# A Picard type theorem for Hyperbolic Gauss Map of CMC-1 Surfaces in Hyperbolic 3-space and de Sitter 3-space

## TL;DR

This paper extends Picard type theorems to the hyperbolic Gauss map of CMC-1 surfaces in hyperbolic and de Sitter spaces, showing limitations on the image omissions similar to classical minimal surface results.

## Contribution

It establishes a Picard type theorem for the hyperbolic Gauss map of CMC-1 surfaces in hyperbolic and de Sitter spaces, generalizing known minimal surface theorems.

## Key findings

- The hyperbolic Gauss map omits at most 3 points for certain CMC-1 surfaces.
- The results apply to surfaces with finite total curvature and regular ends.
- The approach adapts techniques from minimal surface theory to CMC-1 contexts.

## Abstract

In a recent paper Jorge and Mercuri proved that the image of Gauss map of a complete non flat minimal surfaces in R3 with finite total curvature omits at most 2 points. In this work we follow their idea and prove 3a similar result for CMC-1 with finite total curvature in H and CMC-1 faces with finite type and regular ends in S31.