Palais–Smale theory of critical points

Remarks

The proof of the deformation statement follows the flow of $-\nabla f / \norm{\nabla f}^2$ on $f^{-1}[a, b]$. The Palais–Smale condition gives a positive lower bound for $\norm{\nabla f}$ there, and completeness of the metric lets the flow run until it reaches $M^a$. For the minimum, suppose $c = \inf f$ were not a critical value. Then the condition would give $\varepsilon > 0$ with no critical value in $[c, c + \varepsilon]$. The same flow, extended below $c$, would push the nonempty set $M^{c + \varepsilon}$ into a sublevel set $M^{c'}$ with $c' < c$, which is empty. For the third statement, the minimax values $c_k = \inf \set{\sup_A f : A \subseteq M \text{ closed}, \ \operatorname{cat}_M(A) \ge k}$ for $k = 0, \dots, \operatorname{cat}(M)$ are critical values, as in Lusternik–Schnirelmann theory, and equal values force infinitely many critical points.

Example: the energy functional on paths with fixed endpoints in a compact Riemannian manifold, or on its free loop space, satisfies these hypotheses. This gives existence of geodesics and is the setting of the Lyusternik–Fet theorem. Without the Palais–Smale condition the conclusions fail: $f(x) = e^x$ on $\R$ is bounded below and has no critical point, so its infimum $0$ is not attained. The mountain pass theorem is another critical point theorem that rests on the same condition.

Going deeper

For a student

These results are the base of variational methods for nonlinear partial differential equations and for geodesics. To apply them, one checks the Palais–Smale condition for a concrete functional, typically with Sobolev embeddings and compactness theorems. The deformation statement is the engine of all minimax arguments, including the mountain pass theorem and the Lusternik–Schnirelmann bound. The same results hold for $C^1$ functions on complete Finsler manifolds modelled on Banach spaces, with a pseudo-gradient vector field in place of the gradient.