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Large topology asymptotics for spectrally extremal minimal surfaces in $\mathbb{B}^3$ and $\mathbb{S}^3$

Mikhail Karpukhin, Peter McGrath, Daniel Stern

TL;DR

This work develops a unified framework linking spectral optimization (Laplace and Steklov) to the geometry of minimal surfaces in $\mathbb{S}^3$ and free boundary minimal surfaces in $\mathbb{B}^3$, focusing on large Euler characteristic behavior. It provides equivalent planar-domain reformulations for the extremal problems, establishes sharp Neumann and Dirichlet lower bounds, and obtains analogous Steklov estimates, all culminating in varifold-convergence results that confirm conjectures about area-minimizing surfaces and the special role of Lawson's $\xi_{\gamma,1}$. The paper also identifies free boundary analogs of Lawson surfaces, proves stability and convergence results (to equators or disks) in the large-topology limit, and clarifies how topology and symmetry govern area growth and limiting behavior. Overall, the results offer sharp area asymptotics, detailed convergence statements, and a robust platform for future desingularization and doubling constructions in both closed and free-boundary settings.

Abstract

In recent work with Kusner, we developed a method, based on the equivariant optimization of Laplace and Steklov eigenvalues, for producing minimal surfaces of prescribed topology in low-dimensional balls and spheres. We used the method to construct many new minimal embeddings in $\mathbb{S}^3$ with area below $8π$, and many new free boundary minimal embeddings in $\mathbb{B}^3$ with area below $2π$. In this paper, we study the geometry of these surfaces in more detail, with an emphasis on studying sharp area estimates and varifold limits in the large Euler characteristic regime. This allows us to confirm some well-known conjectures regarding the space of low-area minimal surfaces in $\mathbb{S}^3$ in this class of examples and the special role played by Lawson's $ξ_{γ,1}$ surfaces. We also confirm analogous statements in $\mathbb{B}^3$ and identify a family of free boundary minimal surfaces in $\mathbb{B}^3$ most closely resembling $ξ_{γ,1}$.

Large topology asymptotics for spectrally extremal minimal surfaces in $\mathbb{B}^3$ and $\mathbb{S}^3$

TL;DR

This work develops a unified framework linking spectral optimization (Laplace and Steklov) to the geometry of minimal surfaces in and free boundary minimal surfaces in , focusing on large Euler characteristic behavior. It provides equivalent planar-domain reformulations for the extremal problems, establishes sharp Neumann and Dirichlet lower bounds, and obtains analogous Steklov estimates, all culminating in varifold-convergence results that confirm conjectures about area-minimizing surfaces and the special role of Lawson's . The paper also identifies free boundary analogs of Lawson surfaces, proves stability and convergence results (to equators or disks) in the large-topology limit, and clarifies how topology and symmetry govern area growth and limiting behavior. Overall, the results offer sharp area asymptotics, detailed convergence statements, and a robust platform for future desingularization and doubling constructions in both closed and free-boundary settings.

Abstract

In recent work with Kusner, we developed a method, based on the equivariant optimization of Laplace and Steklov eigenvalues, for producing minimal surfaces of prescribed topology in low-dimensional balls and spheres. We used the method to construct many new minimal embeddings in with area below , and many new free boundary minimal embeddings in with area below . In this paper, we study the geometry of these surfaces in more detail, with an emphasis on studying sharp area estimates and varifold limits in the large Euler characteristic regime. This allows us to confirm some well-known conjectures regarding the space of low-area minimal surfaces in in this class of examples and the special role played by Lawson's surfaces. We also confirm analogous statements in and identify a family of free boundary minimal surfaces in most closely resembling .

Paper Structure

This paper contains 16 sections, 47 theorems, 336 equations.

Table of Contents

  1. Introduction
  2. $\bar{\lambda}_1$-extremal surfaces in $\mathbb{S}^3$
  3. $\bar{\sigma}_1$-extremal surfaces in $\mathbb{B}^3$
  4. $\bar{\sigma}_1$-extremal basic reflection surfaces
  5. Organization
  6. Equivalent formulations of the optimization problem
  7. Neumann estimates
  8. Dirichlet lower bounds
  9. Application to symmetric domains
  10. Segmented domains
  11. Lower Bounds in the Steklov setting
  12. Lower bounds for $\sigma^D_1(\Omega)$
  13. Stability and upper bounds
  14. Convergence to equators
  15. Stability and upper bounds in the Steklov setting
  16. ...and 1 more sections

Key Result

Theorem 1.1

KKMS For every orientable basic reflection surface $(M,\Gamma)$ with $\Gamma$ of the form $\Gamma=\langle \tau\rangle\times G$ for a finite reflection group $G$, there exists a metric $g\in \mathrm{Met}_{\Gamma}(M)$ realizing the supremum of $\bar{\lambda}_1$ over all $\Gamma$-invariant metrics $h\in \mathrm{Met}_{\Gamma}(M)$. Moreover, there is an orthogonal representation $\rho: \Gamma\to O(4)$

Theorems & Definitions (96)

  • Theorem 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Conjecture 1.8: cf. Kap:survey
  • Theorem 1.9
  • Corollary 1.10
  • ...and 86 more