Surfaces in $\mathbb{S}^4$ with normal harmonic Gauss maps

TL;DR

This paper studies conformal immersions of Riemann surfaces in the 4-sphere with harmonic Gauss maps, characterizing their properties, integrability, and energy bounds, advancing understanding of geometric structures in differential geometry.

Contribution

It characterizes harmonic Gauss maps for immersions in $b{S}^4$, shows their integrability, and provides energy bounds related to surface genus.

Findings

Harmonic Gauss maps are characterized for conformal immersions in $b{S}^4$.

The normal-harmonic map equation is a completely integrable system.

A lower bound for the normal energy of Gauss maps is established based on genus.

Abstract

We consider conformal immersions of Riemann surfaces in $\bb{S}^4$ and study their Gauss maps with values in the Grassmann bundle $\mathcal{F} = SO_5/T^2 \to \mathbb{S}^4$. The energy of maps from Riemann surfaces into $\mathcal{F}$ is considered with respect to the normal metric on the target and immersions with harmonic Gauss maps are characterized. We also show that the normal-harmonic map equation for Gauss maps is a completely integrable system, thus giving a partial answer of a question posed by Y. Ohnita in \cite{ohnita}. Associated $\mathbb{S}^1$-families of parallel mean curvature immersions in $\mathbb{S}^4$ are considered. A lower bound of the normal energy of Gauss maps is obtained in terms of the genus of the surface.