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

## TL;DR

This paper introduces a Gauss map for hypersurfaces in spheres and complex projective spaces, establishing its harmonicity characterizes constant mean curvature, and explores the geometric and topological implications of this relationship.

## Contribution

It defines a new Gauss map for hypersurfaces in $ ext{S}^7$ and $ ext{CP}^3$, linking harmonicity to CMC and analyzing geometric and topological properties.

## Key findings

- Gauss map is harmonic iff hypersurface has CMC
- Results relate the image of the Gauss map to hypersurface topology
- Similar constructions and results extend to $ ext{CP}^3$

## Abstract

We define a Gauss map $\gamma:M\rightarrow\mathbb{S}^{6}$ 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.