On the maximal number of exceptional values of Gauss maps for various classes of surfaces

Abstract

The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean three-space, weakly complete improper affine spheres in the affine three-space and weakly complete flat surfaces in the hyperbolic three-space. For this purpose, we give an effective curvature bound for a specified conformal metric on an open Riemann surface.2010 Mathematics Subject Classification. Primary 30D35, 53C42 ; Secondary 30F45, 53A10, 53A15.

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

“…The paper is organized as follows: In Section 2, we first give a curvature bound for the conformal metric ds 2 = (1 + |g| 2 ) m |ω| 2 on an open Riemann surface Σ when all of the multiple values of the meromorphic function g are totally ramified (Theorem 2.1). This is a generalization of Theorem 2.1 in [18], and the proof is given in Section 3.1. As a corollary of this theorem, we give a ramification theorem for the meromorphic function g on Σ with the complete conformal metric ds 2 (Corollary 2.2).…”

“…In [18], we revealed a geometric meaning for the maximal number of omitted values of their Gauss maps. To be precise, we gave a curvature bound for the conformal metric ds 2 = (1 + |g| 2 ) m |ω| 2 on an open Riemann surface Σ, where ω is a holomorphic 1-form and g is a meromorphic function on Σ ([18, Theorem 2.1]) and, as a corollary of the theorem, proved that the precise maximal number of omitted values of the nonconstant meromorphic function g on Σ with the complete conformal metric ds 2 is m + 2 ([18, Corollary 2.2, Proposition 2.4]).…”

Gaussian Curvature and Unicity Problem of Gauss Maps of Various Classes of Surfaces

“…Since ds 2 is complete, we may set Remark 2.9. We note that the number 5 + m in Theorem 2.8 is also optimal (see [18]). §3.…”

“…After that, the relations of the omitted properties or ramifications of the Gauss map and the Gaussian curvature of minimal surfaces have been studied (see [14,18,20,30,34,36] for some newest results).…”

“…For instance, Yu [43] showed that the hyperbolic Gauss map of a nonflat complete constant mean curvature one surface in hyperbolic three-space H 3 can omit at most four values, Kawakami and Nakajo [23] obtained that the maximal number of omitted values of the Lagrangian Gauss map of weakly complete improper affine spheres (or improper affine fronts) in the affine three-space R 3 is 3, unless it is an elliptic paraboloid, or Kawakami [19] gave similar results for flat fronts in H 3 . Furthermore, Kawakami [18,20] elucidated the geometric interpretation 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, 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 hyperbolic three-space (also see the review article [21] for instance).…”