Eternal forced mean curvature flows III - Morse homology. 6 1 0 12th January 2016 2 n Graham Smith a J 3 Instituto de Matem´atica, 1 UFRJ, Av. Athos da Silveira Ramos 149, ] Centro de Tecnologia - Bloco C, G Cidade Universit´aria - Ilha do Fund˜ao, D Caixa Postal 68530, 21941-909, . h Rio de Janeiro, RJ - BRASIL t a m Abstract:We complete thetheoretical framework required for the construction ofa Morse [ homology theory for certain types of forced mean curvature flows. The main result of this 1 paper describes theasymptoticbehaviourofthese flowsastheforcing termtendstoinfinity v 7 in a certain manner. This result allows the Morse homology to be explicitely calculated, 3 and will permit us to show in forthcoming work that, for a large family of smooth positive 4 3 functions, F, defined over a (d+1)-dimensional flat torus, there exist at least 2d+1 distinct, 0 locally strictly convex, Alexandrov-embedded hyperspheres of mean curvature prescribed . 1 at every point by F. 0 6 1 Key Words: Morse homology, mean curvature, forced mean curvature flow : v i X AMS Subject Classification: 58C44 (35A01, 35K59, 53C21, 53C42, 53C44, 53C45, r 57R99, 58B05, 58E05) a 1 2 Eternal forced mean curvature flows III - Morse homology 1 - Introdu tion. 1.1 - Prescribed curvature problems. The problem of constructing hypersurfaces of constant curvature subject to geometric or topological restrictions is a standard one of riemannian geometry. The related problem of constructing hypersurfaces whose curvature is prescribed by some function of the ambient space is essentially complementary, in that a full understanding of the one generally entails a full understanding of the other. However, as they are usually more straightforward, prescribed curvature problems often serve as a better testing ground for the development of new ideas, as will be the case here. Consider, therefore, the problem of constructing immersed hyperspheres of prescribed mean curvature inside a (d+1)-dimensional riemannian manifold, M.* It is useful at this stage to introduce some terminology. Thus, Sd will denote the unit sphere inside Rd+1. A smooth immersion, e : Sd M, will be said to be an Alexandrov-embedding whenever → d+1 d+1 it extends to a smooth immersion, e˜ : B M, where B here denotes the closed unit ball inside Rd+1. ˆ(M) will denote the set→of all smooth Alexandrov-embeddings from E Sd into M, and (M) will denote its quotient under the action by reparametrisation of the group of orienEtation-preserving, smooth diffeomorphisms of Sd. The spaces ˆ(M) and E (M) will be furnished respectively with the topology of Ck-convergence for all k, and its E induced quotient topology, making (M), in particular, into a weakly smooth manifold E (c.f. Appendix A). Finally, an element, [e], of (M) will be referred to simply as an E Alexandrov-embedding, and will be identified, at times with a representative element, e, in ˆ(M), and at times with its image in M. E For the purposes of this paper, we may suppose that the extension of any Alexandrov- embedding is actually unique up to diffeomorphism.† Thus, for a smooth function, F : M R, the functional, : (M) R, will be defined by → F E → ([e]) := Vol([e]) d (F e˜)dVol , (1) e˜ F − d+1 ◦ ZB d+1 where e˜ : B M here denotes the extension of e, and dVol denotes the volume e˜ → form that it induces over the closed, unit ball. The integral on the right-hand side will be viewed as a weighted volume of the interior of e, so that the functional, , will be F loosely referred to as the “area-minus-volume” functional. The critical points of this functional are precisely those Alexandrov embedded hyperspheres whose mean curvatures are prescribed at every point by the function, F; that is, those elements, [e], of (M) E such that H F e = 0, (2) e − ◦ * The convention will be adopted throughout this text that the mean curvature of an immersed hypersurface is equal to the arithmetic mean of its principal curvatures, as opposed to their sum. † Although this would seem to be the case whenever the ambient manifold is not home- omorphic to a sphere, we know of no such a result in the literature. In the present paper, we will only be concerned with locally strictly convex Alexandrov-embeddings in flat tori. Since all such maps lift to embeddings bounding convex sets in Euclidean space (c.f. [2]), uniqueness of the extension readily follows in this case. 1 Eternal forced mean curvature flows III - Morse homology where H here denotes the mean curvature of the Alexandrov-embedding, e. e The fact that they arise as critical points suggests that Alexandrov-embedded hy- perspheres of prescribed mean curvature should be amenable to study by differential- topological techniques, which should then yield information about their number, at least for generic data. This idea, which is far from new, has already been used by numerous au- thors to obtain some quite inspiring results in this, and related, settings (c.f., for example [3], [6], [16], [21], [23] and [24]). Of particular relevance to the current discussion, however, isour own, relativelystraightforward, result, [14], which showsthat, under suitablecircum- stances, the number of critical points of is bounded below by the Euler characteristic of F the ambient space. Although this yields existence in certain cases, we have found it rather unsatisfactory, as, on the one hand, it provides little new information in the case where the Euler characteristic of the ambient space vanishes - for example when the ambient space is 3-dimensional - and, on the other, even when this Euler characteristic does not vanish, the number of solutions it yields generally falls fall short of what Morse theory would lead us to expect. Although Morse homology theory would appear to be the natural approach for obtain- ing the best possible lower bounds on the number of critical points of the functional, , F its development in the current setting has proven to be far from trivial, largely due to the technical challenges involved in making any progress in the theory of mean curvature flows. However, the remarkable non-collapsing theorems recently obtained by Ben Andrews et al. (c.f., for example, [4] and [5]) finally allow the construction of a complete Morse homology theory, at least in the case where the ambient manifold, M, is a flat, (d+1)-dimensional torus, and where certain further restrictions are also imposed on elements of (M). E In [19] and [20], we initiated a programme for the study of the Morse homology of the functional, . The objective of the current paper is partly to improve on these results, but F mainly to complete the final theoretical step required to complete the construction (c.f. Theorem 1.3.5, below). The remaining work involved, which, though long and technical, is essentially formal, will be completed in our forthcoming paper, [10]. There we will prove the following result, which we already state here in order to clearly illustrate our motivations. Theorem 1.1.1, In preparation. If d 2 and if Td+1 is a (d + 1)-dimensional torus, then, for generic, smooth functions, ≥ F : Td+1 ]0, [, such that → ∞ Sup D2F(x)(ξ,ξ) < (3 2√2) Inf F(x)3, x∈Td+1,kξk=1 − x∈Td+1 (cid:12) (cid:12) (cid:12) (cid:12) there exist at least 2d+1 distinct Alexandrov-embeddings, [e] (Td+1), of mean curvature ∈ E prescribed at every point by F. Remark: Here a (d + 1)-dimensional torus is taken to be any quotient of Rd+1 by a cocompact lattice. Remark: Intriguingly, Theorem 1.3.1, below, would suggest that the Morse homology itself exhibits some sort of bifurcation behaviour as the forcing term moves beyond being 2 Eternal forced mean curvature flows III - Morse homology subcritical. However, adeeper understanding ofthisliesfarbeyondthescopeofthecurrent paper. Remark: An analogous result should also hold, with suitably modified conditions on F, when the ambient space is 2-dimensional, and also when it is a compact, hyperbolic manifold. 1.2 - Morse homology. In the infinite-dimensional setting, Morse homology alge- braically encodes the relationship between the moduli spaces of solutions of certain non- linear parabolic operators, and the moduli spaces of solutions of those elliptic operators which correspond to their stationary states. However, in order to gain some intuition, it is worthwhile to first review the finite-dimensional theory (c.f. [17] for a complete and thorough exposition). Thus, a smooth function, f : M R, will be said to be of Morse → type whenever every one of its critical points is non-degenerate. It will then be said to be of Morse-Smale type whenever, in addition, every one of its complete gradient flows is non-degenerate in the following sense. Recall that a complete gradient flow, γ : R M, → of f is, by definition, a zero of the non-linear differential operator γ D γ + f(γ(t)). (3) t 7→ ∇ The linearisation of this operator about γ, which maps sections of the pull-back bundle, γ∗TM, to other sections of the same bundle, is given by ξ ξ +Hess(f)(γ(t))ξ. (4) 7→ ∇∂t The Morse property of f ensures that this operator is always of Fredholm type, and the function, f, will then be of Morse-Smale type whenever this operator is surjective for all γ. There is no shortage of functions with this property. Indeed, standard transversality results show that the set of all such functions is generic, that is, of the second category in the sense of Baire. This means that it contains the intersection of a countable family of open, dense subsets of C∞(M), so that, in particular, by the Baire category theorem, it is dense. By the Morse property of f, every one of its critical points is isolated, so that the set, Z, of all its critical points is finite. In particular, for all k, the subset, Z , of Z, defined k to be the set of critical points of Morse index k, is also finite, where the Morse index of a critical point is here defined to be the sum of the multiplicities of all strictly negative eigenvalues of Hess(f) at that point. For all k, the k’th order chain group of the Morse homology of (M,f) will then be defined by C (M,f) := Z [Z ]. k 2 k Equivalently, C (M,f) will be the Z -module of all Z -valued functions over Z . In par- k 2 2 k ticular, the sum of the dimensions of the chain groups is equal to the cardinality of the solution set, Z. We will denote by W the space of all complete gradient flows of f, furnished with the topology of Ck -convergence for all k. By compactness and the Morse property of f, every loc 3 Eternal forced mean curvature flows III - Morse homology complete gradient flow, γ, has well-defined end-points, p and q, in the sense that Lim γ(t) = p, and t→−∞ Lim γ(t) = q. t→+∞ Furthermore, these end-points are always critical points of f, so that W = W , p,q (p,q)∪∈Z×Z where W here denotes the space of all complete gradient flows of f starting at p and p,q ending at q. The Morse-Smale property of f ensures that, for all p,q, W in fact has the p,q structure of a smooth manifold of dimension equal to the difference between the respective Morse indices of p and q. Of particular interest is the case where this difference is equal to 1, and where the space W is therefore a one-dimensional manifold. Indeed, since the p,q differential operator (3) defining elements of W is homogeneous in time, the additive p,q group, R, acts on this space by translation of the time variable, yielding a quotient space which happens to be a compact, zero-dimensional manifold: that is, a finite set of points. The cardinality of this set will then be used to define the boundary operator of the chain complex of (M,f). Indeed, for all p Z , k ∈ ∂ p := [#W /R]q. k p,q q∈XZk−1 The first main theorem of Morse homology theory states that, for all k, the composi- tion, ∂ ∂ , vanishes. It follows that the chain complex, (C (M),∂ ), has a well-defined k−1 k ∗ ∗ ◦ homology, H (M,f) := Ker(∂ )/Im(∂ ), ∗ ∗ ∗+1 and this will be called the Morse homology of (M,f). The second and third main theorems of Morse homology theory then state respectively that the Morse homology is, up to isomorphism, independent of the Morse-Smale function used, and, furthermore, that it is in fact isomorphic to the singular homology of the ambient space, M. Significantly, since the sum of the dimensions of the homology groups yields a lower bound for the sum of the dimensions of the chain groups, it also yields a lower bound for the cardinality of the solution set, Z. In this manner, it is shown that the number of critical points of a generic function, f, is bounded below by the sum of the Betti numbers of the ambient manifold, which is one of the main results of Morse theory. 1.3 - Eternal forced mean curvature flows. Given a riemannian manifold, M, and a smooth, positive function, F : M ]0, [, an eternal forced mean curvature flow → ∞ with forcing term F will be a strongly smooth* curve [e ] : R (M) such that t → E ∂ e ,N +H F e = 0, (5) t t t t t h i − ◦ * c.f. Appendix A for terminology. 4 Eternal forced mean curvature flows III - Morse homology where N is the outward-pointing, unit, normal vector field of the embedding, e , and H t t t is its mean curvature. The eternal forced mean curvature flows with forcing term F are precisely the L2-gradient flows of the “area-minus-volume” functional, , introduced in F Section 1.1. Thus, bearing in mind the discussion of the preceding section, a Morse homol- ogy theory for the pair ( (M), ) should follow once the appropriate properties of eternal E F forced mean curvature flows have been established. The objective of the current paper will be to obtain precisely these properties in the case where M is a (d+1)-dimensional torus, Td+1.† We will first refine what has already been proven in [19] and [20], before stating the new results which complete the theoretical side of the construction. Observe, in particular, that it will be sufficient to obtain results for solutions of the parabolic problem, that is, for eternal forced mean curvature flows, since solutions of the elliptic problem, that is, Alexandrov-embedded hyperspheres of prescribed curvature, arise as special cases of the former, being merely those forced mean curvature flows which are constant. It will first be necessary to introduce geometric restrictions. Thus, for all λ 1, ≥ the subspace (Td+1) of (Td+1) will be identified as follows. First, an embedding, [e], λ E E will be said to be locally strictly convex (or LSC) whenever every one of its principal curvatures is strictly positive. An LSC Alexandrov-embedding, [e], will then be said to be pointwise λ-pinched whenever it has the property that, for every point x Sd, ∈ κ (x) λκ (x), d 1 ≤ where0 < κ (x) κ (x)are, respectively, theleastandgreatestprincipalcurvaturesofthe 1 d ≤ Alexandrov-embedding, e, at thepointx. Nextrecallthat anyLSC Alexendrov-embedding in Rd+1 is actually the boundary of some open, convex set (c.f. [2]). A pointwise λ-pinched Alexandrov-embedding, [e], inRd+1 will thenbe saidto be λ-non-collapsed whenever it has the property that for all x Sd, the Euclidean hypersphere of curvature λκ (x) which is 1 ∈ an interior tangent to [e] at the point x is entirely contained within the closed set bounded by [e].* A pointwise λ-pinched Alexandrov-embedding, [e], in Td+1 is then said to be λ-non-collapsed whenever its lift to Rd+1 has this property. Finally, for all λ 1, the ≥ subspace (Td+1) of (Td+1) will be defined to be the set of all Alexandrov-embeddings, λ E E [e], which are LSC, pointwise λ-pinched, and λ-non-collapsed. Consider now the rational function φ : [1, [ [0, [ given by ∞ → ∞ (t 1) φ(t) := − . (6) t(t+1) Observe that this function tends to 0 at 1 and + and has a unique maximum, equal to ∞ † Throughout this text, a (d+1)-dimensional torus will be taken to be any quotient of Rd+1 by a cocompact lattice. * We alert the reader to the fact that our notion of non-collapsedness compares the curvature of the Euclidean hypersphere to the least principal curvature of the Alexandrov- embedding. Although this may be at first surprising, it makes goods sense, since pointwise λ-pinching ensures that the curvature of the Euclidean hypersphere is actually no less than the greatest principal curvature of the Alexandrov-embedding. 5 Eternal forced mean curvature flows III - Morse homology (3 2√2), at the point (1+√2). There therefore exist two smooth inverses, − λ : [0,(3 2√2)[ [1,1+√2[, and − → Λ : [0,(3 2√2)[ ]1+√2,+ ]. − → ∞ The forcing term, F, will be said to be sub-critical whenever it is strictly positive, and Sup D2(F)(x)(ξ,ξ) (3 2√2) Inf F(x)3. (7) x∈Td+1,kξk=1 ≤ − x∈Td+1 (cid:12) (cid:12) In this case, the constants λ (cid:12)< Λ will be d(cid:12)efined by F F λ := λ Sup D2(F)(x)(ξ,ξ) / Inf F(x)3 , and F x∈Td+1,kξk=1 x∈Td+1 ! (cid:12) (cid:12) (8) (cid:12) (cid:12) Λ := Λ Sup D2(F)(x)(ξ,ξ) / Inf F(x)3 , F x∈Td+1,kξk=1 x∈Td+1 ! (cid:12) (cid:12) and for subcritical F, the subspace (cid:12)(Td+1) will be(cid:12)defined by F E (Td+1) := (Td+1). (9) EF EΛF Embeddings in (Td+1) will be called admissable. Likewise, a forced mean curvature F E flow, [e ], with forcing term, F, will be said to be admissable whenever [e ] is admissable t t for all t. Morse homology will be constructed for the pair ( (Td+1), ). F E F Significantly, the space, (Td+1), of admissable Alexandrov-embeddings is actually F E a closed subset of (Td+1). However, it is of fundamental importance in any differential E topology theory that the space of admissable elements be open. Indeed, otherwise, the perturbative stages of the construction would cease to function. This problem of openness is typically addressed indirectly in the statement of the compactness result, which is the case in our earlier work, [19], where we prove compactness for families of eternal forced mean curvature flows, and where the rather restrictive conditions imposed actuallyserve to ensure that all limits remain within a given open set. The greater generality of the present setting is obtained via the following result, where the problem of openness is addressed via an adaptation of Ben Andrews’ non-collapsing theorem. Indeed, even though the subset, (Td+1), is not open, for subcritical F, the Morse homology of ( (Td+1), ) lies strictly F F E E F in its interior, and is, in particular, separated from the rest of the Morse homology of ( (Td+1), ). An eternal forced mean curvature flow [e ] : R (Td+1) will be said to be t E F → E of bounded type whenever SupDiam([e ]) < . t ∞ t In Section 3, we prove Theorem 1.3.1, Separation. If [e ] : R (Td+1) is an eternal forced mean curvature flow of bounded type with t F → E sub-critical forcing term F, and if [e ] is pointwise Λ -pinched and Λ -non-collapsed for t F F all t, then [e ] is pointwise λ -pinched and λ -non-collapsed for all t. t F F Theorem1.3.1,justifiesthecontextofallthatfollows. First,thefollowingcompactness result is obtained via a straightforward blow-up argument (c.f. Section 2.2). 6 Eternal forced mean curvature flows III - Morse homology Theorem 1.3.2, Compactness. Fix d 2, and let (F ) be a sequence of smooth, positive functions over Rd+1 converging m ≥ in the Ck -sense for all k to the smooth, positive function F . Suppose, furthermore, that loc ∞ 0 < InfF SupF < . m,− m,+ m ≤ m ∞ For all m, let [e ] : R (Rd+1) be an eternal forced mean curvature flow of bounded m,t → E type with forcing term F . Suppose, furthermore, that there exists λ 1 such that, for m ≥ all m, and for all t, [e ] is pointwise λ-pinched and λ-non-collapsed. If there exists a m,t compact subset K Rd+1 such that [e ] K = for all m, then there exists an eternal m,0 ⊆ ∩ 6 ∅ forced mean curvature flow, [e ], towards which the sequence ([e ]) subconverges in ∞,t m,t the Ck -sense for all k. In particular, [e ] is also of bounded type and, for all t, [e ] is loc ∞,t ∞,t pointwise λ-pinched and λ-non-collapsed. As in the finite-dimensional case, the functional, , will be said to be of Morse type F whenever every one of its admissable critical points is non-degenerate. Recalling that Theorem 1.3.2 also applies to sequences of critical points of , straightforward differential- F topological arguments now show (c.f., for example, [14] and [23]) that the functional, , F has this property for generic F. In this case, it readily follows from Theorem 1.3.2, again, that eternal forced mean curvature flows always have well-defined end-points. Theorem 1.3.3, End-points. If is of Morse type, and if [e ] : R (Td+1) is an admissable eternal forced mean t F F → E curvature flow of bounded type, then there exist admissable embeddings, [e ] (Td+1), ± F ∈ E of mean curvature prescribed by F such that Lim [e ] = [e ], and t − t→−∞ Lim [e ] = [e ]. t + t→+∞ However, the main result of this paper actually concerns the asymptotic behaviour of the Morse homology of ( (Td+1), ) as F tends to infinity. Indeed, it is this that will F E F yield an explicit formula for the Morse homology. Consider therefore a smooth function, f : Td+1 R, of Morse-Smale type. For all sufficiently small κ > 0, consider the forcing → term 1 F := (1 κ2f), κ κ − let denote its “area-minus-volume” functional, and let (Td+1) denote the space of all κ κ F E Alexandrov-embedded hyperspheres which are admissable for F , that is κ (Td+1) := (Td+1). Eκ EFκ In [18], building on the work, [25], of Ye, we show (c.f. also [12]), 7 Eternal forced mean curvature flows III - Morse homology Theorem 1.3.4, Concentration: elliptic case. For sufficiently small κ > 0, there exists a strongly smooth map Φ : Td+1 (Td+1) such κ → E that for any critical point, p, of f of Morse index k, the point Φ(p) is a non-degenerate critical point of of Morse index (k +1). Furthermore, has no other critical points κ κ F F in (Td+1). κ E TheMorsehomologyof( (Td+1), )isexplicitlydeterminedoncethecorresponding κ κ E F asymptoticbehaviourfor eternal forcedmeancurvature flowshasbeenproven. Thus, given any two critical points, [e] and [f], of , will denote the space of admissable forced κ [e],[f] F W mean curvature flows of bounded type with forcing term F , starting at [e] and ending κ at [f]. Next, an eternal forced mean curvature flow, [e ] : R (Td+1), will be said to t → E be non-degenerate whenever its linearised mean curvature flow operator is surjective. If every element of is non-degenerate, and if the functional, , is of Morse type, then [e],[f] κ W F it follows by standard techniques of Fredholm theory that has the structure of a [e],[f] W smooth, finite-dimensional manifold. In Sections 4 and 5, making use of the compactness result of Theorem 1.3.2, together with the existence of well-defined end-points established in Theorem 1.3.3, we complement Theorem 1.3.4 by showing Theorem 1.3.5, Concentration: parabolic case. For sufficiently small κ > 0, and for every pair of critical points, p and q, of f such that Index(p) Index(q) = 1, every element of is non-degenerate, and there exists a Φ(p),Φ(q) − W canonical diffeomorphism Φˆ : W . p,q Φ(p),Φ(q) → W Inparticular, there areno other admissableeternalforced meancurvatureflows ofbounded type with forcing term F , starting at Φ(p) and ending at Φ(q). κ Remark: This result follows directly from Theorem 4.4.2 and Lemma 5.4.1, below. In actual fact, in [20], the perturbative part of this result has been shown in the slightly different context of forced mean curvature flows with constant forcing term inside general riemannian manifolds. 1.4 - Constructing the Morse homology. It remains to sketch how the results of Section1.3 serveto construct the Morsehomologyof( (Td+1), ). First, for a subcritical F E F forcing term, F, the elliptic solution space, , is defined to be the set of all critical Z points of ; that is, the set of all admissable, Alexandrov-embedded hyperspheres of mean F curvature prescribed by F. As indicated above, it follows from the compactness result of Theorem 1.3.2 that, for generic F, the “area-minus-volume” functional, , is of Morse F type in the sense that every one of its critical points is non-degenerate. In particular, in this case, consists only of isolated elements, and, by the compactness result of Theorem Z 1.3.2, again, is finite. It follows from the spectral theory of second-order, elliptic operators that every ele- ment, [e], of has finite Morse index, defined here to be the sum of the multiplicities of Z all the strictly negative eigenvalues of the Jacobi operator of at e, (c.f. [14] and [23]). F For all integer k, the finite set, , is then defined to be the set of all critical points of k Z F of Morse index k, and, the k’th order chain group of ( (Td+1), ) is defined by F E F ( (Td+1), ) := Z [ ], k F 2 k C E F Z 8