Embedded minimal surfaces in $\mathbb{S}^3$ and $\mathbb{B}^3$ via equivariant eigenvalue optimization

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 $2π$. Furthermore, we show that the number of minimal surfaces in $\mathbb{S}^3$ of prescribed topology and area below $8π$, and the number of free boundary minimal surfaces in $\mathbb{B}^3$ with prescribed topology and area below $2π$, 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.

Figures (5)

• Figure 1: Conjectural pictures of some of the surfaces given by Theorem \ref{['Tfbms1']}. Image credit: M. Schulz Schulz.
• Figure 2: Conjectural pictures of some of the surfaces given by Theorem \ref{['Tfbms2']}. From left to right: $D_3$-symmetric surface with $7$ boundary components; symmetric surface with boundary components around vertices of the dodecahedron; $\mathbb{Z}_2\times D_{16}$-symmetric surface with $64$ boundary components. Image credit: M. Schulz Schulz.
• Figure 5: Schematic for one quarter of a surface $M(\Gamma, \Omega)$ with $\Gamma = \mathbb{Z}_2 \times *223$ and configuration $1+ 3\rho_2+\rho_1\rho_3$, depicting six fundamental chambers for $\Gamma$.
• Figure 6: A fundamental domain for the action of $\tau$ on $N = N_{\rho_1}(\mathbb{Z}_2,5)$. Fixed-point sets are dashed, with $N^\tau, N^{\rho_2}$, and $N^{\rho_3}$ respectively consisting of the circular curves, horizontal segments, and the vertical segments. A $\mathbb{Z}_2\times D_2$-fundamental domain $U$ is one of the quarters.
• Figure 7: A fundamental domain for $\tau$ on $\hat{N}$ with $f=5$. Pieces on the left are isomorphic to $U$, while pieces on the right are isomorphic to $U^*$, the vertical line indicates the line, where $U$ is attached to $U^*$.

Theorems & Definitions (254)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Pet1KNPPKSminmax
• Theorem 1.5: Pet1
• Proposition 1.6
• Theorem 1.7
• Theorem 1.8: PetridesS
• Theorem 1.9: PetridesS
• Proposition 1.10