Total curvature of complete surfaces in hyperbolic space

TL;DR

This paper establishes a Gauss-Bonnet formula for the extrinsic curvature of complete surfaces in hyperbolic space, linking geometric and conformal invariants under certain asymptotic conditions.

Contribution

It introduces a novel Gauss-Bonnet formula relating extrinsic curvature, geodesic measures, and conformal invariants for surfaces in hyperbolic space.

Findings

Gauss-Bonnet formula for hyperbolic surfaces derived

Relation between geodesic intersection measure and curvature established

Conformal invariant at infinity incorporated into curvature analysis

Abstract

We prove a Gauss-Bonnet formula for the extrinsic curvature of complete surfaces in hyperbolic space under some assumptions on the asymptotic behaviour. The result is given in terms of the measure of geodesics intersecting the surface non-trivially, and of a conformal invariant of the curve at infinity.