Total Curvature of Complete Surfaces in Hyperbolic Space

Abstract

We present a Gauss–Bonnet type formula for complete surfaces in n-dimensional hyperbolic space $$\mathbb{H}^{n}$$ under some assumptions on their asymptotic behaviour. As in recent results for Euclidean submanifolds (see Dillen–Kühnel [4] and Dutertre [5]), the formula involves an ideal defect, i.e., a term involving the geometry of the set of points at infinity.