Critical point theory

Summary

Critical point theory studies the critical points of a differentiable function on a manifold. Its tool is the topology of the manifold and of the sublevel sets of the function. Its main branches are Morse theory, which uses nondegenerate critical points and their indices, and Lusternik–Schnirelmann theory, which uses minimax values. In infinite dimensions it needs a compactness hypothesis such as the Palais–Smale condition. It then gives existence results for geodesics, harmonic maps and nonlinear elliptic equations.