Gauss curvature of Gaussian image of minimal surfaces
type: Item
instance of: scholarly article
title: Gauss curvature of Gaussian image of minimal surfaces
zbMATH Open document ID: 0787.53054
author: Haizhong Li
published in: Kodai Mathematical Journal
publication date: 1993-09-07T00:00:00Z
review text: Let \(M\) be a minimal surface lying linearly full in a real space form \(R^ n(c)\) of constant curvature \(c\). The Gauss map of \(M\) degenerates at points where the Gauss curvature \(K\) of \(M\) equals \(c\). So assume \(c- K\neq 0\) on \(M\), and denote the Gauss curvature of the Gauss map by \(\overline K_ G\). It is known that, if \(n=3\), then \(\overline {K_ G} = 1-c/(c-K)\) and if \(c = 0\) and \(n \geq 4\), then \(\overline {K_ G} \leq 2\) and \(\overline {K_ G} = 2\) if and only if \(M\) is a complex curve in \(\mathbb{C}^ 2\). In this paper, the author extends the latter result. In particular, he proves that \(\overline {K_ G}\leq 2-c/(c-K)\) with equality if and only if \(n = 4\) and \(M\) is superminimal (i.e. the image of the unit circle in the tangent plane by the second fundamental form is a circle in the normal plane).
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 409643
zbMATH Keywords: Gauss map
zbMATH Keywords: complex curve
zbMATH Keywords: superminimal
reviewed by: Franki Dillen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039705
OpenAlex ID: W1989144167
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Wikidata QID: Q125967171
DOI: 10.2996/KMJ/1138039705
description: scientific article; zbMATH DE number 409643