Bounds for the Morse index of free boundary minimal surfaces

TL;DR

This paper establishes bounds on the Morse index of free boundary minimal surfaces, relating it to geometric and topological properties, and extends these bounds to higher-dimensional submanifolds.

Contribution

It introduces new bounds for the Morse index of free boundary minimal surfaces based on area and topology, extending to higher dimensions.

Findings

Area index is controlled by surface area and topology.

In convex Euclidean domains, index is bounded by genus and boundary components.

Provides index bounds for higher-dimensional minimal submanifolds.

Abstract

Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of Fraser-Li, we conclude that the area index of a free-boundary minimal surface in a convex domain of Euclidean three-space, is bounded from above by a linear function of its genus and its number of boundary components. We also prove index bounds for submanifolds of higher dimension.