Vol. 310, No. 1, 2021

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Compactness and generic finiteness for free boundary minimal hypersurfaces, I

Qiang Guang, Zhichao Wang and Xin Zhou

Vol. 310 (2021), No. 1, 85–114
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 nontrivial Jacobi field when it has multiplicity one. In a forthcoming paper, we will construct Jacobi fields when the convergence has higher multiplicity.

Keywords
free boundary minimal surfaces, compactness, Jacobi fields, curvature estimates
Mathematical Subject Classification 2010
Primary: 53A10, 53C42
Milestones
Received: 29 May 2019
Revised: 10 October 2020
Accepted: 5 December 2020
Published: 26 January 2021
Authors
Qiang Guang
Mathematical Sciences Institute
The Australian National University
Canberra, ACT
Australia
Zhichao Wang
Max-Planck Institute for Mathematics
Bonn
Germany
Xin Zhou
Department of Mathematics
University of California Santa Barbara
Santa Barbara, CA
United States
Department of Mathematics
Cornell University
Ithaca, NY
United States