Complete minimal hypersurfaces in $\mathbb H^5$ with constant scalar curvature and zero Gauss-Kronecker curvature

TL;DR

This paper proves that complete minimal hypersurfaces in five-dimensional hyperbolic space with constant scalar curvature and zero Gauss-Kronecker curvature are necessarily totally geodesic, supporting a related conjecture in higher dimensions.

Contribution

It extends the classification of minimal hypersurfaces with constant scalar curvature to five dimensions, confirming they are totally geodesic under specified conditions.

Findings

Complete minimal hypersurfaces with given conditions are totally geodesic.

Supports Cheng-Peng conjecture in five-dimensional hyperbolic space.

Provides partial confirmation of the conjecture in higher dimensions.

Abstract

We show that any complete minimal hypersurface in the five-dimensional hyperbolic space $\mathbb H^5$, endowed with constant scalar curvature and vanishing Gauss-Kronecker curvature, must be totally geodesic. Cheng-Peng [3] recently conjecture that any complete minimal hypersurface with constant scalar curvature in $\mathbb H^4$ is totally geodesic. Our result partially confirms this conjecture in five dimensional setting.