Diameter estimates for surfaces in conformally flat spaces

Abstract

The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three dimensional Riemannian manifold in terms of the integral of the mean curvature and of the length of its boundary. Of particular interest is the application of the inequality to minimal surfaces in the three-sphere and in the hyperbolic space. Here the result implies an a priori estimate for connected solutions of Plateau’s problem, as well as a necessary condition on the boundary data for the existence of such solutions. The proof follows a construction of Miura and uses a diameter bound for closed surfaces obtained by Topping and Wu–Zheng.