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

TL;DR

This work proves that a complete minimal hypersurface in $\mathbb{H}^5$ with constant scalar curvature and zero Gauss-Kronecker curvature must be totally geodesic. The authors deploy Peng-Terng and Simons identities together with the Omori–Yau maximum principle to analyze the cubic trace function $f_3$ of the second fundamental form, performing a detailed case analysis on its extremal values. Their three-way contradiction argument shows rigidity under the given conditions, contributing a higher-dimensional analogue to conjectures about constant-scalar-curvature minimal hypersurfaces in hyperbolic space. The result provides a Bernstein-type rigidity in $\mathbb{H}^5$ under the zero Gauss-Kronecker curvature assumption, partially supporting Cheng-Peng’s conjecture in dimension four extended to five.

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.

Theorems & Definitions (9)

• Conjecture : ChePen24
• Theorem 1
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• proof : Proof of Theorem \ref{['mainthm']}