Strong Minimax Property of Nondegenerate Minimal Submanifolds.

摘要

If N is a riemannian manifold and M is smooth compact submanifold (with or without boundary) that is stationary and strictly stable for the area functional, we show that there is an open subset U of N containing M such that M has less area than any other surface in U that is homologous (in U) to M. Similarly, if M is unstable but has nullity 0, then it is the unique solution of a certain minimax problem in an open set U included in N. The theorems also hold when area is replaced by other parametric elliptic functionals.