Chern-Osserman inequality for minimal surfaces in a Cartan-Hadamard manifold with strictly negative sectional curvatures

TL;DR

This paper establishes a Chern-Osserman inequality relating volume growth and geometric properties of minimal surfaces immersed in negatively curved Cartan-Hadamard manifolds.

Contribution

It extends the Chern-Osserman inequality to minimal surfaces in Cartan-Hadamard manifolds with strictly negative sectional curvatures.

Findings

Proves a volume growth inequality for minimal surfaces in negatively curved spaces.

Provides a geometric bound linking curvature and surface volume growth.

Abstract

We state and prove a Chern-Osserman Inequality in terms of the volume growth for minimal surfaces properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity.