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Appendix An Overview of Morse Theory Morse theory is a beautiful subject that sits between differential geometry, topol- ogy and calculus of variations. It was first developed by Morse [Mor25] in the middle of the 1920s and further extended, among many others, by Bott, Milnor, Palais, Smale, Gromoll and Meyer. The general philosophy of the theory is that the topology of a smooth manifold is intimately related to the number and “type” of critical points that a smooth function defined on it can have. In this brief ap- pendix we would like to give an overview of the topic, from the classical point of view of Morse, but with the more recent extensions that allow the theory to deal with so-called degenerate functions on infinite-dimensional manifolds. A compre- hensive treatment of the subject can be found in the first chapter of the book of Chang [Cha93]. There is also another, more recent, approach to the theory that we are not going to touch on in this brief note. It is based on the so-called Morse complex. This approach was pioneered by Thom [Tho49] and, later, by Smale [Sma61] in his proof of the generalized Poincar´e conjecture in dimensions greater than 4 (see the beautiful book of Milnor [Mil56] for an account of that stage of the theory). The definition of Morse complex appeared in 1982 in a paper by Witten [Wit82]. See the book of Schwarz [Sch93], the one of Banyaga and Hurtubise [BH04] or the survey of Abbondandolo and Majer [AM06] for a modern treatment. We will try to keep our exposition as elementary as possible. In order to do this and avoid subtle technicalities, we will often not give the results in their maximal generality. Nevertheless, keeping in mind the application of Morse theory to the study of the critical points of the Lagrangian action functional, we will stress the regularity that the function under consideration must have, which will be mostly C1 and occasionally C2. M. Mazzucchelli, Critical Point The ory for Lagrangian Systems, Progress in Mathematics 293, 157 DOI 10.1007/978-3-0348-0163-8, © Springer Basel AG 2012 158 Appendix: An Overview of Morse Theory A.1 Preliminaries Throughout this appendix we will denote by M a Hilbert manifold, i.e., a Haus- dorff topological space that is locally homeomorphic to a real separable Hilbert space E with smooth change of charts. If E is finite-dimensional, i.e., E = Rm for some m ∈ N,thenM is just an ordinary m-dimensional smooth manifold. In the context of Morse theory, the most relevant difference between the finite- dimensional and the infinite-dimensional situations is that in this latter case the manifold M is not locally compact. We will come back to this point later on. Throughout this appendix, F : M → R will be a C1 function, unless more regularity will be explicitly required. A point p ∈ M is called a critical point of F when the differential dF(p) vanishes. In this case, the corresponding image F(p) is called a critical value.WedenotebyCritF the set of critical points of F. In what follows, we will only deal with functions F having isolated critical points. We wish to investigate the relationship between the critical points of F and the topological properties (or, more precisely, the homological properties) of the underlying manifold M . Consider an open neighborhood U of a critical point p such that there exists −1 achartφ : U → E of M . We denote by Fφ the function F ◦ φ : E → R.By differentiating, we have −1 dFφ(φ(p)) = dF(p) ◦ dφ (φ(p)). 2 Therefore φ(p) is a critical point of the function Fφ.IfF is C or at least twice Gˆateaux-differentiable, then so is Fφ. We recall that, in these cases, the second Gˆateaux derivative of Fφ at x can be seen as a symmetric bounded bilinear form HessFφ(x):E × E → R given by HessFφ(x)[v, w] = (d(dFφ)(x)v) w, ∀v, w ∈ E. In the above formula, we have denoted by d(dFφ)theGˆateaux derivative of dFφ, i.e., d d(dFφ)(x)v = dFφ(x + εv) ∀x, v ∈ E. dε ε=0 This latter expression coincides with the Fr´echet derivative of dFφ in case F is C2. We define the Hessian of F at a critical point p as the symmetric bilinear form HessF(p):TpM × TpM → R given by HessF(p)[v, w]=HessFφ(x)[vφ, wφ], ∀v, w ∈ TpM , where x = φ(p), vφ =dφ(p)v and wφ =dφ(p)w are in E. It is easy to see that HessF(p) is intrinsically defined, i.e., its definition is independent of the chosen chart φ as long as p is a critical point of F. A.2. The generalized Morse Lemma 159 Remark A.1.1. If M is finite-dimensional and (x1,...,xm) is a system of local coordinates around p, the Hessian of F at the critical point p is given by m ∂2F HessF(p)= (p)dxi ⊗ dxj. ∂xi ∂xj i,j=1 Moreover, if ∇ is any linear connection on M , the Hessian of F at p is given by HessF(p)=∇(dF)(p). We denote by ind(F,p) the supremum of the dimensions of those subspaces of TpM on which HessF(p) is negative-definite, and by nul(F,p) the dimension of the nullspace of HessF(p), i.e., the Hilbert space consisting of all v ∈ TpM such that HessF(p)[v, w] = 0 for all w ∈ TpM .Wecallind(F,p)andnul(F,p) respectively Morse index and nullity of the function F at p. Notice that both may be infinite. Their sum is sometimes called the large Morse index.MorseTheory was initially developed for the so-called Morse functions, which are functions all of whose critical points have nullity equal to zero. Such critical points are called non-degenerate. Nowadays we are able to deal with functions having possibly degenerate critical points. Since the inner product · , · E of E is a non-degenerate bilinear form, there exists a bounded self-adjoint linear operator Hφ = Hφ(p):E → E such that HessF(p)[v, w]= Hφvφ, wφ E , ∀v, w ∈ TpM . By the spectral theorem, this operator induces an orthogonal splitting 0 − + E = Eφ ⊕ Eφ ⊕ Eφ , 0 − + where Eφ is the kernel of Hφ,andEφ [resp. Eφ ] is a subspace of E on which Hφ is negative-definite [resp. positive-definite], i.e., 0 Eφ = {v ∈ E | Hφv =0} , − Hφv, v E < 0, ∀v ∈ Eφ \{0} , + Hφw, w E > 0, ∀w ∈ Eφ \{0} . − 0 Notice that ind(F,p)=dimEφ and nul(F,p)=dimEφ. A.2 The generalized Morse Lemma A starting point for Morse Theory might be the so-called Morse Lemma, originally due to Morse and generalized to infinite-dimensional Hilbert manifolds by Palais [Pal63]. The version that we give here, valid for possibly degenerate functions, is due to Gromoll and Meyer [GM69a] and is sometimes referred to as the splitting 160 Appendix: An Overview of Morse Theory lemma. Since it is a local result, we can assume that our Hilbert manifold M is just the Hilbert space E.Let0 be an isolated critical point of a C2 function F : U → R,whereU is an open subset of E.WedenotebyH : E → E the bounded self- adjoint linear operator associated to the Hessian of F at 0,andbyE0 ⊕E− ⊕E+ of E the orthogonal splitting induced by the spectral theorem as in the previous section. According to this splitting, we will write every vector v ∈ E as v = v0 + v±, where v± ∈ E± := E− ⊕ E+ and v0 ∈ E0. We assume that H is Fredholm, meaning that its image H(E) is closed and that its kernel and cokernel have both finite dimension. In particular nul(F, 0) is finite. Lemma A.2.1 (Generalized Morse Lemma). If V ⊆ U is a sufficiently small open neighborhood of 0, there exist • amapφ :(V , 0) → (U , 0) that is a homeomorphism onto its image, • a C2 function F 0 : V ∩ E0 → R whose Hessian vanishes at the origin, such that F ◦ φ(v)=F 0(v0)+F ±(v±), ∀v = v0 + v± ∈ V , (A.1) where F ± : E± → R is the quadratic form defined by ± ± 1 ± ± 1 ± ± ± ± F (v )= 2 HessF(0)[v , v ]= 2 Hv , v E, ∀v ∈ E . Notice that 0 is a critical point of F ±, and therefore of F 0 as well. If 0 is a non-degenerate critical point of F, namely nul(F, 0) = 0, equation (A.1) reduces to ± 1 F ◦ φ(v)=F(0)+F (v)=F(0)+ Hv, v E . 2 This is a fundamental rigidity result for functions around a non-degenerate critical point: up to a local reparametrization and to an additive constant, they are all non-degenerate quadratic forms. In the general case, equation (A.1) gives us a local representation of a function F as the sum of a non-degenerate quadratic form F ± and of a function F 0 with a totally degenerate critical point at the origin. A.3 Deformation of sublevels At this point, let us recall some standard terminology from topology. Two con- tinuous maps f0 and f1 from a topological space X to another topological space Y are homotopic, which we write as f0 ∼ f1, when there exists a continuous map f :[0, 1] × X → Y , called homotopy, such that f0 = f(0, ·)andf1 = f(1, ·). If we A.3. Deformation of sublevels 161 consider the two maps as maps of topological pairs f0,f1 :(X, A) → (Y,B), i.e., A ⊆ X, B ⊆ Y and f0(A)∪f1(A) ⊆ B, then they are homotopic (as maps of pairs) when there exists a homotopy f as before that further satisfies f(t, A) ⊆ B for all t ∈ [0, 1].
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