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

TL;DR

This paper establishes compactness results for free boundary minimal hypersurfaces with bounded area and Morse index in a compact Riemannian manifold, revealing properties of their limits and Jacobi fields.

Contribution

It proves the compactness of the space of free boundary minimal hypersurfaces with bounded area and Morse index, and analyzes the nature of their limits and Jacobi fields.

Findings

The space of such hypersurfaces is compact under smooth graphical convergence.

Limits of these hypersurfaces inherit non-trivial Jacobi fields when multiplicity is one.

The paper sets the stage for constructing Jacobi fields in higher multiplicity cases in future work.

Abstract

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical convergence away from finitely many points. We show that the limit of a sequence of such hypersurfaces always inherits a non-trivial Jacobi field when it has multiplicity one. In a forthcoming paper, we will construct Jacobi fields when the convergence has higher multiplicity.