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.

Classification of Complete Degenerate Stationary Surfaces in R3,1 Whose Gauss Maps Are Injective

“…Park and M. Ru [22] gave the generalizations of the unicity theorem to minimal surfaces in R n (n > 3). After that, many mathematicians studied the unicity theorem for Gauss maps of minimal surfaces (for more details, see [23][24][25][26]).…”