I was pleased to find, in joint work with Jaigyoung Choe , that the sharp isoperimetric inequality for hyperbolic 2 - space also holds on a two-dimensional minimal surface in hyperbolic n - space, providing its boundary is connected or even just radially connected [35]. Choe has extended this result to a minimal surface whose boundary is radially connected (from some point of the surface) in a Riemannian n-manifold with sectional curvatures less than or equal to -1. See J. reine angewandte Mathematik 506 (1999), 205-214).

[35]. The Sharp Isoperimetric Inequality for Minimal Surfaces with Radially Connected Boundary in Hyperbolic Space (with Jaigyoung Choe), Inventiones Math. 109, 495-503 (1992).