Index of the critical catenoid

TL;DR

This paper proves that the critical catenoid is a minimal surface with index 4 and establishes that all non-flat free boundary minimal surfaces in the unit ball have index at least 4.

Contribution

It demonstrates the index of the critical catenoid and provides a lower bound for the index of other free boundary minimal surfaces in the unit ball.

Findings

Critical catenoid has index 4.

Non-flat free boundary minimal surfaces have index ≥ 4.

Provides insight into stability properties of minimal surfaces.

Abstract

In this article, we show that the critical catenoid, as a free boundary minimal surface of the unit ball in $\mathbb{R}^3$, has index $4$. We also prove that a free boundary minimal surface of the unit ball in $\mathbb{R}^3$, that is not a flat disk, has index at least $4$.