[Paper Review] Existence of infinitely many free boundary minimal hypersurfaces

Zhichao Wang|arXiv (Cornell University)|Jan 14, 2020

Geometric Analysis and Curvature Flows29 references7 citations

TL;DR

This paper establishes the existence of infinitely many almost properly embedded free boundary minimal hypersurfaces in compact Riemannian manifolds with smooth boundary of dimension 3 to 7, resolving the free boundary analog of Yau's conjecture. By adapting Song's min-max approach and leveraging Li-Zhou's regularity theory, the authors prove that the min-max widths grow indefinitely, implying infinitely many distinct minimal hypersurfaces despite potential touching sets on the boundary.

ABSTRACT

In this paper, we prove that in any compact Riemannian manifold with smooth boundary, of dimension at least 3 and at most 7, there exist infinitely many almost properly embedded free boundary minimal hypersurfaces. This settles the free boundary version of Yau's conjecture. The proof uses adaptions of A. Song's work and the early works by Marques-Neves in their resolution to Yau's conjecture, together with Li-Zhou's regularity theorem for free boundary min-max minimal hypersurfaces.

Motivation & Objective

• To resolve the free boundary version of Yau’s conjecture on the existence of infinitely many minimal hypersurfaces in compact Riemannian manifolds with boundary.
• To extend the min-max methods of Song and Marques-Neves to the free boundary setting, where boundary regularity and touching sets pose significant challenges.
• To establish the growth of min-max widths in the free boundary setting, ensuring infinitely many distinct minimal hypersurfaces via width divergence.
• To overcome the difficulty of non-smooth boundary behavior and touching sets by developing new geometric and analytic tools, including a refined embedded Frankel property.

Proposed method

• Adapts Song’s min-max construction for closed manifolds to the free boundary setting using the volume spectrum and Weyl law for min-max widths.
• Applies Li-Zhou’s regularity theorem for free boundary minimal hypersurfaces to ensure almost properly embedded smooth solutions with multiplicity one.
• Introduces a new notion of 'touching set' and develops a modified 'embedded Frankel property' to control overlapping and intersection phenomena.
• Uses a blow-up and approximation argument near the boundary, relying on the convergence of Riemannian metrics and vector fields in the $ε$-regularization framework.
• Employs a sequence of $ε$-regularized metrics and vector fields to control the divergence of the second fundamental form and the mean curvature in the limit.
• Applies a compactness argument based on varifold convergence and stationarity to extract a limit varifold that supports a minimal hypersurface with free boundary.

Experimental results

Research questions

• RQ1 Does every compact Riemannian manifold with smooth boundary of dimension 3 to 7 contain infinitely many free boundary minimal hypersurfaces?
• RQ2 Can the min-max width growth in the free boundary setting be used to produce infinitely many distinct minimal hypersurfaces, even with non-trivial touching sets?
• RQ3 How can the regularity theory for free boundary minimal hypersurfaces be extended to handle non-smooth boundary behavior and singularities?
• RQ4 What modifications to the embedded Frankel property are necessary to control the intersection of minimal hypersurfaces in the presence of boundary tangency?

Key findings

• The min-max widths for free boundary minimal hypersurfaces in compact manifolds of dimension 3 to 7 grow without bound, implying the existence of infinitely many distinct minimal hypersurfaces.
• The authors prove that each min-max width corresponds to an almost properly embedded free boundary minimal hypersurface with multiplicity one, under the assumption of finite multiplicity.
• The proof establishes full regularity of the min-max limit via Li-Zhou’s regularity theorem, even when the boundary is not smooth, by focusing on smooth boundary points.
• A new 'embedded Frankel property' is developed to control the intersection of minimal hypersurfaces and prevent degeneracy in the min-max construction.
• The paper resolves the free boundary version of Yau’s conjecture in dimensions 3 to 7, extending the closed manifold result of Song to the boundary case.
• The method successfully overcomes the challenge of non-empty touching sets by introducing a localized $ε$-regularization and uniform bounds on the second fundamental form and mean curvature.