A duality on $CMC$-1 surfaces in hyperbolic space,and a hyperbolic analogue of the Osserman Inequality

Abstract

We will show the existence of a duality on $CMC$-1 surfaces in hyperbolic 3-space, and we will show an analogue of the Osserman Inequality in terms of dual surfaces. Moreover, we will show that equality holds (in this analogue) if and only if all the ends of the surface are regular and embedded.