Gauss map and the topology of constant mean curvature hypersurfaces of ${\mathbb{S}}^7$ and ${\mathbb{CP}}^3$

Abstract

We define a Gauss map $\gamma :M\to {\mathbb{S}}^6$ of an oriented hypersurface $M$ of the unit sphere ${\mathbb{S}}^7$ and prove that $\gamma$ is harmonic if and only if $M$ has CMC. Results on the geometry and topology of CMC hypersurfaces of ${\mathbb{S}}^7$, under hypothesis on the image of $\gamma$, are then obtained. By a Hopf symmetrization process we define a Gauss map for hypersurfaces of ${\mathbb{CP}}^3$ and obtain similar results for CMC hypersurfaces of this space.

20 pages