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The classical result of Nevanlinna states that two nonconstantmeromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

In this paper, we improve the Rosenbloom's fixed-point theorem and prove a related normality criterion. We also consider the corresponding unicity theorem for transcendental entire functions.

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A Note on a Unicity Theorem for the Gauss Maps of Complete Minimal Surfaces in Euclidean Four-space

Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

On Rosenbloom's fixed-point theorem and related results

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3 results

A Note on a Unicity Theorem for the Gauss Maps of Complete Minimal Surfaces in Euclidean Four-space

Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

On Rosenbloom's fixed-point theorem and related results