.12(iii), we know that(1.6) lim sup i →∞ sup t ∈ [0 , ,x ∈ X F ( ¯Ψ i ( t, x ) , Φ( x )) → . As in the proof of the above Proposition, we can amend ¯Ψ i with the reparametrization maps associatedwith the two pairs (Φ ′ i − , Φ i − ) and (Φ ′ i , Φ i ) , and abuse the notation and still denote them by ¯Ψ i . Then ¯Ψ i is a continuous (in the F -topology) homotopy between Φ i − and Φ i . Note that the reparametriza-tions are done is small cells with sizes converging to zero, so (1.6) still holds true for the amended mapsby Theorem 1.12(iii) again. For given i large enough, to construct the homotopy from Φ i to Φ , we canjust let Ψ i : [0 , ∞ ] × X → C ( M ) be the gluing of all { ¯Ψ j } j ≥ i . Note that by (1.6), Ψ i ( ∞ , · ) = Φ (we can identify [0 , ∞ ] with [0 , in the definition of Ψ i ), and (1.6) holds true with ¯Ψ i replaced by Ψ i .Hence we finish the proof. (cid:3) Proof of the min-max Theorem.
One key ingredient in the Almgren-Pitts theory to prove reg-ularity of min-max varifold is to introduce the “almost minimizing” concept. Given h ∈ S ( g ) , werefer to [52, Section 6] for the detailed notion of h -almost minimizing varifold and related properties.The existence of almost minimizing varifolds follows from a combinatorial argument of Pitts [31, page165-page 174] inspired by early work of Almgren [3]. Pitts’s argument works well in the constructionof min-max PMC hypersurfaces; see [52, Theorem 6.4]. Marques-Neves has generalized Pitts’s com-binatorial argument to a more general form in [28, 2.12], and we can adapt their result to the PMCsetting with no change. We now describe the adaption.Consider a sequence of cubical subcomplexes Y i of I ( m, k i ) with k i → ∞ , and a sequence S = { ϕ i } i ∈ N of maps ϕ i : ( Y i ) → C ( M ) with fineness f ( ϕ i ) = δ i converging to zero. Define L h ( S ) = lim sup i →∞ sup {A h ( ϕ i ( y )) : y ∈ ( Y i ) } , K ( S ) = { V = lim j →∞ | ∂ϕ i j ( y j ) | as varifolds : y j ∈ ( Y i j ) } , and C ( S ) = { V = lim j →∞ | ∂ϕ i j ( y j ) | as varifolds : with lim j →∞ A h ( ϕ i j ( y j )) = L h ( S ) } . We say that an element V ∈ C ( S ) is h -almost minimizing in small annuli with respect to S (c.f.[52, Definition 6.3]), if for any p ∈ M and any small enough annulus A = A r ,r ( p ) centered at p with radii < r < r , there exist sequences { i j } j ∈ N ⊂ { i } i ∈ N and { y j : y j ∈ ( Y i j ) } j ∈ N , such that V = lim j →∞ | ∂ϕ i j ( y j ) | , lim j →∞ A h ( ϕ i j ( y j )) = L h ( S ) , and ϕ i j ( y j ) ∈ A h ( A ; ǫ i , δ i ; M ) (see [52,Definition 6.1]) for some ǫ i , δ i → . The last condition is usually called ( ǫ i , δ i , h ) -almost minimizing .Note that by [52, Proposition 6.5], V is also h -almost minimizing in small annuli in the sense of [52,Definition 6.3].The following is a variant of [28, Theorem 2.13] and [31, Theorem 4.10]. Theorem 1.16.
If no element V ∈ C ( S ) is h -almost minimizing in small annuli with respect to S ,then there exists a sequence ˜ S = { ˜ ϕ i } of maps ˜ ϕ i : Y i ( l i ) → C ( M ) , for some l i ∈ N , such that: • ˜ ϕ i is homotopic to ϕ i with fineness converging to zero as i → ∞ ; • L h ( ˜ S ) < L h ( S ) .Proof. By the assumption of the theorem, for each V ∈ C ( S ) , there exists a p ∈ M , such that for any ˜ r > , there exist r, s > , with ˜ r > r + 2 s > r − s > and ǫ > , such that, if A h ( ϕ i ( y )) > L h ( S ) − ǫ and F ( | ∂ϕ i ( y ) | , V ) < ǫ , then ϕ i ( y ) / ∈ A h ( A r − s,r +2 s ( p ); ǫ, δ ; M ) for any δ > . As inthe proof of [31, Theorem 4.10], we denote c = (3 m ) m . By the compactness of C ( S ) , we can finda uniform ǫ > and I ∈ N , and finitely many points p , · · · , p ν ∈ M , and for each p j , we can find c concentric annuli A j, ⊃⊃ · · · ⊃⊃ A j,c (centered at p j ), such that, if A h ( ϕ i ( y )) > L h ( S ) − ǫ and i > I , then there exists some j ∈ { , · · · , ν } , so that ϕ i ( y ) / ∈ A h ( A j,a ; ǫ, δ ; M ) for all a ∈ { , · · · , c } and for any δ > . From here the construction in [31, Page 165-174] can be applied to S so as toproduce the desired ˜ S . (cid:3) ULTIPLICITY ONE CONJECTURE 13
Now we are ready to prove Theorem 1.7 following closely that of [27, Theorem 3.8]. The onlyadditional thing is to keep track of the volume term R Ω hd H n +1 in A h (Ω) and the values of mapsassumed on Z . Proof of Theorem 1.7.
Let { Φ i } i ∈ N be a pulled-tight min-max sequence for Π . Given Φ i : X →C ( M ) , it has no concentration of mass as it is continuous in the F -topology, so applying Theorem 1.11gives a sequence of maps: φ ji : X ( k ji ) → C ( M ) , with k ji < k j +1 i and a sequence of positive { δ ji } j ∈ N → , satisfying (i) · · · (v) in Theorem 1.11.As Φ i is continuous in the F -topology, by the same reasoning as Theorem 1.11(iii)(iv), we furtherhave that for every x ∈ X ( k ji ) , M ( ∂φ ji ( x )) ≤ M ( ∂ Φ i ( x )) + η ji with η ji → as j → ∞ , and sup { F ( φ ji ( x ) , Φ i ( x )) : x ∈ X ( k ji ) } → , as j → ∞ . Now choose j ( i ) → ∞ as i → ∞ , such that ϕ i = φ j ( i ) i : X ( k j ( i ) i ) → C ( M ) satisfies: • sup { F ( ϕ i ( x ) , Φ i ( x )) : x ∈ X ( k j ( i ) i ) } ≤ a i with a i → as i → ∞ ; • sup { F (Φ i ( x ) , Φ i ( y )) : x, y ∈ α, α ∈ X ( k j ( i ) i ) } ≤ a i ; • the fineness f ( ϕ i ) → as i → ∞ ; • the Almgren extensions Φ j ( i ) i : X → C ( M ) is homotopic to Φ i in the F -topology with homo-topy maps Ψ j ( i ) i , and lim sup i →∞ sup { F (Ψ j ( i ) i ( t, x ) , Φ i ( x )) : t ∈ [0 , , x ∈ X } = 0 , and lim sup i →∞ sup x ∈ X A h (Φ j ( i ) i ( x )) ≤ lim sup i →∞ sup x ∈ X A h (Φ i ( x )) = L h (Π) , by Proposition 1.15.Therefore, if S = { ϕ i } , then L h ( S ) = L h ( { Φ i } ) and C ( S ) = C ( { Φ i } ) . By Theorem 1.16, if noelement V ∈ C ( S ) is h -almost minimizing in small annuli with respect to S , we can find a sequence ˜ S = { ˜ ϕ i } of maps: ˜ ϕ i : X ( k j ( i ) i + l i ) → C ( M ) such that • ˜ ϕ i is homotopic to ϕ i with fineness converging to zero as i → ∞ ; • L h ( ˜ S ) < L h ( S ) .By Proposition 1.14, the Almgren extensions of ϕ i , ˜ ϕ i : Φ j ( i ) i , ˜Φ i : X → C ( M ) , respectively, are homotopic to each other in the F -topology for i large enough, so ˜Φ i is homotopic to Φ i in the F -topology.By assumption (1.5) and (1.4), for i large enough, ˜ ϕ i is the identical to ϕ i ◦ n ( k j ( i ) i + l i , k j ( i ) i ) near Z ( k j ( i ) i + l i ) ; indeed, the deformation process in Theorem 1.16 was only made to those ϕ i ( x ) with A h ( ϕ i ( x )) close to L h ( S ) . Therefore the homotopy maps ˜Ψ i between Φ j ( i ) i and ˜Φ i produced byProposition 1.14 when restricted to Z are just the reparametrization maps described therein. Hence lim sup i →∞ sup { F ( ˜Ψ i ( t, x ) , Φ j ( i ) i ( x )) : t ∈ [0 , , x ∈ Z } = 0 . Therefore { ˜Φ i } i ∈ N ∈ Π . However, by Theorem 1.12 lim sup i →∞ sup {A h ( ˜Φ i ( x )) : x ∈ X } ≤ L h ( ˜ S ) < L h ( S ) = L h (Π) . This is a contradiction. So some V ∈ C ( S ) = C ( { Φ i } ) is h -almost minimizing in small annuli withrespect to S , and hence is h -almost minimizing in small annuli in the sense of [52, Definition 6.3].To finish the proof, we need to show that V has c -bounded first variation, and then [52, Theorem7.1 and Proposition 7.3] give the regularity of V and the existence of Ω . Indeed, by Definition 1.9, V either has c -bounded first variation or belongs to | ∂ Φ | ( Z ) . Being h -almost minimizing in small annuliimplies that V has c -bounded first variation away from finitely many points by [52, Lemma 6.2]. If V ∈ | ∂ Φ | ( Z ) , then the proof of [19, Theorem 4.1] implies that k V k has at most r n − -volume growthnear these bad points, so the first variation extends across these points, and hence V has c -boundedfirst variation in M . (Note that even if V ∈ | ∂ Φ | ( Z ) , the associated Ω / ∈ Φ ( Z ) , as Ω may be equalto M \ Φ ( z ) for some z ∈ Z .) So we finish the proof. (cid:3)
2. C
OMPACTNESS OF
PMC
HYPERSURFACES WITH BOUNDED M ORSE INDEX
Now we present an adaption of Sharp’s compactness theorem [35] (for minimal hypersurfaces)to the PMC setting and necessary modifications of the proof. Given a closed Riemannian manifold ( M n +1 , g ) and h ∈ S ( g ) , denote by P h the class of smooth, closed, almost embedded hypersurfaces Σ ⊂ M , such that Σ is represented as the boundary of some open subset Ω ⊂ M (in the sense ofcurrent), and the mean curvature of Σ with respect to the outer normal of Ω is prescribed by h , i.e. H Σ = h | Σ . In the following we will sometime abuse the notation and identify Σ with Ω .Note that when h ∈ S ( g ) , the min-max PMC hypersurfaces produced in Theorem 1.7 satisfy theabove requirements. Indeed, such Σ = ∂ Ω is a critical point of the weighted A h functional (1.1): A h (Ω) = Area(Σ) − Z Ω h d H n +1 . The second variation formula for A h along normal vector field X = ϕν ∈ X ( M ) is given by δ A h | Ω ( X, X ) = Z Σ ( |∇ ϕ | − ( Ric M ( ν, ν ) + | A Σ | + ∂ ν h ) ϕ ) dµ Σ . The classical Morse index for Σ is defined as the number of negative eigenvalues of the the abovequadratic form. However, since we will deal with hypersurfaces with self-touching, a weaker versionof index is needed. We adopt a concept used by Marques-Neves [27, Definition 4.1]. As we will see,this weaker index works well for proving both compactness theory and Morse index upper bound. Definition 2.1.
Given Σ ∈ P h with Σ = ∂ Ω , k ∈ N and ǫ ≥ , we say that Σ is k -unstable in an ǫ -neighborhood if there exists < c < an a smooth family { F v } v ∈ B k ⊂ Diff( M ) with F = Id , F − v = F − v for all v ∈ B k (the standard k -dimensional ball in R k ) such that, for any Ω ′ ∈ B F ǫ (Ω) ,the smooth function: A h Ω ′ : B k → [0 , ∞ ) , A h Ω ′ ( v ) = A h ( F v (Ω ′ )) ULTIPLICITY ONE CONJECTURE 15 satisfies: • A h Ω ′ has a unique maximum at m (Ω ′ ) ∈ B kc / √ (0) ; • − c Id ≤ D A h Ω ′ ( u ) ≤ − c Id for all u ∈ B k .Since Σ is a critical point of A h , necessarily m (Ω) = 0 . Remark . If a sequence Ω i converges to Ω in the F -topology, then A h Ω i tends to A h Ω in the smoothtopology. Thus if a Σ ∈ P h is k -unstable in a -neighborhood, then it is k -unstable in an ǫ -neighborhoodfor some ǫ > . Definition 2.3.
Given a Σ ∈ P h and k ∈ N , we say that its Morse index is bounded (from above) by k , denoted as index(Σ) ≤ k, if it is not j -unstable in -neighborhood for any j ≥ k + 1 .All the above concepts can be localized to an open subset U ⊂ M by using Diff( U ) in place of Diff( M ) . If Σ has index equal to 0 in U , we say Σ is weakly stable in U . Proposition 2.4. If Σ ∈ P h is smoothly embedded with no self-touching, then Σ is k -unstable (in -neighborhood) if and only if its classical Morse index is ≥ k .Proof. The proof is the same as [27, Proposition 4.3]. (cid:3)
We have the following curvature estimates as a variant of [52, Theorem 3.6] (with relatively weakerstability assumptions).
Theorem 2.5 (Curvature estimates for weakly stable PMC) . Let ≤ ( n + 1) ≤ , and U ⊂ M be anopen subset. Let Σ ∈ P h be weakly stable in U with Area(Σ) ≤ C , then there exists C dependingonly on n, M, k h k C , C , such that | A Σ | ( x ) ≤ C dist M ( x, ∂U ) for all x ∈ Σ . Proof.
The curvature estimates follow from standard blowup arguments together with the BernsteinTheorem [34, Theorem 2] and [33, Theorem 3]. In particular, being weakly stable in U means that forany ambient vector field X ∈ X ( U ) which generates the flow φ Xt , we have(2.1) d dt (cid:12)(cid:12)(cid:12) t =0 A h ( φ Xt (Ω)) ≥ . Assume the conclusion were false, then there exists a sequence of weakly stable hypesurfaces { Σ i } i ∈ N with prescribing functions { h i } i ∈ N satisfying uniform bounds, but sup U dist M ( · , ∂U ) | A Σ i | ( · ) →∞ . By the standard blowup process (c.f. [41]), one can take a sequence of rescalings of Σ i whichconverges locally in C ,α and graphically to a non-flat minimal hypersurface Σ ∞ in R n +1 . Note thatthe rescalings of { h i } converges to 0 locally uniformly in C . By the almost embedded assumptionand the maximum principle for minimal hypersurfaces ([8]), Σ ∞ is embedded and hence is 2-sided.By the classical monotonicity formula and area upper bound assumption on { Σ i } , Σ ∞ has polynomialvolume growth. The key observation is that (2.1) is preserved under locally C ,α convergence, andhence Σ ∞ is a stable minimal hypersurface. Therefore it has to be flat by the Bernstein Theorem, butthis is a contradiction. (cid:3) Given h ∈ S ( g ) , < Λ ∈ R and I ∈ N , let P h (Λ , I ) := { Σ ∈ P h : Area(Σ) ≤ Λ , index(Σ) ≤ I } . Theorem 2.6 (Compactness for PMC’s with bounded index) . Let ( M n +1 , g ) be a closed Riemannianmanifold of dimension ≤ ( n + 1) ≤ . Assume that { h k } k ∈ N is a sequence of smooth functions in S ( g ) such that lim k →∞ h k = h ∞ in smooth topology. Let { Σ k } k ∈ N be a sequence of hypersurfacessuch that Σ k ∈ P h k (Λ , I ) for some fixed Λ > and I ∈ N . Then,(i) Up to a subsequence, there exists a smooth, closed, almost embedded hypesurface Σ ∞ withprescribed mean curvature h ∞ , such that Σ k → Σ ∞ (possibly with integer multiplicity) in thevarifold sense, and hence also in the Hausdorff distance by monotonicity formula.(ii) There exists a finite set of points Y ⊂ M with Y ≤ I , such that the convergence of Σ k → Σ ∞ is locally smooth and graphical on Σ ∞ \ Y .(iii) If h ∞ ∈ S ( g ) , then the multiplicity of Σ ∞ is 1, and Σ ∞ ∈ P h ∞ (Λ , I ) .(iv) Assuming Σ k = Σ ∞ eventually and h k = h ∞ = h ∈ S ( g ) for all k such that every Σ ∈ P h is properly embedded with no self-touching, then Y = ∅ , and the nullity of Σ ∞ with respect to δ A h is ≥ .(v) If h ∞ ≡ , then the classical Morse index of Σ ∞ satisfies index(Σ ∞ ) ≤ I (without countingmultiplicity).Remark . One main goal of this result is to use PMC hypersurfaces with prescribing functions in S ( g ) to approximate PMC’s with prescribing functions lying in C ∞ ( M ) \S ( g ) . Therefore, it is naturalto not assume h ∞ ∈ S ( g ) . Indeed for some h ∈ C ∞ ( M ) , a PMC Σ associated with h may havetouching set containing a relative open subset W ⊂ Σ , where h vanishes. For such hypersurfaces, A h is defined by viewing Σ as an Alexandrov immersed hypersurface, and so does the weak index. Proof.
The proof follows essentially the same way as [35, Theorem 2.3] once we use Theorem 2.5 toreplace [35, Theorem 2.1]; we will provide necessary modifications.
Part 1:
We first have the following variant of [35, Lemma 3.1]. Given any collection of I + 1 pairwise disjoint open sets { U i } I +1 i =1 , we have that Σ k (we drop the sub-index k in this paragraph)is weakly stable in U i for some ≤ i ≤ I + 1 . Indeed, suppose this were false, then Σ = ∂ Ω is at least -unstable in each U i , hence there exist c i ∈ (0 , and { F it } t ∈ [ − , ⊂ Diff( U i ) with F i − t = ( F it ) − , such that − c i ≤ d dt A h ( F it (Ω)) ≤ − c i . Now for v = ( v , · · · , v I +1 ) ∈ B I +1 , let F v ( x ) = F v I +1 ◦ · · · ◦ F v ( x ) . Since { U i } are pairwise disjoint, it is easy to see that c = min { c i } and { F v } give an ( I + 1) -unstable pair for Σ , and hence is a contradiction.This fact together with Theorem 2.5 imply that (up to a subsequence) Σ k converges locally smoothlyand graphically to an almost embedded hypersurface Σ ∞ of prescribed mean curvature h ∞ (possiblywith integer multiplicity) away from at most I points, which we denote by Y . Since as varifolds Σ k have uniformly bounded first variation, by Allard’s compactness theorem [1], Σ k also converges asvarifolds to an integral varifold represented by Σ ∞ .Now we prove that Σ ∞ extends smoothly as an almost embedded hypersurface across the singularpoints Y , i.e. Y are removable. By the argument in [35, Claim 2, page 326], for each y i ∈ Y , thereexists some r i > such that Σ ∞ is weakly stable in B r i ( y i ) \{ y i } in the following sense. Denote Ω ∞ as the weak limit of Ω k as Caccioppoli sets where Σ k = ∂ Ω k . The associated functional for Σ ∞ is A h ∞ (Σ ∞ ) = Area(Σ ∞ ) − R Ω ∞ h ∞ d H n +1 . Note that the touching set of Σ ∞ may contain anopen subset W ⊂ Σ ∞ and hence ∂ Ω ∞ = Σ ∞ \{ touching set of Σ ∞ } may only be a proper subset of Σ ∞ . Nevertheless, we say Σ ∞ is weakly stable, if for any X ∈ X ( B r i ( y i ) \{ y i } ) with the associatedflow { φ Xt : t ∈ [ − ǫ, ǫ ] } , d dt (cid:12)(cid:12) t =0 A h ∞ ( φ Xt (Σ ∞ )) ≥ . Note that if this were not true for some X ∈ X ( B r i ( y i ) \{ y i } ) , as A h k ( φ Xt (Σ k )) converges to A h ∞ ( φ Xt (Σ ∞ )) smoothly as functions of t ,then Σ k is not weakly stable in B r i ( y i ) \{ y i } for k sufficiently large. Following [35, Claim 2, page ULTIPLICITY ONE CONJECTURE 17 Σ ∞ . Since Σ ∞ has bounded first variation, thenby a classical removable singularity result, Theorem B.1, we get the smooth extension. Up to here, wehave finished proving (i) and (ii). Part 2: If h ∞ ∈ S ( g ) , [52, Theorem 3.20] implies that Σ ∞ has multiplicity 1, and is a boundaryof some open set Ω ∞ ; (note that when h ∞ ∈ S ( g ) , only case (2) of [52, Theorem 3.20] will happen).In fact, fix a point p ∈ Σ ∞ where Σ ∞ is properly embedded. If the limit Σ ∞ has multiplicity ≥ ,then for i sufficiently large and inside a neighborhood of p , Σ i consists of several sheets with normalpointing to the same side of Σ ∞ , but this can not happen when Σ i bounds a region Ω i . We refer to theproof of [52, Theorem 3.20] for more details.If index(Σ ∞ ) > I , then there exist c ∈ (0 , and { F v : v ∈ B I +1 } ⊂ Diff( M ) such that − c Id ≤ D A h ∞ ( F v (Ω ∞ )) ≤ − c Id for all v ∈ B I +1 . Since Σ k = ∂ Ω k converges to Σ ∞ smoothlyaway from finitely many points, we know that Ω k converges to Ω ∞ in the F -topology as Caccioppolisets, then the sequence v → A h k ( F v (Ω k )) converges to v → A h ∞ ( F v (Ω ∞ )) smoothly as functions on B I +1 . Therefore, for k large enough, − c Id ≤ D A h k ( F v (Ω k )) ≤ − c Id , so Σ k is ( I + 1) -unstable,which is a contradiction. This finishes the proof of (iii). Part 3:
Assuming Σ k = Σ ∞ eventually and h k = h ∞ = h ∈ S ( g ) such that every element in P h is properly embedded, we know Y = ∅ by multiplicity 1 convergence and the Allard regularitytheorem [1]. Next we will produce a Jacobi field for the second variation δ A h along Σ ∞ ; this impliesthe nullity is ≥ .By (1.3), the Jacobi operator associated with δ A h along a PMC Σ ∈ P h is L h Σ ϕ = −△ Σ ϕ − (cid:0) Ric M ( ν, ν ) + | A Σ | + ∂ ν h (cid:1) ϕ. The smooth graphical convergence of Σ k → Σ implies that for k sufficiently large, Σ k can be writtenas a graph u k in the normal bundle of Σ ∞ , and u k → uniformly in smooth topology. Subtracting themean curvature operators between Σ k and Σ ∞ , we get: h ( x, u k ) − h ( x,
0) = H Σ k − H Σ ∞ = L Σ ∞ u k + o ( u k ) , where L Σ ∞ u = −△ u − (cid:0) Ric M ( ν, ν )+ | A Σ | (cid:1) u is the Jacobi operator for second variation of area, andthe second equation follows from [37] and [35, page 331]; (note that though the calculation in [35, page331] is done assuming h ≡ , it does not depend on h ). The left hand side equals to ∂ ν h ( x, t ( x ) u k ) · u k by the mean value theorem. Let ˜ u k = u k / k u k k L (Σ ∞ ) be the renormalizations, then standard ellipticestimates imply that ˜ u k converges smoothly to a nontrivial ϕ ∈ C ∞ (Σ ∞ ) such that ∂ ν h · ϕ = L Σ ∞ ϕ .This is the same as L h Σ ∞ ϕ = 0 , so we finish proving (iv). Part 4:
Assuming h ∞ ≡ , then Σ ∞ is an embedded minimal hypersurface. Assume without lossof generality that Σ ∞ is connected with multiplicity m ∈ N . Suppose the Morse index index(Σ ∞ ) ≥ I + 1 , then by similar argument as in (iii), we can deduce a contradiction. In particular, by [27,Proposition 4.3], there exist c ∈ (0 , and { F v : v ∈ B I +1 } ⊂ Diff( M ) such that − c Id ≤ D Area( F v (Σ ∞ )) ≤ − c Id for all v ∈ B I +1 . Since Σ k converges to m · Σ ∞ as varifolds, and since h k → uniformly, we know that A h k ( F v (Ω k )) converges to m · Area( F v (Σ ∞ )) smoothly as functionson B I +1 . Therefore, for k large enough, Ω k is ( I + 1) -unstable, which is a contradiction. So we finishproving (v). (cid:3) There is also a theorem analogous to the above one in the setting of changing ambient metrics on M ; see [35, Theorem A.6] for a similar result for minimal hypersurfaces. The proof proceeds thesame way when one realizes that the constant C in Theorem 2.5 depends only on the k g k C when g is allowed to change. Theorem 2.8.
Let M n +1 be a closed manifold of dimension ≤ ( n + 1) ≤ , and { g k } k ∈ N be asequence of metrics on M that converges smoothly to some limit metric g . Let { h k } k ∈ N be a sequenceof smooth functions with h k ∈ S ( g k ) that converges smoothly to some limit h ∞ ∈ C ∞ ( M ) . Let { Σ k } k ∈ N be a sequence of hypersurfaces with Σ k ∈ P h k (Λ , I ) for some fixed Λ > and I ∈ N . Thenthere exists a smooth, closed, almost embedded hypersurface Σ ∞ with prescribing mean curvature h ∞ , such that all properties (i)(ii)(iii) in the above theorem are satisfied.
3. M
ORSE INDEX UPPER BOUND
In this part, we will establish Morse index upper bound for min-max PMC hypersurfaces obtainedin Theorem 1.7. We will follow closely the strategy of Marques-Neves [27, Theorem 1.2], where theyproved Morse index upper bound for min-max minimal hypersurfaces. Recall that the Morse index ofan almost embedded PMC hypersurface Σ is given in Definition 2.3. Theorem 3.1.
Let ( M n +1 , g ) be a closed Riemannian manifold of dimension ≤ ( n + 1) ≤ , and h ∈ S ( g ) which satisfies R M h ≥ . Given a k -dimensional cubical complex X and a subcomplex Z ⊂ X , let Φ : X → C ( M ) be a map continuous in the F -topology, and Π be the associated ( X, Z ) -homotopy class of Φ . Suppose (3.1) L h (Π) > max x ∈ Z A h (Φ ( x )) . Then there exists a nontrivial, smooth, closed, almost embedded hypersurface Σ n ⊂ M , such that • Σ is the boundary of some Ω ∈ C ( M ) where its mean curvature with respect to the unit outernormal of Ω is h , i.e. H Σ = h | Σ , • A h (Ω) = L h (Π) , • index(Σ) ≤ k . Preliminary lemmas.
Let h ∈ S ( g ) . Assume that Σ = ∂ Ω ∈ P h is k -unstable in an ǫ -neighborhood, ǫ > . Let { F v } v ∈ B k be the associated smooth family given in Definition 2.1.The first lemma is a counterpart of [27, Lemma 4.4]. Lemma 3.2.
There exists ¯ η = ¯ η ( ǫ, Σ , { F v } ) > , such that if Ω ∈ C ( M ) with F (Ω , Ω ) ≥ ǫ satisfies A h ( F v (Ω)) ≤ A h (Ω) + ¯ η for some v ∈ B k , then F ( F v (Ω) , Ω ) ≥ η .Proof. Assume by contradiction that there exist Ω i , F (Ω i , Ω ) ≥ ǫ satisfying A h ( F v i (Ω i )) ≤ A h (Ω i ) + 1 i for some v i ∈ B k , but F ( F v i (Ω i ) , Ω ) ≤ i .Denote v = lim v i , and pass to the limit as i → ∞ , then Ω i → F − v (Ω ) in F -metric, and A h (Ω ) ≤A h ( F − v (Ω )) , which implies that v = 0 ; hence Ω i → Ω in the F -metric, which is a contradiction. (cid:3) ULTIPLICITY ONE CONJECTURE 19
For each Ω ∈ B F ǫ (Ω ) , consider the one-parameter flow { φ Ω ( · , t ) : t ≥ } ⊂ Diff( B k ) generatedby the vector field u → − (1 − | u | ) ∇A h Ω ( u ) , u ∈ B k . When u ∈ B k is fixed, the function t → A h Ω ( φ Ω ( u, t )) is non-increasing.The following lemma is a variant of [27, Lemma 4.5], and the proof is recorded in Appendix C. Lemma 3.3.
For any δ < / there exists T = T ( δ, ǫ, Ω , { F v } , c ) ≥ such that for any Ω ∈ B F ǫ (Ω ) and v ∈ B k with | v − m (Ω) | ≥ δ , we have A h Ω ( φ Ω ( v, T )) < A h Ω (0) − c and | φ Ω ( v, T ) | > c . Deformation theorem.
Taking a min-max sequence { Φ i } i ∈ N , we will prove a deformation the-orem as an adaption of [27, Theorem 5.1] to our setting. Recall that P h denotes the class of smooth,closed, almost embedded hypersurface Σ ⊂ M represented as boundary Σ = ∂ Ω , and of prescribedmean curvature h .Fix a σ > such that L h − sup x ∈ Z A h (Φ ( x )) > σ . Denote X i,σ = { x ∈ X, such that A h (Φ i ( x )) ≥ L h − σ } . Note that when i is sufficiently large, X i,σ ⊂ X \ Z .Now we present the deformation theorem, and the proof follows closely that of [27, Theorem 5.1].Given two subsets A, B ⊂ C ( M ) , we denote F ( A, B ) := inf { F (Ω A , Ω B ) : Ω A ∈ A, Ω B ∈ B } . Theorem 3.4.
Suppose that(a)
Σ = ∂ Ω ∈ P h is ( k + 1) -unstable;(b) K ⊂ C ( M ) is a subset, so that F ( { Ω } , K ) > and F (Φ i ( X i,σ ) , K ) > for all i ≥ i ;(c) A h (Ω) = L h .Then there exist ¯ ǫ > , j ∈ N , and another sequence { Ψ i } i ∈ N , Ψ i : X → ( C ( M ) , F ) , so that(i) Ψ i is homotopic to Φ i in the F -topology for all i ∈ N and Ψ i | Z = Φ i | Z for i ≥ j ;(ii) L h ( { Ψ i } ) ≤ L h ;(iii) F (Ψ i ( X i,σ ) , B F ¯ ǫ (Ω) ∪ K ) > for all i ≥ j .Proof. Denote d = F ( { Ω } , K ) > .By (a), Σ is ( k + 1) -unstable in some ǫ -neighborhood. Let { F v } v ∈ B k +1 , c be the associated familyand constant as in Definition 2.1. By possibly changing ǫ , { F v } , c , we can assume that(3.2) inf { F ( F v (Ω ′ ) , K ) , v ∈ B k +1 } > d , for all Ω ′ ∈ B F ǫ (Ω) . Let X ( k i ) be a sufficiently fine subdivision of X so that F (Φ i ( x ) , Φ i ( y )) < δ i for any x, y belong-ing to the same cell in X ( k i ) with δ i = min { − ( i + k +2) , ǫ/ } . We can also assume that | m (Φ i ( x )) − m (Φ i ( y )) | < δ i for any x, y with F (Φ i ( x ) , Ω) ≤ ǫ , F (Φ i ( y ) , Ω) ≤ ǫ , and belonging to the same cell in X ( k i ) .For η > , let U i,η be the union of all cells σ ∈ X ( k i ) so that F (Φ i ( x ) , Ω) < η for all x ∈ σ . Then U i,η is a subcomplex of X ( k i ) . If a cell β / ∈ U i,η , then F (Φ i ( x ′ ) , Ω) ≥ η for some x ′ ∈ β . Therefore, F (Φ i ( x ) , Ω) ≥ η − δ i for all x ∈ β . By (c) (after possibly shrinking ǫ ), we can assume U i, ǫ ⊂ X i,σ . For each i ∈ N and x ∈ U i, ǫ , we simply denote A hi,x = A h Φ i ( x ) , m i ( x ) = m (Φ i ( x )) and φ i,x = φ Φ i ( x ) . The function m i : U i, ǫ → B k +1 is continuous, and the two families {A hi,x } x ∈ U i, ǫ , { φ i,x } x ∈ U i, ǫ are continuous in x . Following [27, 5.1] we can define a continuous map ˆ H i : U i, ǫ × [0 , → B k +11 / i (0) , so that ˆ H i ( x,
0) = 0 for all x ∈ U i, ǫ and(3.3) inf x ∈ U i, ǫ | ˆ H i ( x, − m i ( x ) | ≥ η i > , for some η i > . The construction here is the same so we omit details. The crucial ingredient is the fact that U i, ǫ hasdimension less than or equal to k while the image set B k +1 has dimension k + 1 .Let c : [0 , ∞ ) → [0 , be a cutoff function which is non-increasing, equals to in a neighborhoodof [0 , ǫ/ , and in a neighborhood of [7 ǫ/ , + ∞ ) . For y / ∈ U i, ǫ , F (Φ i ( y ) , Ω) ≥ ǫ − δ i ≥ ǫ/ .Hence c ( F (Φ i ( y ) , Ω)) = 0 , for all y / ∈ U i, ǫ . Consider the map H i : X × [0 , → B k +12 − i (0) defined as H i ( x, t ) = ˆ H i ( x, c ( F (Φ i ( x ) , Ω)) t ) , if x ∈ U i, ǫ and H i ( x, t ) = 0 , if x ∈ X \ U i, ǫ . Then H i is continuous.With η i as given in (3.3), let T i = T ( η i , ǫ, Ω , { F v } , c ) ≥ be given by Lemma 3.3. Now we set D i : X → B k +1 such that D i ( x ) = φ i,x ( H i ( x, , c ( F (Φ i ( x ) , Ω)) T i ) , if x ∈ U i, ǫ and D i ( x ) = 0 , if x ∈ X \ U i, ǫ . Then D i is continuous.Define Ψ i : X → C ( M ) , Ψ i ( x ) = F D i ( x ) (Φ i ( x )) . In particular, Ψ i ( x ) = Φ i ( x ) , if x ∈ X \ U i, ǫ . Hence Ψ i | Z = Φ i | Z for i sufficiently large.Note that the map D i is homotopic to the zero map in B k +1 , so Ψ i is homotopic to Φ i in the F -topology for all i ∈ N . Up to here, we proved (i). Claim 1: L h ( { Ψ } i ∈ N ) ≤ L h .By the non-increasing property of t → A hi,x ( φ i,x ( u, t )) , we have that for all x ∈ X , A h (Ψ i ( x )) ≤ A h ( F H i ( x, (Φ i ( x ))) . Using the fact that H i ( x, ∈ B k +11 / i (0) for all x ∈ X and that k F v − Id k C → uniformly as v → ,we have that(3.4) lim i →∞ sup x ∈ X (cid:12)(cid:12) A h (Φ i ( x )) − A h ( F H i ( x, (Φ i ( x ))) (cid:12)(cid:12) = 0 , and this finishes proving Claim 1. ULTIPLICITY ONE CONJECTURE 21
Claim 2:
There exists ¯ ǫ > , such that for all sufficiently large i , F (Ψ i ( X ) , Ω) > ¯ ǫ .There are three cases. If x ∈ X \ U i, ǫ , then Ψ i ( x ) = Φ i ( x ) and so F (Ψ i ( x ) , Ω) ≥ ǫ .If x ∈ U i, ǫ \ U i, ǫ/ , then F (Φ i ( x ) , Ω) ≥ ǫ . The non-increasing property of t → A hi,x ( φ i,x ( u, t )) implies A h (Ψ i ( x )) = A h ( F D i ( x ) (Φ i ( x ))) ≤ A h ( F H i ( x, (Φ i ( x ))) . From (3.4), we have that for i large enough, A h ( F H i ( x, (Φ i ( x ))) ≤ A h (Φ i ( x )) + ¯ η, for all x ∈ X, where ¯ η = ¯ η ( ǫ, Ω , { F v } ) > is given by Lemma 3.2. Combining the two inequalities with Lemma3.2 applied to Φ i ( x ) , v = D i ( x ) , we get F (Ψ i ( x ) , Ω) ≥ η .Finally when x ∈ U i, ǫ/ , c ( F (Φ i ( x ) , Ω)) = 1 . Hence by Lemma 3.3 (with δ = η i , Ω = Φ i ( x ) , v = H i ( x, ) we have A h (Ψ i ( x )) = A hi,x ( φ i,x ( H i ( x, , T i )) < A hi,x (0) − c
10 = A h (Φ i ( x )) − c . Note that there exists ¯ γ = ¯ γ (Ω , c ) so that A h (Ω ′ ) ≤ A h (Ω) − c
20 = ⇒ F (Ω ′ , Ω) ≥ γ. By assumption (c), we can choose i sufficiently large so that sup x ∈ X A h (Φ i ( x )) ≤ A h (Ω) + c . So A h (Ψ i ( x )) ≤ A h (Ω) − c . This implies that F (Ψ i ( x ) , Ω) ≥ γ , and hence ends the proof of Claim 2. Claim 3:
For all i , F (Ψ i ( X i,σ ) , K ) > .If x ∈ X i,σ \ U i, ǫ , then Ψ i ( x ) = Φ i ( x ) and so F (Ψ i ( X i,σ \ U i, ǫ ) , K ) > . If x ∈ U i, ǫ , then F (Φ i ( x ) , Ω) ≤ ǫ , and by (3.2) we have F (Ψ i ( x ) , K ) ≥ d . So we finish proving Claim 3, and hencethe theorem. (cid:3) Proof of Morse index upper bound.
Let M n +1 be a closed manifold of dimension ≤ ( n +1) ≤ . A pair ( g, h ) consisting of a Riemannian metric g and a smooth function h ∈ C ∞ ( M ) iscalled a good pair , if • h ∈ S ( g ) , i.e. h is Morse and the zero set { h = 0 } is a smooth embedded hypersurface in M with mean curvature H vanishing to at most finite order, and • g is bumpy for P h , i.e. every Σ ∈ P h is properly embedded (no self-touching), and is nonde-generate (nullity equal to zero).Denote S as the class of smooth functions h ∈ C ∞ ( M ) such that h is Morse and the zero set { h = 0 } is a smooth embedded hypersurface. S is open and dense in C ∞ ( M ) , and is independent ofthe choice of a metric; (see [52, Proposition 3.8]). Lemma 3.5.
Given h ∈ S , the set of Riemannian metrics g on M with ( g, h ) as a good pair is genericin the Baire sense. Proof.
By the proof of [52, Proposition 3.8], we know that the set of metrics g under which { h = 0 } has mean curvature vanishing to at most finite order is an open and sense subset. In particular, opennessfollows as small smooth perturbations of g will bound the order of vanishing of H { h =0 } . To showdenseness, note that it is proved in [52, Proposition 3.8] for any h ∈ S and any metric g , one canfirst perturb g slightly so that { h = 0 } is not a minimal hypersurface, and then there exists a flow { F t : t ∈ ( − ǫ, ǫ ) } ⊂ Diff( M ) supported near { h = 0 } , such that the zero set of h ◦ ( F t ) − has meancurvature vanishing to at most finite order for t > . That is to say the zero set { h = 0 } satisfies therequirement for the pull-back metrics F ∗ t g .In a series of celebrated papers [42, 44, 45], White proved that for a fixed h ∈ S , the set of metricsunder which all closed, simple immersed PMC’s are non-degenerate and self-transverse is generic inthe Baire sense. In fact, White proved in [42, Section 7] that the set of metrics under which all closed,simple immersed CMC hypersurfaces are non-degenerate is generic, and the proof is the same in asmooth neighborhood of an arbitrary pair ( g, h ) when h ∈ S ( g ) , hence the result follows as the setof g where h ∈ S ( g ) is open and dense. In [45, Theorem 33], White further proved self-transverseproperty for a generic set of metrics. Our almost embedded hypersurfaces are simple immersed. Sofor such generic metrics, almost embedded PMC’s are properly embedded.To finish the proof, we take the intersection of the two generic sets of metrics, which is still genericin the Baire sense. (cid:3) The following theorem is a counterpart of [27, Theorem 6.1], and the proof follows closely. Weremark that by Theorem 2.6(iv), if ( g, h ) is a good pair, then there are only finitely many elements in P h (Λ , I ) . Theorem 3.6.
Assume that ( g, h ) is a good pair and let { Φ i } i ∈ N be a min-max sequence of Π suchthat L h ( { Φ i } i ∈ N ) = L h (Π) = L h and (3.1) is satisfied.There exists a smooth, closed, properly embedded hypersurface Σ = ∂ Ω ∈ C ( { Φ i } i ∈ N ) such that Σ ∈ P h with L h (Π) = A h (Ω) , and index(Σ) ≤ k. Proof.
By the finiteness remark above, it suffices to show that, for every r > , there is a ˜Σ = ∂ ˜Ω ∈ P h such that F ([ ˜Σ] , C ( { Φ i } i ∈ N )) < r , L h (Π) = A h ( ˜Ω) , and index( ˜Σ) ≤ k. Denote by W the set of all ˜Σ = ∂ ˜Ω ∈ P h with A h ( ˜Ω) = L h and by W ( r ) the set { Σ ∈ W : F ([Σ] , C ( { Φ i } i ∈ N )) ≥ r } . Lemma 3.7.
There exist i ∈ N and ¯ ǫ > such that F (Φ i ( X ) , W ( r )) > ¯ ǫ for all i ≥ i .Proof. Suppose by contradiction for some subsequence { j } ⊂ { i } , x j ∈ X , ˜Σ j = ∂ ˜Ω j ∈ W ( r ) sothat lim j →∞ F (Φ j ( x j ) , ˜Ω j ) = 0 . Since A h ( ˜Ω j ) ≡ L h , we have lim j →∞ A h (Φ j ( x j )) = L h . Hence a subsequence | ∂ Φ j ( x j ) | willconverge as varifolds to some V ∈ C ( { Φ i } i ∈ N ) , which is a contradiction to F ( | ∂ ˜Ω i | , C ( { Φ i } i ∈ N )) ≥ r . (cid:3) Denote W k +1 as the collection of elements in W with index greater than or equal to ( k + 1) .As ( g, h ) is a good pair, this set is countable by the remark above the theorem, and we can write ULTIPLICITY ONE CONJECTURE 23 W k +1 \ B F ¯ ǫ ( W ( r )) = { Σ , Σ , · · · } , where Σ i = ∂ Ω i . Note that by possibly perturbing ¯ ǫ , we canmake sure W k +1 ∩ ∂ B F ¯ ǫ ( W ( r )) = ∅ .Using Theorem 3.4 (we can take X i,σ to be X ) with K = B F ¯ ǫ ( W ( r )) and Σ = Σ , we find ¯ ǫ > , i ∈ N , and { Φ i } i ∈ N so that • Φ i is homotopic to Φ i in the F -topology for all i ∈ N and Φ i | Z = Φ i | Z for i ≥ i ; • L h ( { Φ i } i ∈ N ) ≤ L h ; • F (Φ i ( X ) , B F ¯ ǫ (Ω ) ∪ B F ¯ ǫ ( W ( r ))) > for i ≥ i . • no Ω j belongs to ∂ B F ¯ ǫ (Ω ) .We consider Σ now. If Ω / ∈ B F ¯ ǫ (Ω ) , we apply Theorem 3.4 with K = B F ¯ ǫ (Ω ) ∪ B F ¯ ǫ ( W ( r )) , Σ = Σ , and find ¯ ǫ > , i ∈ N , and { Φ i } i ∈ N so that • Φ i is homotopic to Φ i in the F -topology for all i ∈ N and Φ i | Z = Φ i | Z for i ≥ i ; • L h ( { Φ i } i ∈ N ) ≤ L h ; • F (Φ i ( X ) , B F ¯ ǫ (Ω ) ∪ B F ¯ ǫ (Ω ) ∪ B F ¯ ǫ ( W ( r ))) > for i ≥ i ; • no Ω j belongs to ∂ B F ¯ ǫ (Ω ) ∪ ∂ B F ¯ ǫ (Ω ) .If F (Ω , Ω ) < ¯ ǫ , we skip it and repeat the construction with Σ .By induction there are two possibilities. We can find for all l ∈ N a sequence { Φ li } i ∈ N , ¯ ǫ l > , i l ∈ N , and Σ j l ∈ W k +1 \ B F ¯ ǫ ( W ( r )) for some subsequences { j l } ⊂ N so that • Φ li is homotopic to Φ i in the F -topology for all i ∈ N and Φ li | Z = Φ i | Z for i ≥ i l ; • L h ( { Φ li } i ∈ N ) ≤ L h ; • F (Φ li ( X ) , ∪ lq =1 B F ¯ ǫ q (Ω j q ) ∪ B F ¯ ǫ ( W ( r ))) > for i ≥ i l ; • { Ω , · · · , Ω l } ⊂ ∪ lq =1 B F ¯ ǫ q (Ω j q ) ; • no Ω j belongs to ∂ B F ¯ ǫ q (Ω j q ) for all q = 1 , · · · , l .Or the process ends in finitely many steps. That means we can find some m ∈ N , a sequence { Φ mi } i ∈ N , ¯ ǫ , . . . , ¯ ǫ m > , i m ∈ N and Σ j , · · · , Σ j m ∈ W k +1 \ B F ¯ ǫ ( W ( r )) so that • Φ mi is homotopic to Φ i in the F -topology for all i ∈ N and Φ mi | Z = Φ i | Z for i ≥ i m ; • L h ( { Φ mi } i ∈ N ) ≤ L h ; • F (Φ mi ( X ) , ∪ mq =1 B F ¯ ǫ q (Ω j q ) ∪ B F ¯ ǫ ( W ( r ))) > for i ≥ i m . • { Ω j : j ≥ } ⊂ ∪ mq =1 B F ¯ ǫ q (Ω j q ) .In the first case we choose an increasing sequence p l ≥ i l so that sup x ∈ X A h (Φ lp l ) ≤ L h + 1 l , and set Ψ l = Φ lp l . In the second case we set p l = l and Ψ l = Φ ml . The sequence { Ψ l } l ∈ N satisfies that(i) Ψ l is homotopic to Φ p l in the F -topology, and Ψ l | Z = Φ p l | Z for all l ;(ii) L h ( { Ψ l } l ∈ N ) ≤ L h ;(iii) given any subsequence { l j } ⊂ { l } , x j ∈ X , if lim j →∞ A h (Ψ l j ( x j )) = L h , then { Ψ l j ( x j ) } j ∈ N does not converge in F -topology to any element in W k +1 ∪ W ( r ) .The Min-max Theorem 1.7 applied to { Ψ l } i ∈ N implies that W\ ( W k +1 ∪ W ( r )) is not empty andthis proves the theorem. (cid:3) Now we can use the previous theorem and the Compactness Theorem 2.8 to prove Theorem 3.1.
Proof of Theorem 3.1.
Given ( g, h ) as in the theorem, then h ∈ S ( g ) ⊂ S . By Lemma 3.5 there existsa sequence of metrics { g j } j ∈ N converging smoothly to g such that ( g j , h ) is a good pair for all j ∈ N .If L hj = L hj (Π , g j ) is the h -width of Π with respect to g j , then the sequence { L hj } j ∈ N tends to the h -width L h (Π , g ) with respect to g , and for j large enough (3.1) is satisfied with g j in place of g . For each j large enough, the previous theorem gives a properly embedded closed hypersurface Σ j = ∂ Ω j ∈ P h with A h j (Ω j ) = L hj and index(Σ j ) ≤ k (with respect to g j ). Let Σ ∞ = ∂ Ω ∞ be the limit of { Σ j } j ∈ N given in Theorem 2.8, then the locally smooth convergence implies that A h (Ω ∞ ) = L h (Π , g ) and index(Σ ∞ ) ≤ k . (cid:3)
4. M IN - MAX HYPERSURFACES ASSOCIATED WITH SWEEPOUTS OF BOUNDARIES HAVEMULTIPLICITY ONE IN A BUMPY METRIC
We present our first multiplicity one result. In particular, we will prove that the min-max minimalhypersurfaces associated with sweepouts of boundaries of Caccioppoli sets are two-sided and havemultiplicity one in a bumpy metric. We will approximate the area functional by the weighted A ǫh -functionals for some prescribing function h when ǫ → . We know by Section 1 that the min-maxPMC hypersurfaces are two-sided with multiplicity one, and we will prove that the limit minimalhypersurfaces (when ǫ → ) are also two-sided and have multiplicity one by choosing the right pre-scribing function h .Recall that a Riemannian metric g is said to be bumpy if every smooth closed immersed minimalhypersurface is non-degenerate. White proved that the set of bumpy metrics is generic in the Bairesense [42, 44]. Theorem 4.1 (Multiplicity one theorem for sweepouts of boundaries) . Let ( M n +1 , g ) be a closedRiemannian manifold of dimension ≤ ( n + 1) ≤ . Let X be a k -dimensional cubical complex and Z ⊂ X be a subcomplex, and Φ : X → C ( M ) be a map continuous in the F -topology. Let Π be theassociated ( X, Z ) -homotopy class of Φ . Assume that (4.1) L (Π) > max x ∈ Z M ( ∂ Φ ( x )) , where we let h ≡ in Section 1.1.If g is a bumpy metric, then there exists a disjoint collection of smooth, connected, closed, embedded,two-sided, minimal hypersurfaces Σ = ∪ Ni =1 Σ i , such that L (Π) = N X i =1 Area(Σ i ) , and index(Σ) = N X i =1 index(Σ i ) ≤ k. In particular, each component of Σ is two-sided and has exactly multiplicity one.Proof. Pick a h ∈ S ( g ) with R M h ≥ (to be fixed at the end), and ǫ > small enough so that L (Π) − max x ∈ Z M ( ∂ Φ ( x )) > ǫ sup M | h | · Vol( M ) . Note that we have for each Ω ∈ C ( M ) (4.2) M ( ∂ Ω) − ǫ sup M | h | · Vol( M ) ≤ A ǫh (Ω) ≤ M ( ∂ Ω) + ǫ sup M | h | · Vol( M ) . ULTIPLICITY ONE CONJECTURE 25
The above two inequalities imply that if we consider the A ǫh -functional in place of the mass M -functional for the ( X, Z ) -homotopy class Π , we have L ǫh (Π) > max x ∈ Z A ǫh (Φ ( x )) . Note that when h ∈ S ( g ) , ǫh also belongs to S ( g ) . Therefore Theorem 3.1 applies to Π and producesa nontrivial, smooth, closed, almost embedded hypersurface Σ ǫ , such that • Σ ǫ is the boundary for some Ω ǫ ∈ C ( M ) where its mean curvature with respect to the unitouter normal ν (of Ω ǫ ) is ǫ · h , i.e. H Σ ǫ = ǫ · h | Σ ǫ ; • A ǫh (Ω ǫ ) = L ǫh (Π) ; • index(Σ ǫ ) ≤ k .We denote L = L (Π) and L ǫ = L ǫh (Π) . In the following, we proceed the proof by parts. Part 1 : L ǫ → L when ǫ → . Proof : From (4.2), it is easy to see L − ǫ sup M | h | Vol( M ) ≤ L ǫ ≤ L + ǫ sup M | h | Vol( M ) . Part 2 : By Theorem 2.6, there exists a subsequence { ǫ k } → , such that Σ k = Σ ǫ k converges tosome smooth, closed, embedded, minimal hypersurface Σ ∞ (with integer multiplicity) in the sense ofTheorem 2.6(i)(ii). We denote Y as the set of points where the convergence fails to be smooth. Inparticular, by (4.2) and Part 1 and Theorem 2.6(v), we have M (Σ ∞ ) = L , and index(Σ ∞ ) ≤ k. That is to say that Σ ∞ is a min-max minimal hypersurface associated with Π .Without loss of generality, we assume from Part 3 to Part 8 that Σ ∞ has only one connected compo-nent. If Σ ∞ is 2-sided with the multiplicity equal to one, then we are done; otherwise we may assumethat either the multiplicity m > or Σ ∞ is 1-sided. Part 3 : We first assume that Σ ∞ is 2-sided. We will implicitly use exponential normal coordinatesof Σ ∞ with respect to one fixed unit normal of Σ ∞ . By the local, smooth graphical convergence Σ k → Σ ∞ away from Y , we know that there exists an exhaustion by compact domains { U k ⊂ Σ ∞ \Y} and some small δ > , so that for k large enough, Σ k ∩ ( U k × ( − δ, δ )) can be written as a set of m -normal graphs { u k , · · · , u mk : u ik ∈ C ∞ ( U k ) } over U k , and such that u k ≤ u k ≤ · · · ≤ u mk , and u ik → , in smooth topology as k → ∞ . Since Σ k is the boundary of some set Ω k , by the Constancy Theorem (applied to Ω k in U k × ( − δ, δ ) ),we know that the unit outer normal ν k of Ω k will alternate orientations along these graphs. In particular,if ν k restricted to the graph of u ik points upward (or downward), then ν k restricted to the graph of u i +1 k will point downward (or upward). Part 4 : We first deal with an easier case: m is an odd number . Hence m ≥ . In this case ν k restricted to the bottom ( u k ) and top ( u mk ) sheets point to the same side of Σ ∞ , and without loss of generality we may assume that ν k points upward therein. That means: H | Graph( u mk ) ( x ) = ǫ k h ( x, u mk ( x )) , and H | Graph( u k ) ( x ) = ǫ k h ( x, u k ( x )) , for x ∈ U k . Here and in the following the sign convention is made so that H | Graph( u ) is defined with respect to theupward pointing normal of Graph( u ) , and hence the linearized operator is positively definite.Note that since ǫh ∈ S ( g ) , by the Strong Maximum Principle [52, Lemma 3.12] (applied to twosheets of the same orientation), we know u mk ( x ) − u k ( x ) > , for all x ∈ U k . Now by subtracting the above two equations, and using the fact H | Graph( u mk ) − H | Graph( u k ) = L Σ ∞ ( u mk − u k ) + o ( u mk − u k ) (see [35, page 331] and part 3 in the proof of Theorem 2.6), we have(4.3) L Σ ∞ ( u mk − u k ) + o ( u mk − u k ) = ǫ k · ∂ ν h ( x, v k ( x )) · ( u mk ( x ) − u k ( x )) , where v k ( x ) = t ( x ) u mk ( x ) + (1 − t ( x )) u k ( x ) for some t ( x ) ∈ [0 , .Now it is a standard argument to produce a nontrivial positive Jacobi field on Σ ∞ \Y . Let us presentthe details for completeness. Write h k = u mk − u k , and pick a fixed point p ∈ U . Let ˜ h k = h k /h k ( p ) ,then ˜ h k ( p ) = 1 . By standard Harnack and elliptic estimates, ˜ h k will converge locally smoothly to apositive function ϕ on any fixed U ⊂ U k , and by a diagonalization process, we can extend ϕ to Σ ∞ \Y ,and such that L Σ ∞ ϕ = 0 , outside Y . Part 5 : Next we use White’s local foliation argument [41] to prove that ϕ extends smoothly across Y ,and this will contradict the bumpy assumption of g .Fix y ∈ Y . We use the exponential normal coordinates ( x, z ) ∈ Σ ∞ × [ − δ, δ ] . Let ǫ > beas given in Proposition D.1. Fix a small radius < η < ǫ , and choose k large enough such that k u k k ,α , k u mk k ,α ≪ ǫη near ∂B nη ( y ) so that some extensions of them to the whole B nη ( y ) have C ,α -norms bounded by ǫη . Let v k,t , v mk,t : B nη ( y ) → R , t ∈ [ − η, η ] , be the PMC local foliations associatedwith ǫ k h , H Graph( v ik,t ) ( x ) = ǫ k h ( x, v ik,t ( x )) , i = 1 , m, x ∈ B nη ( y ) , and v ik,t ( x ) = u ik ( x ) + t, i = 1 , m, x ∈ ∂B nη ( y ) . By the Hausdorff convergence of Σ k → Σ ∞ and the Strong Maximum Principle [52, Lemma 3.12](applied to Graph( u k ) and { Graph( v k,t ) } , Graph( u mk ) and { Graph( v mk,t ) } ), we have u mk ( x ) − u k ( x ) ≤ v mk, ( x ) − v k, ( x ) , when x ∈ U k ∩ B nη ( y ) . By subtracting the mean curvature equations for
Graph( v ik, ) , i = 1 , m , we get an equation similarto (4.3), L Σ ∞ ( v mk, − v k, ) + o ( v mk, − v k, ) = ǫ k · ∂ ν h ( x, v k ( x )) · ( v mk, ( x ) − v k, ( x )) . Note that the two graphs
Graph( v ik, ) , i = 1 , m must be disjoint by the Strong Maximum Principle.By elliptic estimates via the weak maximum principle [13, Theorem 3.7], we have for η small enoughand k sufficiently large and a uniform C > so that, max B nη ( v mk, − v k, ) ≤ C max ∂B nη ( v mk, − v k, ) . ULTIPLICITY ONE CONJECTURE 27
This implies max U k ∩ B nη ( u mk ( x ) − u k ( x )) ≤ C max ∂B nη ( u mk ( x ) − u k ( x )) . Hence max U k ∩ B nη ˜ h k ≤ C max ∂B nη ˜ h k , so ϕ is uniformly bounded and hence extends smoothly across y . Part 6 : We now take care the more interesting case: m is an even number . Hence m ≥ . In this case ν k restricted to the bottom ( u k ) and top ( u mk ) sheets point to different side of Σ ∞ , and without loss ofgenerality we may assume that ν k points downward on top sheet, and upward on bottom sheet. Thatmeans: H | Graph( u mk ) ( x ) = − ǫ k h ( x, u mk ( x )) , and H | Graph( u k ) ( x ) = ǫ k h ( x, u k ( x )) , for x ∈ U k . Note that u mk ( x ) − u k ( x ) ≥ , for all x ∈ U k , but it may take zeros in a co-dimension 1 subset by [52, Proposition 3.17].Again by subtracting the above two equations, and using the fact H | Graph( u mk ) − H | Graph( u k ) = L Σ ∞ ( u mk − u k ) + o ( u mk − u k ) , we have(4.4) L Σ ∞ ( u mk − u k ) + o ( u mk − u k ) = − ǫ k · ( h ( x, u k ( x ) + h ( x, u mk ( x )) . Fix a point p ∈ U , and we discuss the renormalization in two cases. Again write h k = u mk − u k . Case 1: lim sup k →∞ h k ( p ) ǫ k = + ∞ . Consider renormalizations ˜ h k ( x ) = h k ( x ) /h k ( p ) . Then by thesame reasoning as Part 4, ˜ h k converges locally smoothly to a nontrivial function ϕ ≥ on Σ ∞ \Y , andsuch that L Σ ∞ ϕ = 0 , outside Y . Case 2: lim sup k →∞ h k ( p ) ǫ k < + ∞ . Consider renormalizations ˜ h k ( x ) = h k ( x ) /ǫ k . Then again bythe same reasoning, ˜ h k converges locally smoothly to a nonnegative ϕ ≥ on Σ ∞ \Y , and such that L Σ ∞ ϕ = − h | Σ ∞ , outside Y . Part 7 : We will follow a slightly different local foliation argument to prove removable singularity for ϕ . We inherit all notations in Part 5. Without loss of generality, we may assume sup M | h | = 1 . Let v k,t , v mk,t : B nη → R , t ∈ [ − η, η ] , be the CMC local foliations associated with − ǫ k and ǫ k respectively, H Graph( v mk,t ) ( x ) = ǫ k , and H Graph( v k,t ) ( x ) = − ǫ k , x ∈ B nη ( y ) , and v ik,t ( x ) = u ik ( x ) + t, i = 1 , m, x ∈ ∂B nη ( y ) . By the same reasoning as Part 5 using the Strong Maximum Principle for varifolds by White [43],we get max U k ∩ B nη ( u mk ( x ) − u k ( x )) ≤ max B nη ( v mk, ( x ) − v k, ( x )) . Note that slightly different with Part 5, we have L Σ ∞ ( v mk, − v k, ) + o ( v mk, − v k, ) = 2 ǫ k . By [13, Theorem 3.7], we have for η small enough, k large enough and for some uniform C > U k ∩ B nη ( u mk ( x ) − u k ( x )) ≤ C (cid:0) max ∂B nη ( u mk ( x ) − u k ( x )) + ǫ k (cid:1) . Then for both Case 1 and Case 2, this implies that ϕ is uniformly bounded and hence extends smoothlyacross Y .Note that if we flip the orientations of the top and bottom sheets, then in Case 2 the limit of renor-malizations of heights will converge to a solution of L Σ ∞ ϕ = 2 h | Σ ∞ , where ϕ ≥ . Note that in theprevious case, we can just flip the sign of ϕ , and obtain L Σ ∞ ϕ = 2 h | Σ ∞ , where ϕ ≤ . Part 8 : Now we briefly record the case when Σ ∞ is only one-sided. Then the convergence of Σ k must have multiplicity at least 2; otherwise the convergence will be smooth by the Allard regularitytheorem [1], and hence all Σ k will be 1-sided for k sufficiently large, which is a contradiction. Denote π : ˜Σ ∞ → Σ ∞ as the 2-sided double cover of Σ ∞ , and τ : ˜Σ ∞ → ˜Σ ∞ the deck transformation map.By the same argument for the 2-sided case applied to the double cover ˜Σ ∞ , we can either construct anon-trivial Jacobi field ϕ on ˜Σ ∞ with ϕ ◦ τ = ϕ and L ˜Σ ∞ ϕ = 0; or a smooth function ϕ on ˜Σ ∞ with ϕ ◦ τ = ϕ , such that ϕ does not change sign, and L ˜Σ ∞ ϕ = 2 h | Σ ∞ ◦ π. By [44], the first case cannot happen in a bumpy metric.Summarizing the discussion, we proved that if g is bumpy, then each connected 2-sided component Σ o of Σ ∞ with multiplicity bigger than one must carry a smooth solution ϕ to the equation(4.5) L Σ o ϕ = 2 h | Σ o ; and the double cover ˜Σ u of each 1-sided component Σ u of Σ ∞ must carry a smooth solution ϕ (4.6) L ˜Σ u ϕ = 2 h | Σ u ◦ π, and ϕ ◦ τ = ϕ. Moreover, in both cases ϕ does not change sign. Part 9 : We will show that for a nicely chosen h ∈ S ( g ) , the (unique) solutions to (4.5) and (4.6) mustchange sign. Thus there is no 1-sided component, and the multiplicity for 2-sided component must beone. Lemma 4.2 (Key Lemma) . Assume that g is bumpy. Given L > and k ∈ N , there exists h ∈ S ( g ) ,such that if Σ is a smooth, connected, closed, embedded minimal hypersurface with Area(Σ) ≤ L , and index(Σ) ≤ k, then the solution of (4.5) (when Σ is 2-sided) or (4.6) (when Σ is 1-sided) must change sign.Proof. As g is bumpy, by the compactness analysis of Sharp [35], there are only finitely many such Σ with Area(Σ) ≤ L and index(Σ) ≤ k , and we can denote them as { Σ , · · · , Σ L } . If Σ i is 1-sided,we use π i : ˜Σ i → Σ i to denote the 2-sided double cover, and τ i : ˜Σ i → ˜Σ i to denote the decktransformation map.On each Σ i , we can choose two disjoint open subsets U + i and U − i ⊂ Σ i , so that the collectionof subsets { U ± i } i =1 , ··· ,L are pairwise disjoint. Moreover, by possibly changing U ± i , we can make ULTIPLICITY ONE CONJECTURE 29 sure that the pre-image π − i ( U + i ) , π − i ( U − i ) are diffeomorphic to two disjoint copies of U + i , U − i respectively. In that case, we will denote the two copies as ˜ U + i, , ˜ U + i, , and ˜ U − i, , ˜ U − i, . That is π − i ( U + i ) = ˜ U + i, ∪ ˜ U + i, , and π − i ( U − i ) = ˜ U − i, ∪ ˜ U − i, . For each i ∈ { , · · · , L } such that Σ i is 2-sided, we can choose an arbitrary pair of nontrivialsmooth functions f + i ∈ C ∞ c ( U + i ) , f − ∈ C ∞ c ( U − i ) such that f + i ≥ , and f + i ( p + i ) > at some p + i ∈ U + i , and f − i ≤ , and f − i ( p − i ) < at some p − i ∈ U i . Let h + i ∈ C ∞ c ( U + i ) and h − i ∈ C ∞ c ( U − i ) be defined by: h + i = L Σ i f + i , h − i = L Σ i f − i . If Σ i is 1-sided, we choose ˜ f ± i, ∈ C ∞ c ( ˜ U ± i, ) , ˜ f ± i, ∈ C ∞ c ( ˜ U ± i, ) in the same way, and we can makesure they are the same under deck transformation: ˜ f ± i, ◦ τ = ˜ f ± i, . In particular, ˜ f + i, ≥ , and ˜ f + i, > somewhere in ˜ U + i, , and ˜ f − i, ≤ , and ˜ f − i, < somewhere in ˜ U − i, . Then we define h ± i, , h ± i, in the same manner, so obviously h ± i, ◦ τ = h ± i, , and they pass to twofunctions h + i ∈ C ∞ c ( U + i ) , and h − i ∈ C ∞ c ( U − i ) . We can extend each h ± i to a function defined on Σ i by letting it be zero outside U ± i . Using the factthat the set of smooth functions S ( g ) is open and dense in C ∞ ( M ) , we can choose a h ∈ S ( g ) so that h is as close to h ± i as we want in any C k,α -norm when restricted to Σ i . We may need to flip the sign of h to make R M h ≥ , but the following argument proceeds the sameway. Since all { Σ i : i = 1 , · · · , L } are non-degenerate (the Jacobi operator is an isomorphism), weknow that if L Σ i ϕ = 2 h | Σ i when Σ i is 2-sided, or L ˜Σ i ϕ = 2 h | Σ i ◦ π i when Σ is 1-sided , then ϕ is as close to f ± i or ˜ f ± i,j ( j = 1 , ) as we want in C k +2 ,α -norm when restricted to Σ i or ˜Σ i . Then ϕ must change sign, and this is what we want to prove. (cid:3) Note that by Part 2, all connected components of a min-max minimal hypersurface must satisfy thearea and index bound in Lemma 4.2. So we finish the proof of the theorem. (cid:3)
Remark . Indeed, we can obtain more information. Since Σ ∞ has multiplicity one, the Allardregularity theorem [1] implies that the convergence Σ k → Σ ∞ is smooth everywhere, and hence Σ k isproperly embedded for k large. Remark . Without assuming that g is bumpy, our proof says that if the multiplicity of a 2-sidedcomponent is greater than 2, or if the multiplicity for a 1-sided component is greater than 1, then thereexists a nontrivial, nonnegative Jacobi field. Let us point out the necessary details for 2-sided case, andthe 1-sided case follows the same way. Indeed, we only need to focus on the case when the multiplicity m is even and m ≥ ; and moreover, we can focus on Case 2 in Part 6. Using notations in Part 6 and7, we consider the height difference between the two pairs ( u k , u m − k ) and ( u k , u mk ) , h ak = u m − k − u k , h bk = u mk − u k . Then both h ak , h bk > and satisfy equations of type (4.3) since the graphs of the two pairs have outernormals pointing to the same side. Consider the renormalizations: ˜ h ak = h ak /ǫ k and ˜ h bk = h bk /ǫ k . Then ˜ h ak , ˜ h bk ≤ ˜ h k , and ˜ h ak + ˜ h bk ≥ ˜ h k . Note that the limit of ˜ h k can not be identically zero, as then h | Σ ∞ ≡ , violating the assumption h ∈ S ( g ) . Then the above two inequalities and standard elliptic estimates imply that at least one limitof the two sequences { ˜ h ak } k ∈ N and { ˜ h bk } k ∈ N must be a smooth, nontrivial, nonnegative Jacobi field.Part of the proof of the theorem can be summarized as the following multiplicity one convergenceresult, which we believe has its independent interests. Theorem 4.5 (Multiplicity one convergence) . Let ( M n +1 , g ) be a closed manifold of dimension ≤ ( n + 1) ≤ with a bumpy metric g . Given L > , I ∈ N , then there exists a smooth function h : M → R , h ∈ S ( g ) , such that:Let { Σ k } k ∈ N be a sequence of smooth, closed, almost embedded hypersurfaces, and { ǫ k } k ∈ N → ,such that • Σ k is the boundary of some open set Ω k , and the mean curvature of Σ k with respect to theouter normal of Ω k is prescribed by ǫ k h ; • Area(Σ k ) ≤ L , and index(Σ k ) ≤ I .Then up to a subsequence { Σ k } k ∈ N converges smoothly to a smooth, closed, embedded, two-sided,minimal hypersurface Σ ∞ with multiplicity one.
5. A
PPLICATION TO VOLUME SPECTRUM
In this part, we will show how to apply the result in Section 4 to study volume spectrum introducedby Gromov, Guth, and Marques-Neves. In particular, we will prove that in a bumpy metric, the volumespectrum can be realized by the area of min-max minimal hypersurfaces produced by Theorem 4.1.To do this, we will carefully pick a sequence of sweepouts of mod 2 cycles, and open the parameterspace so as to produce sweepouts of boundaries of Caccioppoli sets, whose relative homotopy classessatisfy (4.1). As the space of Caccioppli sets forms a double cover of the space of mod 2 cycles, theparameter-space-opening process is achieved by lifting to the double cover.We first recall the definition of volume spectrum following [28, Section 4]. Let ( M n +1 , g ) be aclosed Riemannian manifold. Let X be a cubical subcomplex of I m = [0 , m for some m ∈ N .Given k ∈ N , a continuous map Φ : X → Z n ( M, Z ) is a k -sweepout if Φ ∗ (¯ λ k ) = 0 ∈ H k ( X, Z ) , where ¯ λ ∈ H ( Z n ( M, Z ) , Z ) = Z is the generator. Φ is said to be admissible if it has no concen-tration of mass. Denote by P k as the set of all admissible k -sweepouts. Then ULTIPLICITY ONE CONJECTURE 31
Definition 5.1.
The k -width of ( M, g ) is ω k ( M, g ) = inf Φ ∈P k sup { M (Φ( x )) : x ∈ dmn(Φ) } , where dmn(Φ) is the domain of Φ .It was proved in [28, Theorem 5.1 and 8.1] that there exists some constant C = C ( M, g ) , such that C k n +1 ≤ ω k ( M, g ) ≤ Ck n +1 . Assume from now on that the dimension satisfies ≤ ( n + 1) ≤ . It was later observed byMarques-Neves in [27] that one can restrict to a subclass of P k in the definition of ω k ( M, g ) . Inparticular, let ˜ P k denote those elements Φ ∈ P k which is continuous under the F -topology, and whosedomain X = dmn(Φ) has dimension k (and is identical to its k -skeleton). Then ω k ( M, g ) = inf Φ ∈ ˜ P k sup { M (Φ( x )) : x ∈ dmn(Φ) } . They also proved in [27] that for each k ∈ N there exists a disjoint collection of smooth, connected,closed, embedded minimal hypersurfaces { Σ ki : i = 1 , · · · , l k } with integer multiplicities { m ki : i =1 , · · · , l k } ⊂ N , such that ω k ( M, g ) = l k X i =1 m ki · Area(Σ ki ) , and l k X i =1 index(Σ ki ) ≤ k. Now we are going to state and prove our main theorem.
Theorem 5.2 (Theorem A) . If g is a bumpy metric and ≤ ( n + 1) ≤ , then for each k ∈ N , thereexists a disjoint collection of smooth, connected, closed, embedded, two-sided minimal hypersurfaces { Σ ki : i = 1 , · · · , l k } , such that ω k ( M, g ) = l k X i =1 Area(Σ ki ) , and l k X i =1 index(Σ ki ) ≤ k. That is to say, the min-max minimal hypersurfaces are all two-sided and have multiplicity one.Proof. If g is bumpy, then there are only finitely many closed, embedded, minimal hypersurfaces with Area ≤ Λ and index ≤ I for given Λ > , I ∈ N by Sharp’s result [35]. Using the Morse index upperbound estimates for min-max theory by Marques-Neves [27], we have Lemma 5.3.
Suppose g is bumpy, then for each k ∈ N , there exists a k -dimensional cubical complex X k and a map Φ ,k : X k → Z n ( M, F , Z ) continuous in the F -topology with Φ ,k ∈ ˜ P k , such that L (Π k ) = ω k ( M, g ) , where Π k = Π(Φ ,k ) is the class of all maps Φ : X k → Z n ( M, F , Z ) continuous in the F -topologythat are homotopic to Φ ,k in flat topology.Proof. From definition we know that ω k ( M, g ) = inf { L (Π(Φ)) , Φ ∈ ˜ P k } . By area and index upper bounds and the finiteness result, the infimum is achieved. (cid:3)
Now we fix k ∈ N and omit the sub-index k in the following. Take Π = [Φ : X → Z n ( M, F , Z )] with L (Π) = ω k . The following result is an outcome of the proof of [27, Theorem 6.1]. Lemma 5.4.
Suppose g is bumpy. Then there exists a pull-tight (see [27, 3.7] ) min-max sequence { Φ i } i ∈ N of Π such that if Σ ∈ C ( { Φ i } i ∈ N ) has support a smooth, closed, embedded minimal hyper-surface, then k Σ k ( M ) = ω k ( M, g ) , and index( support of Σ) ≤ k. We proceed the proof by the following four steps.
Step 1 : In this and the next step, we show how to find another min-max sequence, still denoted as { Φ i } i ∈ N , such that for i sufficiently large, either | Φ i ( x ) | is close to a regular min-max minimal hyper-surface, or the mass M (Φ i ( x )) is strictly less than ω k ( M, g ) . We recall the following observation by [28, Claim 6.2]. Let S be the set of all stationary integralvarifolds with Area ≤ ω k whose support is a smooth closed embedded minimal hypersurface with index( support ) ≤ k . Consider the set T of all mod 2 flat cycles T ∈ Z n ( M, Z ) with M ( T ) ≤ ω k and such that either T = 0 or the support of T is a smooth closed embedded minimal hypersurfacewith index ≤ k . By the bumpy assumption, both sets S and T are finite. Moreover, Lemma 5.5 (Claim 6.2 in [28]) . For every ¯ ǫ > , there exists ǫ > such that T ∈ Z n ( M, Z ) with F ( | T | , S ) ≤ ǫ = ⇒ F ( T, T ) < ¯ ǫ. We also need another observation by [28, Corollary 3.6]. Denote S by the unit circle. Lemma 5.6 (Corollary 3.6 in [28]) . If ¯ ǫ is sufficiently small, depending on T , then every map Φ : S → Z n ( M, Z ) with Φ( S ) ⊂ B F ¯ ǫ ( T ) = { T ∈ Z n ( M, Z ) : F ( T, T ) < ¯ ǫ } is homotopically trivial. let { Φ i } i ∈ N be chosen as in Lemma 5.4. We choose ¯ ǫ as Lemma 5.6 and ǫ by Lemma 5.5. Take asequence { k i } i ∈ N → ∞ , such that sup { F (Φ i ( x ) , Φ i ( y )) : α ∈ X ( k i ) , x, y ∈ α } ≤ ǫ/ . Consider Z i to be the cubical subcomplex of X ( k i ) consisting of all cells α ∈ X ( k i ) so that F ( | Φ i ( x ) | , S ) ≥ ǫ, for every vertex x in α .Hence F ( | Φ i ( x ) | , S ) ≥ ǫ/ for all x ∈ Z i .Consider this sub-coordinating sequence { Φ i | Z i } i ∈ N . L ( { Φ i | Z i } ) and C ( { Φ i | Z i } ) are defined inthe same way as in Section 1.1 with A h replaced by M . Lemma 5.7.
We have the following dichotomy: • no element V ∈ C ( { Φ i | Z i } i ∈ N ) is Z -almost minimizing in small annuli (see [28, 2.10] ), • or (5.1) L ( { Φ i | Z i } i ∈ N ) < L (Π) = ω k . Proof.
Suppose that (5.1) does not hold, then L ( { Φ i | Z i } i ∈ N ) = L (Π) . As { Φ i } i ∈ N is pull-tight, weknow that every V ∈ C ( { Φ i | Z i } i ∈ N ) is stationary. If V is also Z -almost minimizing in small annuli,then V is regular by the regularity of Pitts [31, Theorem 7.11] and Schoen-Simon [33, Theorem 4];(see also [28, Theorem 2.11] for the adaption to Z -coefficients). By Lemma 5.4, V ∈ S , which is acontradiction. (cid:3) ULTIPLICITY ONE CONJECTURE 33
Let Y i = X \ Z i . It then follows that(5.2) F ( | Φ i ( x ) | , S ) ≤ ǫ, for all x ∈ Y i . We also denote B i = Y i ∩ Z i . In fact, B i is the topological boundary of Y i and Z i . For later purpose,we also consider the set B i = the union of all cells α ∈ Z i such that α ∩ B i = ∅ . B i can be thought of the “thickening” of B i inside Z i .Let λ = Φ ∗ i (¯ λ ) ∈ H ( X, Z ) . Consider the inclusion maps i : Y i → X and i : Z i → X . It thenfollows from (5.2), Lemma 5.5 and Lemma 5.6 that i ∗ ( λ ) = 0 ∈ H ( Y i , Z ) . Then by [28, Claim 6.3], (Φ i ) | Z i is a ( k − -sweepout, i.e. i ∗ ( λ k − ) = 0 ∈ H k − ( Z i , Z ) . Now we let Y ′ i = Y i ∪ B i and Z ′ i = Z i \ B i , and i ′ i : Y ′ i → X and i ′ : Z ′ i → X be the inclusion maps.Note that (5.2) is satisfied with Y i , ǫ replaced by Y ′ i , ǫ respectively, so by similar reasoning we have ( i ′ ) ∗ ( λ ) = 0 ∈ H ( Y ′ i , Z ) , and ( i ′ ) ∗ ( λ k − ) = 0 ∈ H k − ( Z ′ i , Z ) . Step 2 : The strategy is to follow the idea in the proof of Theorem 1.7 and apply [28, Theorem 2.13] (seealso Theorem 1.16) to deform { Φ i } i ∈ N so as to decrease L ( { (Φ i ) | Z i } i ∈ N ) and make (5.1) be satisfied. If (5.1) holds true, then we are done for this step. So let us assume that(5.3) L ( { Φ i | Z i } i ∈ N ) = L (Π) = ω k . By Lemma 5.7 and our assumption (5.3), we know that no element V ∈ C ( { Φ i | Z i } i ∈ N ) is Z -almost minimizing in small annuli.Since Φ i : X → Z n ( M, F , Z ) has no concentration of mass as it is continuous in F -topology, wecan apply [28, Theorem 3.9] (the counterpart of Theorem 1.11 for maps to Z n ( M, Z ) ) to produce asequence of maps φ ji : X ( k i + k ji ) → Z n ( M, Z ) , with k ji ∈ N and k ji < k j +1 i for all j ∈ N and a sequence of positive { δ ji } j ∈ N → , such that(i) the fineness f ( φ ji ) ≤ δ ji ;(ii) sup {F ( φ ji ( x ) − Φ i ( x )) : x ∈ X ( k i + k ji ) } ≤ δ ji ; (iii) for some sequence l ji → ∞ with l ji < k ji M ( φ ji ( x )) ≤ sup { M (Φ i ( y )) : x, y ∈ α, for some α ∈ X ( k i + l ji ) } + δ ji . As Φ i is continuous in F -topology, we get from property (iii) that for all x ∈ X ( k i + k ji ) , M ( φ ji ( x )) ≤ M (Φ i ( x )) + η ji with η ji → as j → ∞ . Applying [26, Lemma 4.1] with S = Φ i ( X ) , we get by (ii) that(iv) sup { F ( φ ji ( x ) , Φ i ( x )) : x ∈ X ( k i + k ji ) } → , as j → ∞ . We can choose j ( i ) → ∞ as i → ∞ (then k j ( i ) i → ∞ ) such that ϕ i = φ j ( i ) i : X ( k i + k j ( i ) i ) →Z n ( M, Z ) satisfies: • sup { F ( ϕ i ( x ) , Φ i ( x )) : x ∈ X ( k i + k j ( i ) i ) } ≤ a i with a i → as i → ∞ ; • sup { F (Φ i ( x ) , Φ i ( y )) : x, y ∈ α, α ∈ X ( k i + k j ( i ) i ) } ≤ a i ; • the fineness f ( ϕ i ) → as i → ∞ ; • the Almgren extension Φ j ( i ) i : X → Z n ( M, M , Z ) (see [28, 3.10] for definition, and itis continuous in the M -topology) is homotopic to Φ i in the flat topology (by [28, Corollary3.12]), and sup { F (Φ j ( i ) i ( x ) , Φ i ( x )) : x ∈ X } → as i → ∞ (by [28, 3.10]).If we let S = { ϕ i } i ∈ N be a discrete sweepout, then we have L ( S ) = L ( { Φ i } i ∈ N ) and C ( S ) = C ( { Φ i } i ∈ N ) . Moreover, consider the restrictions of ϕ i to Z i ( k j ( i ) i ) : S Z = { ϕ i : Z i ( k j ( i ) i ) → Z n ( M, Z ) } . Similarly we have L ( S Z ) = L ( { Φ i | Z i } i ∈ N ) = L (Π) , and C ( S Z ) = C ( { Φ i | Z i } i ∈ N ) . As no V ∈ C ( S Z ) is Z -almost minimizing in small annuli, by [28, Theorem 2.13] (which is areformulation of Almgren-Pitts combinatorial argument [31, Theorem 4.10]), we can find a sequence ˜ S Z = { ˜ ϕ i } of maps: ˜ ϕ i : Z i ( k j ( i ) i + l i ) → Z n ( M, Z ) , and a sequence of homotopies ψ i : I ( l i ) × Z i ( k j ( i ) i + l i ) → Z n ( M, Z ) , such that • ψ i ([0] , x ) = ϕ i ◦ n ( k j ( i ) i + l i , k j ( i ) i )( x ) and ψ i ([1] , x ) = ˜ ϕ i ( x ) ; • the fineness of ψ i tends to zero as i → ∞ ; • lim sup i →∞ sup { M ( ψ i ( t, x )) : ( t, x ) ∈ I ( l i ) × Z i ( k j ( i ) i + l i ) } = L ( S Z ); (note that this property was not explicitly listed in [28, Theorem 2.13], but it follows from theconstruction in [31, Theorem 4.10]).) • L ( ˜ S Z ) < L ( S Z ) .Now we construct a new sequence S ∗ = { ϕ ∗ i } i ∈ N with ϕ ∗ i : X ( k i + k j ( i ) i + l i ) → Z n ( M, Z ) , defined as • ϕ ∗ i ( x ) = ϕ i ◦ n ( k j ( i ) i + l i , k j ( i ) i )( x ) , when x ∈ Y i ( k j ( i ) i + l i ) ; • ϕ ∗ i ( x ) = ψ i ( t ( x ) , x ) , where x ∈ B i ( l i ) and t ( x ) = min { − l i · d ( x, B i ∩ Y i ) , } ∈ I ( l i ) ;(here d is the distance function restricted to B i ( l i ) ; see Appendix A); • ϕ ∗ i ( x ) = ˜ ϕ i ( x ) , when x ∈ Z ′ i ( k j ( i ) i + l i ) ; (note that t ( x ) ≥ when x ∈ Z ′ i ∩ B i ).By the construction, we see that • ϕ ∗ i is homotopic to ϕ i with fineness tending to zero as i → ∞ ; • L ( S ∗ ) = L (Π) ; • lim sup i →∞ sup { M ( ϕ ∗ i ( x )) : x ∈ Z ′ i ( k j ( i ) i + l i ) } ≤ L ( ˜ S Z ) < L (Π) . ULTIPLICITY ONE CONJECTURE 35
Consider the Almgren’s extension of ϕ ∗ i : Φ ∗ i : X → Z n ( M, M , Z ) . Then(a) Φ ∗ i is homotopic to Φ j ( i ) i and hence to Φ i in the flat topology by [28, 3.11]; and by [28, 3.10](b) sup { F (Φ ∗ i ( x ) , Φ i ( x )) : x ∈ Y i } → ;(c) L ( { Φ ∗ i } ) = L ( S ∗ ) = L (Π) ;(d) lim sup i →∞ sup { M (Φ ∗ i ( x )) : x ∈ Z ′ i } ≤ L ( ˜ S Z ) < L (Π) . By summarizing what we have done (and abusing the notation Y i = Y ′ i and Z i = Z ′ i ), we producedanother min-max sequence { Φ ∗ i } i ∈ N ⊂ Π such that(1) X can be decomposed to Y i and Z i with Z i = X \ Y i , and for i large enough, i ∗ ( λ ) = 0 ∈ H ( Y i , Z ) , and i ∗ ( λ k − ) = 0 ∈ H k − ( Z i , Z ) . (2) L ( { Φ ∗ i } ) = L ( { Φ i } ) = L (Π) ;(3) lim sup i →∞ sup { M (Φ ∗ i ( x )) : x ∈ Z i } < L (Π) . Note that both Y i and Z i are nonempty for i large enough by (1)(3). Step 3 : Now we want to produce sweepouts in C ( M ) by lifting to the double cover ∂ : C ( M ) →Z n ( M, Z ) so as to produce sweepouts satisfying the assumption of Theorem 4.1. We abuse notation and still write { Φ ∗ i } as { Φ i } . Since (Φ i ) ∗ (¯ λ ) = 0 ∈ H ( X, Z ) = Z , thereexist a double cover π : ˜ X → X with deck transformation map τ : ˜ X → ˜ X , and the lifting maps: ˜Φ i : ˜ X → ( C ( M ) , F ) , satisfying ∂ ˜Φ i = Φ i ◦ π . Indeed, the cohomological condition implies that the induced maps (Φ i ) ∗ : π ( X ) → π ( Z n ( M, Z )) = Z are surjective; see [28, Definition 4.1 (i)]. So the kernel of (Φ i ) ∗ isa subgroup of π ( X ) with index 2. Then the existence of such liftings follows from [20, Proposition1.36 and Proposition 1.33].Note that i ∗ λ = 0 ∈ H ( Y i , Z ) , so the pre-image of Y i is disconnected, and is a disjoint union oftwo copies of Y i : ˜ Y i = π − ( Y i ) = Y + i ∪ Y − i , where both Y + i and Y − i are homeomorphic to Y i . In fact, the cohomological condition implies thatevery closed curve γ : S → Y i lies in the kernel of (Φ i ) ∗ , so the lift ˜ γ of γ to ˜ X is still a closed curve.This means that ˜ Y i is disconnected as we want.Denote ˜ Z i , ˜ B i and ˜ B i the pre-images of Z i , B i , B i under π respectively. Then ˜ B i = B + i ∪ B − i isalso a disjoint union of two copies of B i . Lemma 5.8.
For i large enough, if ˜Π i is the ( ˜ X, ˜ Z i ) -homotopy class associated with ˜Φ i , then we have L ( ˜Π i ) ≥ L (Π) > max x ∈ ˜ Z i M ( ∂ ˜Φ i ( x )) . Proof.
Fix i large, so that sup x ∈ Z i M (Φ i ( x )) < L (Π) , and we will omit the sub-index in the following proof. If the conclusion were not true, then we can find a sequence of maps { ˜Ψ j : ˜ X → ( C ( M ) , F ) } j ∈ N ⊂ ˜Π , such that lim sup j →∞ sup { M ( ∂ ˜Ψ j ( x )) : x ∈ X } < L (Π) , and homotopy maps { H j : [0 , × ˜ X → C ( M ) } which are continuous in the flat topology, H j (0 , · ) =˜Ψ j , H j (1 , · ) = ˜Φ , and lim sup j →∞ sup { F ( H j ( t, x ) , ˜Φ( x )) : t ∈ [0 , , x ∈ ˜ Z } = 0 . We construct a new sequence of maps { ˜Ψ ∗ j } j ∈ N defined as • ˜Ψ ∗ j ( x ) = ˜Ψ j ( x ) , if x ∈ Y + , and ˜Ψ ∗ j ( x ) = ˜Ψ j ◦ τ ( x ) , if x ∈ Y − ; • ˜Ψ ∗ j ( x ) = H j ( t ( x ) , x ) , where t ( x ) = min { dist( x, B + ∩ Y + ) , } if x ∈ B + , and ˜Ψ ∗ j ( x ) = H j ◦ τ ( x ) , if x ∈ B − ; (here dist is the distance function by viewing B as a cube complex insome I ( m, l ) ); • ˜Ψ ∗ j ( x ) = ˜Φ( x ) , if x ∈ ˜ Z ′ ; (note that t ( x ) ≥ for x ∈ ˜ Z ′ ∩ ( B + ∩ B − ) ).Note that though ˜Ψ ∗ j themselves may not be continuous as maps to C ( M ) , Ψ ∗ j can be passed to quotientas continuous maps from X to Z n ( M, Z ) . This is essentially where we used the structures of ˜ Y and ˜ B , that is, ( Y + , Y − ) and ( B + , B − ) are pairwise disjoint.Denote the quotient maps of { ˜Ψ ∗ j } j ∈ N by { Ψ ∗ j = ∂ ◦ ˜Ψ ∗ j : X → Z n ( M, Z ) } j ∈ N . We have • Ψ ∗ j is homotopic to Φ in the flat topology; • lim sup j →∞ sup { M (Ψ ∗ j ( x )) : x ∈ X } < L (Π) = ω k ( M, g ) (by the three above inequalities).This will lead to a contradiction with the definition of k -width once we prove that Ψ ∗ j is an admis-sible k -sweepout when j is sufficiently large. Indeed, the only thing left is to show that Ψ ∗ j has noconcentration of mass. This follows from the third inequality above. So we finish the proof. (cid:3) Step 4 : We are ready to finish the proof of Theorem 5.2.
For i large enough as in Lemma 5.8, Theorem 4.1 applied to ˜Π i gives a disjoint collection of smooth,connected, closed, embedded, 2-sided, minimal hypersurfaces Σ i = ∪ N i j =1 Σ i,j , such that L ( ˜Π i ) = N i X j =1 Area(Σ i,j ) , and index(Σ i ) ≤ k. Note also that L ( ˜Π i ) ≤ L (Φ i ) → L (Π) = ω k . Counting the fact that there are only finitely manysmooth, closed, embedded minimal hypersurfaces with Area ≤ ω k +1 and index ≤ k , for i sufficientlylarge we have L ( ˜Π i ) = L ( ˜Π i +1 ) = · · · = ω k . Hence we finish the proof of Theorem 5.2. (cid:3)
Remark . By the course of the above proof, in a bumpy metric, the min-max minimal hypersurfacesassociated with any homotopically nontrivial sweepouts of mod-2 cycles are always two-sided andhave multiplicity one. In fact, if
Φ : X → Z n ( M, Z ) is homotopically nontrivial, then the inducedmap Φ ∗ : π ( X ) → π ( Z n ( M, Z )) = Z must be surjective. Otherwise by [20, Proposition 1.33] Φ can be lifted to a map ˜Φ : X → C ( M ) which is then homotopically trivial as C ( M ) is contractible. ULTIPLICITY ONE CONJECTURE 37
With these topological information, the above proof works the same way and implies the two-sidednessand multiplicity one for min-max minimal hypersurfaces associated with
Π(Φ) .A PPENDIX
A. C
UBICAL COMPLEX STRUCTURES
Here we recall several cubical complex structures in [28, 2.1].For each k ∈ N , I (1 , k ) denotes the cubical complex on the unit interval I = [0 , whose 1-cellsand 0-cells (which are also called vertices) are, respectively, [0 , − k ] , [3 − k , · − k ] , · · · , [1 − − k , and [0] , [3 − k ] , · · · , [1 − − k ] , [1] . We then denote by I ( m, k ) the cell complex on I m : I ( m, k ) = I (1 , k ) ⊗ · · · ⊗ I (1 , k ) m times . Then α = α ⊗ · · · ⊗ α m is a q -cell of I ( m, k ) if and only if α i is a cell of I (1 , k ) for each i , and P mi =1 dim( α i ) = q . We often identify a q -cell α with its support α × · · · × α m ⊂ I m . The distancefunction d on I ( m, k ) is defined as d ( x, y ) = 3 k P ki =1 | x i − y i | , x, y ∈ I ( m, k ) , [31, 4.1(1)(e)].Let X ⊂ I m be a cubical subcomplex. The cubical complex X ( k ) is the union of all cells of I ( m, k ) whose support is contained in some cell of X . We use the notation X ( k ) q to denote the setof all q -cells in X ( k ) , and particularly X ( k ) to denote the set of vertices in X ( k ) . Two vertices x, y ∈ X ( k ) are adjacent if they belong to a common cell in X ( k ) .Let Y ⊂ I ( m, k ) be a cubical subcomplex. Similarly, the cubical complex Y ( l ) is the union of allcells of I ( m, k + l ) whose support is contained in some cell of Y . Y ( k ) q is defined in the same way.Given k, l ∈ N , we define n ( k, l ) : X ( k ) → X ( l ) so that n ( i, j )( x ) is the element in X ( l ) thatis closest to x ; (see [31, page 141]).A PPENDIX
B. R
EMOVING SINGULARITY FOR WEAKLY STABLE
PMCWe record the following standard removable singularity result.
Theorem B.1.
Let ( M n +1 , g ) be a closed Riemannian manifold of dimension ≤ ( n + 1) ≤ . Given h ∈ C ∞ ( M ) and Σ ⊂ B ǫ ( p ) \{ p } an almost embedded hypersurface with ∂ Σ ∩ B ǫ ( p ) \{ p } = ∅ ,assume that Σ has prescribing mean curvature h , and Σ is weakly stable in B ǫ ( p ) \{ p } as in Theorem2.6, Part 1 of proof. If Σ represents a varifold of bounded first variation in B ǫ ( p ) , then Σ extendssmoothly across p as an almost embedded hypersurface in B ǫ ( p ) .Proof. Given any sequence of positive λ i → , consider the blowups { µ p,λ i (Σ) ⊂ µ p,λ i ( M ) } , where µ p,λ i ( x ) = x − pλ i . Since Σ has bounded first variation, µ p,λ i (Σ) converges (up to a subsequence) to astationary integral rectifiable cone C in R n +1 = T p M . By weakly stability and Theorem 2.5 (whichworks well for the notion of weak stability of Σ ∞ ), the convergence is locally smooth and graphicalaway from the origin, so C is an integer multiple of some embedded minimal hypercone; moreover, C is weakly stable, and hence is stable as an embedded minimal hypersurface away from . Therefore C is an integer multiple of some n -plane P by Simons’s classification [38], i.e. C = m · P where m = Θ n (Σ , p ) . Note that a priori C may not be unique.By the locally smooth and graphical convergence, there exists σ > small enough, such thatfor any < σ ≤ σ , Σ has an m -sheeted, ordered (in the sense of [52, Definition 3.2]), graphicaldecomposition in the annulus A σ/ ,σ ( p ) = B σ ( p ) \ B σ/ ( p ) : Σ ∩ A σ/ ,σ ( p ) = ∪ mi =1 Σ i ( σ ) . Here each Σ i ( σ ) is a graph over A σ/ ,σ ( p ) ∩ P for some n -plane P ⊂ T p M . We can continue each Σ i ( σ ) all the way to B σ ( p ) \{ p } , and we denote the continuation by Σ i .Each Σ i can be extended as a varifold across p with uniformly bounded first variation (since Σ i ⊂ Σ satisfies the area decay estimates, area(Σ i ∩ B σ ( p )) ≤ Cσ n ). We claim that the density satisfies Θ n (Σ i , p ) = 1 for each i . In fact, Θ n (Σ i , p ) ≥ as any blowups of Σ i converges to an n -plane, but m = Θ n (Σ , p ) = P mi =1 Θ n (Σ i , p ) . Now applying the Allard regularity theorem [1] to each Σ i , we getthat Σ i extends as a C ,α hypersurface across p . Higher regularity of Σ i follows from the prescribingmean curvature equation and elliptic regularity. (cid:3) A PPENDIX
C. P
ROOF OF L EMMA B be some compact topological space with ∈ B , and { f ω ∈ C ∞ ( B k ) : ω ∈ B } be a family of smooth functions defined on B k , such that ω → f ω is a continuous map in thesmooth topology on C ∞ ( B k ) . Moreover we assume • f ω has a unique maximum m ( ω ) ∈ B kc / √ , and m (0) = 0 ; • − c Id ≤ D f ω ( u ) ≤ − c Id , for all u ∈ B k and for some c ∈ (0 , .So for each ω ∈ B , we have(C.1) f ω ( m ( ω )) − c | u − m ( ω ) | ≤ f ω ( u ) ≤ f ω ( m ( ω )) − c | u − m ( ω ) | for all u ∈ B k .For each f ω , consider the one-parameter flow { φ ω ( · , t ) : t ≥ } ⊂ Diff( B k ) generated by thevector field u → − (1 − | u | ) ∇ f ω ( u ) , u ∈ B k . For fixed u ∈ B k , the function t → f ω ( φ ω ( u, t )) is non-increasing.The prototype of [27, Lemma 4.5] is the following lemma, and the proof is essentially the same astherein so we omit it. Lemma C.1.
For any δ < , there exists T = T ( δ, B , { f ω } , c ) ≥ such that for any ω ∈ B and v ∈ B k with | v − m ( ω ) | ≥ δ , we have f ω ( φ ω ( v, T )) < f ω (0) − c and | φ ω ( v, T ) | > c . Now we are ready to prove Lemma 3.3. Note that the ball B F ǫ (Ω ) is not compact under the F -topology, so to apply Lemma C.1, we need to introduce a compactification of B F ǫ (Ω ) . Proof of Lemma 3.3.
Given a F -Cauchy sequence { Ω i } ⊂ B F ǫ (Ω ) , we denote ( V ∞ , Ω ∞ ) ∈ V n ( M ) ×C ( M ) as the limit such that V ∞ = lim i →∞ | ∂ Ω i | as varifolds and Ω ∞ = lim i →∞ Ω i as Caccioppolisets. If we define A h ∞ ( v ) = k ( F v ) V ∞ k ( M ) − Z F v (Ω ∞ ) h d H n +1 , for each v ∈ B k , Then A h Ω i converges smoothly to A h ∞ as functions in C ∞ ( B k ) .Now take B as the union of B F ǫ (Ω ) with the limits of the form ( V ∞ , Ω ∞ ) , f Ω = A h Ω and f ( V ∞ , Ω ∞ ) = A h ∞ , then Lemma 3.3 follows from Lemma C.1. (cid:3) ULTIPLICITY ONE CONJECTURE 39 A PPENDIX
D. E
XISTENCE OF LOCAL
PMC
FOLIATIONS
We recall the following classical result of White [41, Appendix and Remark 2]. Note that the A h -functional can be locally expressed as the integration of an elliptic integrand. Proposition D.1.
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