Non-degeneracy of critical points of the squared norm of the second fundamental form on manifolds with minimal boundary

TL;DR

This paper proves that for a generic Riemannian metric on a manifold with minimal boundary, the squared norm of the second fundamental form is a Morse function, with non-degenerate critical points, especially within conformal classes.

Contribution

It establishes the generic non-degeneracy of critical points of the squared second fundamental form on manifolds with minimal boundary, including within conformal classes.

Findings

Squared norm of second fundamental form is Morse for generic metrics.

Non-degeneracy holds within conformal classes.

Results apply to manifolds with minimal boundary.

Abstract

Let $(M,\bar g)$ be a compact Riemannian manifold with minimal boundary such that the second fundamental form is nowhere vanishing on $\partial M$. We show that for a generic Riemannian metric $\bar g$, the squared norm of the second fundamental form is a Morse function, i.e. all its critical points are non-degenerate. We show that the generality of this property holds when we restrict ourselves to the conformal class of the initial metric on $M$.