A symmetry condition for genus zero free boundary minimal surfaces attaining the first eigenvalue of one

TL;DR

This paper proves that genus zero free boundary minimal surfaces in the 3-ball with multiple boundary components and symmetric reflection planes have the first Steklov eigenvalue equal to one, confirming a conjecture for symmetric cases.

Contribution

It establishes a symmetry condition under which the first Steklov eigenvalue equals one for genus zero free boundary minimal surfaces.

Findings

Surfaces with n boundary components and n reflection symmetries have eigenvalue one.

Supports Fraser and Li's conjecture in symmetric cases.

Provides geometric conditions linking symmetry and spectral properties.

Abstract

An embedded free boundary minimal surface in the 3-ball has a Steklov eigenvalue of one due to its coordinate functions. Fraser and Li conjectured that whether one is the first nonzero Steklov eigenvalue. In this paper, we show that if an embedded free boundary minimal surface of genus zero, with $n$ boundary components, in the 3-ball has $n$ distinct reflection planes, then one is the first eigenvalue of the surface.