# A new eigenvalue problem for free boundary minimal submanifolds in the unit ball

## TL;DR

This paper introduces a new eigenvalue problem related to free boundary minimal surfaces in the unit ball, revealing that the normal components are eigenfunctions with eigenvalue -2, with implications for spectral analysis and surface characterization.

## Contribution

It proposes a novel eigenvalue problem for the Jacobi operator and links it to the geometry of free boundary minimal surfaces, extending spectral understanding.

## Key findings

- Normal components are eigenfunctions with eigenvalue -2
- Implications for spectral properties of free boundary minimal surfaces
- Potential for characterizing surfaces via spectral data

## Abstract

The purpose of the present paper is to show that the components of the unit normal of any minimal surface with free boundary in the unit ball, are eigenfunctions associated with the eigenvalue $-2$, for some (new) natural eigenvalue problem for the Jacobi operator; this fact has analytic (spectral) consequences for free boundary minimal surfaces in the unit $3$-ball of index $4$, and might prove useful in order to characterize these. This is also in strong analogy with the case of minimal surfaces in $S^3$.