Area estimates and rigidity of capillary $H-$surfaces in three-manifolds with boundary

TL;DR

This paper establishes area bounds and rigidity results for capillary H-surfaces in three-manifolds with boundary, revealing geometric properties and existence results for special surfaces in hyperbolic three-manifolds.

Contribution

It provides new area estimates and rigidity theorems for capillary H-surfaces, and demonstrates the existence of totally geodesic surfaces under certain conditions.

Findings

Derived area bounds for capillary H-surfaces.

Proved rigidity theorems when bounds are attained.

Established existence of totally geodesic surfaces in hyperbolic manifolds.

Abstract

We obtain a bound for the area of a capillary $H-$surface in a three-manifold with umbilic boundary and controlled sectional curvature. We then analyze the geometry when this area bound is realized, and obtain rigidity theorems. As a side product, we obtain existence of totally geodesic embedded surfaces in hyperbolic three-manifolds under the assumption of the existence of a $H-$surface realizing the area bound, in particular, under the existence of a totally umbilic $H-$surface.