Lusternik–Schnirelmann theory of critical points

Summary

Lusternik–Schnirelmann theory gives lower bounds for the number of critical points of a function from the topology of its domain, without any nondegeneracy assumption. Its main result is that a $C^1$ function on a closed manifold has at least $\operatorname{cat}(M) + 1$ critical points, where $\operatorname{cat}(M)$ is the normalized Lusternik–Schnirelmann category. The critical values are obtained by minimax over closed sets of increasing category. It is a branch of critical point theory, complementary to Morse theory, and it extends to infinite dimensions under the Palais–Smale condition.