Nondegeneracy of critical points of the mean curvature of the boundary for Riemannian manifolds

TL;DR

This paper proves that for a generic Riemannian metric on a compact manifold with boundary, all critical points of the boundary's mean curvature are nondegenerate, ensuring stability and genericity of these points.

Contribution

It establishes that nondegeneracy of critical points of boundary mean curvature is a generic property under Riemannian metrics.

Findings

Critical points of boundary mean curvature are nondegenerate for generic metrics

Nondegeneracy holds for a dense set of Riemannian metrics

Results apply to compact manifolds with boundary

Abstract

Let $M$ be a compact smooth Riemannian manifold of finite dimension $n+1$ with boundary $\partial M$and $\partial M$ is a compact $n$-dimensional submanifold of $M$. We show that for generic Riemannian metric $g$, all the critical points of the mean curvature of $\partial M$ are nondegenerate.