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 δqmpuq 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
ϕipDmi
q
Φ´1pc
´ εq X
ϕipDmi
q
ϕipBDmi
q
and
Φc´ε
Y
Ťk
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
Kq
such that
Φpξpuqq
1
2
Aw, w ` Φpv ` ηpvqq, u P B.
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