Compactness and generic finiteness for free boundary minimal hypersurfaces, I

Zhou, Xin; Wang, Zhichao; Zhou, Xin

doi:10.2140/pjm.2021.310.85

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Compactness and generic finiteness for free boundary minimal hypersurfaces, I

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Wang, Zhichao
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.2140/pjm.2021.310.85
(Publisher version)

https://doi.org/10.48550/arXiv.1803.01509
(Preprint)

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Citation

Zhou, X., Wang, Z., & Zhou, X. (2021). Compactness and generic finiteness for free boundary minimal hypersurfaces, I. Pacific Journal of Mathematics, 310(1), 85-114. doi:10.2140/pjm.2021.310.85.

Cite as: https://hdl.handle.net/21.11116/0000-0008-6822-D

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.