Characterization of embedded free boundary surfaces in the ball

TL;DR

This paper characterizes embedded free boundary minimal surfaces in the 3D ball with index 4, proving that the critical catenoid is the unique such surface, extending previous foundational results.

Contribution

It establishes the uniqueness of the critical catenoid among embedded free boundary minimal surfaces with index 4 in the ball.

Findings

The critical catenoid is the only embedded free boundary minimal surface with index 4 in the ball.

The result extends previous work by Fraser, Schoen, and Tran.

Provides a classification result for minimal surfaces with specific index in the ball.

Abstract

We prove that the only embedded free boundary minimal surface $\Sigma$ in $\mathbb{B}^3$ with index $4$ is the critical catenoid. This extends fundamental work of A. Fraser and R. Schoen, as well as the work of H. Tran.