GAUSSIAN CURVATURE AND UNICITY PROBLEM OF GAUSS MAPS OF VARIOUS CLASSES OF SURFACES

Abstract

In this article, we establish a new estimate for the Gaussian curvature of open Riemann surfaces in Euclidean three-space with a specified conformal metric regarding the uniqueness of the holomorphic maps of these surfaces. As its applications, we give new proofs on the unicity problems for the Gauss maps of various classes of surfaces, in particular, minimal surfaces in Euclidean three-space, constant mean curvature one surfaces in the hyperbolic three-space, maximal surfaces in the Lorentz–Minkowski three-space, improper affine spheres in the affine three-space and flat surfaces in the hyperbolic three-space.