Index bounds for free boundary minimal surfaces of convex bodies
Pam Sargent

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 tends to infinity as its genus or the number of boundary components tends to infinity.
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