Index Characterization for Free Boundary Minimal Surfaces

TL;DR

This paper develops a method to compute the Morse index of free boundary minimal submanifolds using related fixed boundary and Dirichlet-to-Neumann problems, with an application to characterizing the critical catenoid.

Contribution

It introduces a novel approach to determine the Morse index of free boundary minimal surfaces based on auxiliary problems, providing a new characterization of the critical catenoid.

Findings

Morse index of a free boundary minimal annulus is 4 if and only if it is the critical catenoid.

Provides a computational framework linking fixed boundary problems and Jacobi fields to Morse index.

Enhances understanding of stability properties of free boundary minimal surfaces.

Abstract

In this paper, we compute the Morse index for a free boundary minimal submanifold from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. As an application, we show that the Morse index of a free boundary minimal annulus is equal to 4 if and only if it is the critical catenoid.