Complete minimal hypersurfaces in a hyperbolic space $H^4(-1)$

Qing-Ming Cheng and Yejuan Peng

Vol. 338 (2025), No. 2, 251–265

DOI: 10.2140/pjm.2025.338.251

Abstract

We study $n$-dimensional complete minimal hypersurfaces in the hyperbolic space $H^{n+1}(-1)$ of constant curvature $-1$. We prove that a $3$-dimensional complete minimal hypersurface with constant scalar curvature in $H^4(-1)$ satisfies $S\le \frac{21}{29}$ by making use of the generalized maximum principle, where $S$ denotes the squared norm of the second fundamental form of the hypersurface.

Keywords

minimal hypersurfaces, hyperbolic space, constant scalar curvature, the generalized maximum principle

Mathematical Subject Classification

Primary: 53C40, 53C42

Milestones

Received: 4 August 2024

Revised: 23 April 2025

Accepted: 3 July 2025

Published: 24 August 2025