0.3. CRITICAL GROUPS 5 Now suppose that W is a Hilbert space pH, p¨, ¨qq and Φ P C2pH, Rq. Then the Hessian A Φ2puq is a self-adjoint operator on H for each u. When u is a critical point the dimension of the negative space of A is called the Morse index of u and is denoted by mpuq, and m˚puq mpuq`dim ker A is called the large Morse index. We say that u is nondegenerate if A is invertible. The Morse lemma describes the local behavior of the functional near a nondegenerate critical point. Lemma 0.6 (Morse Lemma). If u is a nondegenerate critical point of Φ, then there is a local diffeomorphism ξ from a neighborhood U of u into H with ξpuq 0 such that Φpξ´1pvqq Φpuq ` 1 2 Av, v , v P ξpU q. Morse lemma in Rn was proved by Morse [85]. Palais [92], Schwartz [123], and Nirenberg [88] extended it to Hilbert spaces when Φ is C3. Proof in the C2 case is due to Kuiper [62] and Cambini [23]. A direct consequence of the Morse lemma is Theorem 0.7. If u is a nondegenerate critical point of Φ, then CqpΦ,uq δ qmpuq G. The handle body theorem describes the change in topology as the level sets pass through a critical level on which there are only nondegenerate critical points. Theorem 0.8 (Handle Body Theorem). If c is an isolated critical value of Φ for which there are only a finite number of nondegenerate critical points ui, i 1,...,k, with Morse indices mi mpuiq, and Φ satisfies pPSq, then there are an ε ą 0 and homeomorphisms ϕi from the unit disk Dmi in Rmi into H such that Φc´ε X ϕipDmiq Φ´1pc ´ εq X ϕipDmi q ϕipBDmiq and Φc´ε Y Ť k i“1 ϕipDmi q is a deformation retract of Φc`ε. The references for Theorems 0.3, 0.7, and 0.8 are Morse [85], Pitcher [106], Milnor [84], Rothe [114, 115, 117], Palais [92], Palais and Smale [91], Smale [124], Marino and Prodi [79], Schwartz [123], Mawhin and Willem [81], and Chang [28]. The splitting lemma generalizes the Morse lemma to degenerate critical points. Assume that the origin is an isolated degenerate critical point of Φ and 0 is an isolated point of the spectrum of A Φ2p0q. Let N ker A and write H N N K, u v ` w. Lemma 0.9 (Splitting Lemma). There are a ball B Ă H centered at the origin, a local homeomorphism ξ from B into H with ξp0q 0, and a map η P C1pB X N, N K q such that Φpξpuqq 1 2 Aw, w ` Φpv ` ηpvqq, u P B.
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