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Min-max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications

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

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Fulltext (public)

arXiv:1907.12064.pdf
(Preprint), 336KB

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Citation

Guang, Q., Li, M.-M.-c., Wang, Z., & Zhou, X. (in press). Min-max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications. Mathematische Annalen, Published Online - Print pending. doi:10.1007/s00208-020-02096-0.


Cite as: http://hdl.handle.net/21.11116/0000-0007-822E-3
Abstract
For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,\partial M)$ where $3\leq (n+1)\leq 7$, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates to prove that for almost every (in the $C^\infty$ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in $M$. If $\partial M$ is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.