On cylindrically bounded $H$-Hypersurfaces of $\mathbb{H}^{n}\times \mathbb{R}$

TL;DR

This paper investigates the geometric properties of $H$-hypersurfaces in hyperbolic space cross a line, showing that those confined in a cylinder with specific Ricci curvature decay have mean curvature exceeding a certain bound.

Contribution

It establishes a new lower bound on the mean curvature of cylindrically bounded $H$-hypersurfaces with Ricci curvature decay in hyperbolic space cross a line.

Findings

$|H| > (n-1)/n$ for the considered hypersurfaces

Ricci curvature with strong quadratic decay influences mean curvature bounds

Hypersurfaces contained in a vertical cylinder are characterized by this curvature condition

Abstract

We show that $H$-hypersurfaces of $\mathbb{H}^{n}\times \mathbb{R}$ contained in a vertical cylinder and with Ricci curvature with strong quadratic decay have mean curvature $| H| > (n-1)/n$.