Ramification estimates for the hyperbolic Gauss map

TL;DR

This paper establishes optimal bounds on the number of exceptional and ramified values of the hyperbolic Gauss map for certain constant mean curvature surfaces in hyperbolic space, and explores related value distribution problems.

Contribution

It provides the best possible upper bounds for the hyperbolic Gauss map's ramification and extends results to the de Sitter space setting.

Findings

Optimal upper bounds on exceptional values for hyperbolic Gauss map

Partial results on Osserman problem for algebraic surfaces

Analysis of value distribution in de Sitter space

Abstract

We give the best possible upper bound on the number of exceptional values and the totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic constant mean curvature one surfaces in the hyperbolic three-space and some partial results on the Osserman problem for algebraic case. Moreover, we study the value distribution of the hyperbolic Gauss map for complete constant mean curvature one faces in de Sitter three-space.