On the Gauss Map of Anisotropic Minimal Surfaces and applications to the Morse Index estimates

TL;DR

This paper investigates the Gauss map of anisotropic minimal surfaces, proving the discreteness of its critical set and deriving bounds for the Morse index using conformal geometric techniques.

Contribution

It introduces a local analysis of the Gauss map for anisotropic minimal surfaces, establishing its critical set as discrete and linking the Gauss map to Morse index estimates.

Findings

Discreteness of the critical set of the Gauss map

Gauss map as a branched covering to the sphere

Bounds for Morse index of anisotropic minimal surfaces

Abstract

In the paper, we study the Gauss map of a completely immersed anisotropic minimal surface with respect to convex parametric integrand in $\mathbb{R}^3$. By a local analysis, we prove the discreteness of the critical set of the Gauss map of an anisotropic minimal surface. In particular, we may consider the Gauss map as a branched covering map from an anisotropic minimal surface to the unit sphere. As a consequence, we may obtain an upper and a lower estimate for the Morse index of an anisotropic minimal surface by applying some conformal geometric technics to the Gauss map.