On Steklov Eigenspaces for Free Boundary Minimal Surfaces in the Unit Ball

TL;DR

This paper introduces new methods to compare coordinate function spans with Steklov eigenspaces in free boundary minimal surfaces, proving the critical catenoid's uniqueness among certain symmetric annuli.

Contribution

Develops novel techniques to relate coordinate spans and Steklov eigenspaces, confirming the critical catenoid's uniqueness and extending results to various symmetric free boundary minimal surfaces.

Findings

Proves that the coordinate span equals the first Steklov eigenspace for certain symmetric free boundary annuli.

Shows the critical catenoid is the unique embedded free boundary minimal annulus with antipodal symmetry.

Extends the equality of spans and eigenspaces to a broad class of symmetric free boundary minimal surfaces.

Abstract

We develop new methods to compare the span $\mathcal{C}(\Sigma)$ of the coordinate functions on a free boundary minimal submanifold $\Sigma$ embedded in the unit $n$-ball $\mathbb{B}^n$ with its first Steklov eigenspace $\mathcal{E}_{\sigma_1}(\Sigma)$. Using these methods, we show that $\mathcal{C}(A)=\mathcal{E}_{\sigma_1}(A)$ for any embedded free boundary minimal annulus $A$ in $\mathbb{B}^3$ invariant under the antipodal map, and thus prove that $A$ is congruent to the critical catenoid. We also confirm that $\mathcal{C}=\mathcal{E}_{\sigma_1}$ for any free boundary minimal surface embedded in $\mathbb{B}^3$ with the symmetries of many known or expected examples, including: examples of any positive genus from stacking at least three disks; two infinite families of genus $0$ examples with dihedral symmetry, as well as a finite family with the various Platonic symmetries; and examples of any genus by desingularizing several disks that meet at equal angles along a diameter of the ball.