Min-max theory for free boundary minimal hypersurfaces. II: General Morse index bounds and applications. (English) Zbl 1466.53074
For any smooth Riemannian metric on an \((n+1)\)-dimensional compact manifold with boundary \((M,\partial M)\) where \(3\leq n+1\leq 7\), the authors establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. They apply their Morse index estimates to prove that for almost every (in the \(C^{\infty}\) Baire sense) Riemannian 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, they show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Their results prove a conjecture of Yau for generic metrics in the free-boundary setting.
For Part I, see [the second and last author, “Min-max theory for free boundary minimal hypersurfaces. I: Regularity theory”, Preprint, arXiv:1611.02612].
For Part I, see [the second and last author, “Min-max theory for free boundary minimal hypersurfaces. I: Regularity theory”, Preprint, arXiv:1611.02612].
Reviewer: Atsushi Fujioka (Osaka)
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
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