On the index of the critical M\"obius band in $\mathbb B^4$

TL;DR

This paper proves that the Morse index of the critical M"obius band in 4D Euclidean ball is 5, and explores spectral and energy index relationships, also providing a new proof for the catenoid's index in 3D.

Contribution

It establishes the Morse index of the critical M"obius band in $\

Findings

Morse index of the critical M"obius band in $\

Comparison theorem between spectral and energy index

New proof of the catenoid's index in $\

Abstract

In this paper we prove that the Morse index of the critical M\"obius band in the $4-$dimensional Euclidean ball $\mathbb B^4$ equals 5. It is conjectured that this is the only embedded non-orientable free boundary minimal surface of index 5 in $\mathbb B^4$. One of the ingredients in the proof is a comparison theorem between the spectral index of the Steklov problem and the energy index. The latter also enables us to give another proof of the well-known result that the index of the critical catenoid in $\mathbb B^3$ equals 4.