Abstract
For any smooth Riemannian metric on an (n+1)-dimensional compact manifold with boundary (M,partial M) where 3le (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.
Highlights
In his celebrated 1982 Problem Section, S.-T
In a very recent work [13], the second and the last author developed a version of min–max theory for manifolds with boundary and proved up-to-the-boundary regularity for the free boundary minimal hypersurfaces produced by their theory, completing the program set out by Almgren in the hypersurface case
Later in [17], they established the general Morse index upper bounds for minimal hypersurfaces produced by Almgren–Pitts theory for compact manifolds without boundary
Summary
In his celebrated 1982 Problem Section, S.-T. Yau raised the following question: Question 1.1 ([33]) Does every closed three-dimensional Riemannian manifold (M3, g) contain infinitely many (immersed) minimal surfaces?. They showed that in any closed manifold (Mn+1, g) there exists at least one embedded closed minimal hypersurface, which is smooth except possibly along a singular set of Hausdorff codimension at least 7 These fascinating results partly motivated Question 1.1 asked by Yau. In a very recent work [13], the second and the last author developed a version of min–max theory for manifolds with boundary and proved up-to-the-boundary regularity for the free boundary minimal hypersurfaces produced by their theory, completing the program set out by Almgren in the hypersurface case. Later in [17], they established the general Morse index upper bounds for minimal hypersurfaces produced by Almgren–Pitts theory for compact manifolds without boundary (the one-parameter case was studied earlier in the works of Marques–Neves [15] and the last author [34,35]). We present a few applications of our general Morse index upper bounds
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