Index bounds for free boundary minimal surfaces of convex bodies

TL;DR

This paper establishes a relationship between eigenvalues of the Hodge Laplacian and the Jacobi operator for free boundary minimal hypersurfaces in convex bodies, leading to new bounds on their index based on topology.

Contribution

It introduces a novel relationship between eigenvalues of key operators and derives index bounds for free boundary minimal hypersurfaces in convex bodies.

Findings

Index tends to infinity with genus or boundary components in $ ext{R}^3$

New bounds relate topology to stability properties

Eigenvalue relationships provide tools for geometric analysis

Abstract

In this paper, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in $\mathbb{R}^3$ tends to infinity as its genus or the number of boundary components tends to infinity.