Free boundary minimal surfaces of unbounded genus
By Daniel Ketover. 2016
56 citations recorded by the source corpus.
For each integer $g\geq 1$ we use variational methods to construct in the unit $3$-ball $B$ a free boundary minimal surface $\Sigma_g$ of symmetry group $\mathbb{D}_{g+1}$. For $g$ large, $\Sigma_g$ has three boundary components and genus $g$. As $g\rightarrow\infty$ the surfaces $\Sigma_g$ 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 $\epsilon$-regularity theorem for free boundary minimal surfaces in $B$.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6b1d420a50ac983d2a9a43b229894f32c75b0e98
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In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Compactness analysis for free boundary minimal hypersurfaces (2017, methodology): “Ketover [23] proposed an equivariant min-max approach to generate free boundary minimal surfaces with given discrete symmetry group, thereby producing examples with arbitrarily large genus.”
- Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc (2017, methodology): “[11] Fraser-Schoen constructed new examples of embedded free boundary minimal surfaces with genus zero and arbitrary number of boundary components. Recently, more examples were constructed by Ketover [37] using min-max method. The main result of this article is Theorem 6.2 which clearly implies the ”