Index estimates for harmonic Gauss maps

TL;DR

The paper extends Palmer’s genus‑based index bounds for harmonic Gauss maps to constant mean curvature surfaces embedded in 3‑dimensional Lie groups with bi‑invariant metrics. It defines a generalized Gauss map $\mathcal N$ via left translations and proves a Palmer‑style lower bound $Ind_{\mathcal E}(\mathcal N) \ge g(\Sigma)$ for closed surfaces (with $H\neq0$ in the Abelian case), and a bound $Ind_{\mathcal E}(\mathcal N) \ge \dim \mathcal H^1(\Sigma)/2$ for complete noncompact surfaces. The approach combines harmonic map second variation, the invariant shape operator $\mathcal A$, and harmonic vector fields as test variations, together with cut‑off techniques for noncompact settings. The results connect the topology of the surface (genus or $L^2$‑cohomology) to the stability of the harmonic Gauss map, and relate energy stability to area stability, offering a framework to compare indices in Lie‑group ambient geometries and prompting further questions about broader contexts and bounds. These findings generalize classic Ruh–Vilms/Palmer phenomena to homogeneous 3‑manifolds and highlight the role of $L^2$‑cohomology in index theory for noncompact CMC surfaces.

Abstract

Let $Σ$ 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 $Σ$ is bounded below by its topological genus. We also obtain index estimates in the case of complete non compact surfaces.

Theorems & Definitions (12)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• proof
• Lemma 5.1
• proof