A strong minimax property of nondegenerate minimal submanifolds. (English) Zbl 0808.49037

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 \(U\subset N\). The theorems also hold when area is replaced by other parametric elliptic functionals.
Reviewer: B.White (Stanford)


49Q20 Variational problems in a geometric measure-theoretic setting
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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