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Embedded minimal surfaces in $\mathbb{S}^3$ and $\mathbb{B}^3$ via equivariant eigenvalue optimization
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abstract
In 1970, Lawson solved the topological realization problem for minimal surfaces in the sphere, showing that any closed orientable surface can be minimally embedded in $\mathbb{S}^3$. The analogous problem for surfaces with boundary was posed by Fraser and Li in 2014, and it has attracted much attention in recent years, stimulating the development of many new constructions for free boundary minimal surfaces. In this paper, we resolve this problem by showing that any compact orientable surface with boundary can be embedded in $\mathbb{B}^3$ as a free boundary minimal surface with area below $2\pi$. Furthermore, we show that the number of minimal surfaces in $\mathbb{S}^3$ of prescribed topology and area below $8\pi$, and the number of free boundary minimal surfaces in $\mathbb{B}^3$ with prescribed topology and area below $2\pi$, grow at least linearly with the genus. This is achieved via a new method for producing minimal surfaces of prescribed topology in low-dimensional balls and spheres, based on the optimization of Laplace and Steklov eigenvalues in the presence of a discrete symmetry group. As a key ingredient, we develop new techniques for proving the existence of maximizing metrics, which can be used to resolve the existence problem in many symmetric situations and provide at least partial existence results for classical eigenvalue optimization problems.
Forward citations
Cited by 6 Pith papers
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Genus two embedded minimal surfaces in $\mathbb{S}^3$ with dihedral symmetry
The Lawson surface ξ2,1 is the unique closed embedded minimal surface of genus 2 in S^3 whose isometry group contains the bidihedral group D4h.
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Free boundary minimal surfaces in products of balls
For any rectangular prism, a genus-0 free boundary minimal surface with one boundary curve on each face exists.
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New Homogeneous Solutions for the One-Phase Free Boundary Problem
New gluing constructions yield λ1-extremal domains in S^2 with arbitrarily many boundary components, and analogous solutions in S^3, answering a question of Jerison-Kamburov and disproving Souam's conjecture.
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A new family of minimal surfaces of even genus in the three-dimensional sphere
For each n at least 2, an equivariant min-max procedure yields a new embedded minimal surface Gamma_n in S^3 with genus 2n or 2n-2, area just above the sphere's, full symmetry group G_n for n at least 4, and Morse ind...
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Large topology asymptotics for spectrally extremal minimal surfaces in $\mathbb{B}^3$ and $\mathbb{S}^3$
For spectrally extremal minimal surfaces, Lawson surfaces have least area at large genus, and generic surfaces converge to a double equator, with analogous results in the ball.
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Free boundary minimal M\"obius band in spherical caps
For each r in (0,π/2), a free boundary minimal Möbius band immersed by first Steklov eigenfunctions exists in the four-dimensional spherical cap, and any such immersion is intrinsically rotationally symmetric.
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