Index bounds for closed minimal surfaces in 3-manifolds with the Killing property

TL;DR

This paper establishes a lower bound on the combined Morse index and nullity of closed minimal surfaces in certain 3-manifolds, linking these spectral properties to the surface's genus.

Contribution

It provides the first index bounds for minimal surfaces in 3-manifolds with a Killing property, including Lie groups with bi-invariant metrics.

Findings

Sum of Morse index and nullity is bounded below by a constant times genus

Applicable to 3-manifolds with orthonormal Killing frames including Lie groups

Advances understanding of stability and spectral properties of minimal surfaces

Abstract

Let $\Sigma$ be a closed minimal surface immersed in a Riemannian 3-manifold carrying an orthonormal Killing frame. This class of ambient spaces includes Lie groups with a bi-invariant metric. In this paper, we prove that the sum of the Morse index and the nullity of $\Sigma$ is bounded from below by a constant times its genus.