Morse theory on Hilbert manifolds

Remarks

For such $f$ the two basic theorems of Morse theory hold. If $a < b$ and $f^{-1}[a, b]$ contains no critical point, then the sublevel set $M^a = \set{f \le a}$ is a deformation retract of $M^b$. If $f^{-1}[a, b]$ contains exactly one critical point, of index $\lambda$, in its interior, then $M^b$ has the homotopy type of $M^a$ with a $\lambda$-cell attached.

Example: let $(N, h)$ be a compact Riemannian manifold, let $p, q \in N$, and let $M$ be the Hilbert manifold of paths $\omega \colon [0, 1] \to N$ of Sobolev class $H^1$ with $\omega(0) = p$ and $\omega(1) = q$. The energy $E(\omega) = \frac{1}{2} \int_0^1 \abs{\omega'}^2 \, dt$ is bounded below and satisfies condition (C). Its critical points are the geodesics from $p$ to $q$, of finite index, and they are nondegenerate when $q$ is not conjugate to $p$ along any geodesic.

Non-example: on $M = \R$ the function $f(x) = \arctan x$ is bounded below and has no critical points, but condition (C) fails along $x_n = -n$, since $f(x_n)$ is bounded and $f'(x_n) \to 0$. Accordingly $f$ does not attain its infimum $-\pi/2$, while a function that is bounded below and satisfies condition (C) attains its infimum at a critical point.