Index estimates for harmonic Gauss maps

TL;DR

This paper establishes lower bounds on the energy index of harmonic Gauss maps for constant mean curvature surfaces in 3D Lie groups, linking geometric analysis with topological properties.

Contribution

It provides the first index estimates for harmonic Gauss maps in the setting of surfaces in Lie groups with bi-invariant metrics.

Findings

Energy index is bounded below by the surface's genus.

Index estimates are extended to complete non-compact surfaces.

Results connect geometric analysis with topological invariants.

Abstract

Let $\Sigma$ denote a closed surface with constant mean curvature in $\mathbb{G}^3$, a 3-dimensional Lie group equipped with a bi-invariant metric. For such surfaces, there is a harmonic Gauss map which maps values to the unit sphere within the Lie algebra of $\mathbb{G}$. We prove that the energy index of the Gauss map of $\Sigma$ is bounded below by its topological genus. We also obtain index estimates in the case of complete non compact surfaces.