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

TL;DR

This paper investigates the geometry and topology of minimal surfaces in 3D balls and spheres, confirming conjectures about their area bounds and the significance of Lawson's surfaces, especially in the large Euler characteristic limit.

Contribution

It provides detailed area estimates, varifold limit analysis, and confirms conjectures about low-area minimal surfaces, highlighting the role of Lawson's surfaces in these classes.

Findings

Confirmed conjectures on low-area minimal surfaces in $ ext{S}^3$

Identified a family of free boundary minimal surfaces in $ ext{B}^3$

Analyzed varifold limits in the large Euler characteristic regime

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 $8\pi$, and many new free boundary minimal embeddings in $\mathbb{B}^3$ with area below $2\pi$. 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 $\xi_{\gamma,1}$ 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 $\xi_{\gamma,1}$.