# Area Estimates and Rigidity of Non-compact $H$-Surfaces in 3-Manifolds

## TL;DR

This paper establishes area estimates for non-compact constant mean curvature surfaces in negatively curved 3-manifolds and explores rigidity conditions, showing that equality cases often imply a hyperbolic structure, with exceptions for minimal surfaces.

## Contribution

It provides new area bounds for non-compact H-surfaces and characterizes rigidity in certain cases, extending understanding of geometric properties in negatively curved 3-manifolds.

## Key findings

- Area estimates for non-compact H-surfaces with finite topology and area
- Rigidity results showing isometry to hyperbolic Fuchsian manifolds under equality
- Counter-example demonstrating lack of rigidity for minimal surfaces at equality

## Abstract

For appropriately values of $H$, we obtain an area estimate for a complete non-compact $H$-surface of finite topology and finite area, embedded in a three-manifold of negative curvature. Moreover, in the case of equality and under additional assumptions, we prove that a neighbourhood of the mean convex side of the surface must be isometric to a hyperbolic Fuchsian manifold. Also, we show by an counter-example that although that area estimate holds for minimal surfaces, one does not have rigidity for equality in this case.