On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto's theorem
type: Item
instance of: scholarly article
title: On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto's theorem
zbMATH Open document ID: 0666.53003
author: Xiaokang Mo
author: Robert Osserman
published in: Journal of Differential Geometry
publication date: 1990-01-01T00:00:00Z
review text: The main results of this paper are: Theorem 1. Let S be a complete minimal surface in \({\mathbb{R}}^ 3\). If the Gauss map g takes on 5 distinct values only a finite number of times, then S has finite total curvature. Theorem 2. Let S be a complete minimal surface in \({\mathbb{R}}^ 3\), not a plane. If the Gauss map g of S omits 4 points on the unit sphere, then every other point must be covered infinitely often. Theorem 3. Let S be a complete minimal surface in \({\mathbb{R}}^ 4\), not a plane. If each of the factors \(g_ k\) of the Gauss map of S omits 4 distinct points, then each of the \(g_ k\) must cover every other point infinitely often. If one of the \(g_ k\) is constant, then the other must cover every point infinitely often with at most 3 exceptions. - Examples are given to show that all of these statements are sharp.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 4090273
zbMATH Keywords: minimal surface
zbMATH Keywords: finite total curvature
zbMATH Keywords: Gauss map
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214444316
OpenAlex ID: W1563569788
Wikidata QID: Q115182290
DOI: 10.4310/JDG/1214444316
description: scientific article; zbMATH DE number 4090273