#### Vol. 310, No. 1, 2021

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### 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