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

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

2011-03-12

arxiv.org/abs/1103.2485v1

Mathematics

Differential Geometry

26 pages, this version v1

Scientific paper

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.

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