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

Abstract

For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,\partial M)$ where $3\le (n+1)\le 7$, 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.

arXiv comment25 pages; Comments are welcome!

References in corpus (2)

• Existence of infinitely many minimal hypersurfaces in closed manifolds
• Compactness and generic finiteness for free boundary minimal hypersurfaces (I)

Cited by in corpus (4)

• Free boundary minimal surfaces in the unit ball : recent advances and open questions
• Existence of infinitely many free boundary minimal hypersurfaces
• Multiplicity one for min-max theory in compact manifolds with boundary and its applications
• Locally Maximizing Metric of Width on Manifolds with Boundary