Compactness and generic finiteness for free boundary minimal hypersurfaces (I)
Differential Geometry
2021-01-27 v3
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.
Keywords
Cite
@article{arxiv.1803.01509,
title = {Compactness and generic finiteness for free boundary minimal hypersurfaces (I)},
author = {Qiang Guang and Zhichao Wang and Xin Zhou},
journal= {arXiv preprint arXiv:1803.01509},
year = {2021}
}
Comments
We fixed a gap in the construction of Jacobi fields and split the original paper into two parts. The followup paper(the second part) will deal with the construction of Jacobi fields when the convergence has higher multiplicity