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, ∂ M) where 3 ≤ (n + 1) ≤ 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∞ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If ∂ 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 Show more