A note on surfaces with prescribed oriented Euclidean Gauss map

Ricardo Sa Earp,Eric Toubiana

doi:10.1155/ijmms.2005.537

Abstract

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

Highlights

Throughout this paper we will consider surfaces immersed in Euclidean space thinking that, locally, the surface is immersed into the upper half-space R3+ = {(u, v, w), w > 0}
There are two important metrics in R3+: the standard Euclidean metric and the hyperbolic metric given by d s2 = (1/w2) · (d u2 + d v2 + d w2)
Our main goal is to show how certain geometric quantities relating to the oriented Euclidean Gauss map, the hyperbolic Gauss map, and the coordinate functions for conformal immersions of a planar domain into the upper half-space model of hyperbolic space, can be used to infer that the oriented Euclidean Gauss map determines locally a conformal immersion in Euclidean space with nonvanishing mean curvature, up to a homothety and Euclidean translation

Summary

A NOTE ON SURFACES WITH PRESCRIBED ORIENTED EUCLIDEAN GAUSS MAP

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss map G and formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper

Introduction

Proof of the theorem