Free boundary minimal surfaces of unbounded genus

2016/12/27 by Daniel Ketover, Ketover, Daniel · 6 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1612.08691

Abstract

For each integer g≥ 1 we use variational methods to construct in the unit 3-ball B a free boundary minimal surface Σ_g of symmetry group \mathbbD_g+1. For g large, Σ_g has three boundary components and genus g. As g→∞ the surfaces Σ_g converge as varifolds to the union of the disk and critical catenoid. These examples are the first with genus greater than 1 and were conjectured to exist by Fraser-Schoen. We also construct several new free boundary minimal surfaces in B with the symmetry groups of the cube, tetrahedron and dodecahedron. Finally, we prove that free boundary minimal surfaces isotopic to those of Fraser-Schoen can be constructed variationally using an equivariant min-max procedure. We also prove an ε-regularity theorem for free boundary minimal surfaces in B.

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• Self-expanders of positive genus
• Existence and classification of S¹-invariant free boundary annuli and Möbius bands in \mathbbBⁿ
• Free boundary minimal hypersurfaces with least area

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