Compactness and generic finiteness for free boundary minimal hypersurfaces (II)

TL;DR

This paper proves that sequences of free boundary minimal hypersurfaces with bounded area and Morse index in a compact manifold have non-trivial Jacobi fields in the limit, using a new Harnack inequality for minimal graphs.

Contribution

It establishes the existence of non-trivial Jacobi fields in the limit of such hypersurfaces, advancing understanding of their compactness and stability properties.

Findings

Limit hypersurfaces inherit non-trivial Jacobi fields

Uniform bounds lead to compactness results

New Harnack inequality for minimal graphs with holes

Abstract

Given a compact Riemannian manifold with boundary, we prove that the limit of a sequence of embedded, almost properly embedded free boundary minimal hypersurfaces, with uniform area and Morse index upper bound, always inherits a non-trivial Jacobi field. To approach this, we prove a one-sided Harnack inequality for minimal graphs on balls with many holes.