On the index of minimal surfaces with free boundary in a half-space

TL;DR

This paper investigates the Morse index of free boundary minimal surfaces in a half-space, improving estimates and addressing questions about their non-existence, with implications for the classification of such surfaces.

Contribution

It refines the relationship between Neumann and Dirichlet indices and resolves an open question on the non-existence of certain minimal surfaces with free boundary.

Findings

Improved estimates relating Neumann and Dirichlet indices.

Proved non-existence of index two embedded minimal surfaces with free boundary.

Provided a simplified proof for lower bounds on the index of Costa deformations.

Abstract

We study the Morse index of minimal surfaces with free boundary in a half-space. We improve previous estimates relating the Neumann index to the Dirichlet index and use this to answer a question of Ambrozio, Buzano, Carlotto, and Sharp concerning the non-existence of index two embedded minimal surfaces with free boundary in a half-space. We also give a simplified proof of a result of Chodosh and Maximo concerning lower bounds for the index of the Costa deformation family.