Mean curvature and compactification of surfaces in a negatively curved Cartan-Hadamard manifold

TL;DR

This paper establishes a Chern-Osserman-type inequality relating volume growth and mean curvature for complete surfaces immersed in negatively curved Cartan-Hadamard manifolds, advancing understanding of geometric inequalities in such spaces.

Contribution

It introduces a new inequality connecting volume growth and mean curvature for surfaces in negatively curved Cartan-Hadamard manifolds, extending classical results to this setting.

Findings

Proves a Chern-Osserman-type inequality in negatively curved manifolds.

Relates volume growth to mean curvature bounds.

Provides geometric bounds for immersed surfaces in Cartan-Hadamard spaces.

Abstract

We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a negative quantity $K_{N}\leq b<0$