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

TL;DR

This paper investigates complete minimal hypersurfaces in hyperbolic space $H^{4}(-1)$, establishing an upper bound on the squared norm of the second fundamental form for hypersurfaces with constant scalar curvature.

Contribution

It provides a new bound on the second fundamental form for 3D minimal hypersurfaces with constant scalar curvature in hyperbolic space, using the Generalized Maximum Principle.

Findings

Established that $S \,\leq\, \frac{21}{29}$ for the hypersurfaces studied.

Applied the Generalized Maximum Principle to derive geometric inequalities.

Focused on the case of 3-dimensional hypersurfaces in $H^{4}(-1)$.

Abstract

In this paper, we study $n$-dimensional complete minimal hypersurfaces in a 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 $S\leq \frac{21}{29}$ by making use of the Generalized Maximum Principle, where $S$ denotes the squared norm of the second fundamental form of the hypersurface.