Comparison between Second Variation of Area and Second Variation of Energy of a Minimal Surface

TL;DR

This paper compares the Morse indices of minimal surfaces as critical points of area and energy functionals, providing bounds on their stability and index in various geometric contexts.

Contribution

It establishes a relationship between the Morse indices of minimal surfaces for area and energy functionals, offering new bounds based on topology and ambient geometry.

Findings

The difference in Morse indices is at most the dimension of Teichmuller space.

Provides upper bounds on the index of minimal surfaces in Euclidean space.

Bounds the index in Riemannian manifolds using surface genus and ambient geometry.

Abstract

The conformal parameterisation of a minimal surface is harmonic. Therefore, a minimal surface is a critical point of both the energy functional and the area functional. In this paper, we compare the Morse index of a minimal surface as a critical point of the area functional with its Morse index as a critical point of the energy functional. The difference between these indices is at most the real dimension of Teichmuller space. This comparison allows us to obtain surprisingly good upper bounds on the index of minimal surfaces of finite total curvature in Euclidean space of any dimension. We also bound the index of a minimal surface in an arbitrary Riemannian manifold by the area and genus of the surface, and the dimension and geometry of the ambient manifold.