Constant Mean Curvature Surfaces in Hyperbolic Space

Abstract

This chapter is devoted to the study of compact surfaces with constant mean curvature in hyperbolic space $\mathbb{H}^{3}$. Although many questions have its origin in Euclidean space, there are some differences, emphasizing the variety of umbilical surfaces of $\mathbb{H}^{3}$ because besides totally geodesic surfaces and spheres, there are equidistant surfaces and horospheres. We shall investigate if the umbilical surfaces are the only possible examples with boundary a circle. The tangency principle and the hyperbolic version of the flux formula will give us a certain control and information about the shape of a compact cmc surface of $\mathbb{H}^{3}$ in terms of the geometry of its boundary. When the boundary is a circle, we shall prove that a compact H-surface spanning a circle and |H|≤1 must be umbilical. Finally, we end the chapter studying cmc surfaces bounded by a circle under the hypothesis that the surface is a topological disk or stable.