Embeddedness of proper minimal submanifolds in homogeneous spaces
type: Item
instance of: scholarly article
title: Embeddedness of proper minimal submanifolds in homogeneous spaces
zbMATH Open document ID: 1282.53055
author: Sung-Hong Min
published in: Journal of Differential Geometry
publication date: 2013-11-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.4140
full work available at URL: https://projecteuclid.org/euclid.jdg/1381931737
reviewed by: Vladimir Yu. Rovenskij
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 6231016
zbMATH Keywords: minimal submanifold
zbMATH Keywords: embeddedness
zbMATH Keywords: geodesic
zbMATH Keywords: Jordan curve
zbMATH Keywords: total curvature
zbMATH Keywords: Möbius transformation
MaRDI profile type: Publication
OpenAlex ID: W2963889475
Wikidata QID: Q115169117
arXiv ID: 1011.4140
DOI: 10.4310/JDG/1381931737
review text: Let \(\Gamma_{2m+1}\) be a piecewise geodesic Jordan curve with \(2m+1\) vertices in \(\mathbb R^n\), \(\mathbb H^n\) or \(\mathbb S^n\), where \(m\geq 2\) is an integer. The author proves three embeddedness results.NEWLINENEWLINE (i) The total curvature of \(\Gamma_{2m+1}\subset\mathbb R^n\) is \(<2m\pi\). In particular, the total curvature of \(\Gamma_5<4\pi\) and thus any minimal surface \(\Sigma\subset\mathbb R^n\) bounded by \(\Gamma_5\) is embedded. For \(\Gamma_5\subset\mathbb H^n\), any minimal surface \(\Sigma\subset\mathbb H^n\) bounded by \(\Gamma_5\) is embedded. If \(\Gamma_5\) is in a geodesic ball of radius \(\pi/4\) in \(\mathbb S^n_+\), then \(\Sigma\subset S^n_+\) is also embedded. As a consequence, \(\Gamma_5\) is an unknot in \(\mathbb R^3\), \(\mathbb H^3\), and \(S^3_+\).NEWLINENEWLINE (ii)~Let \(\Sigma\) be an \(m\)-dimensional proper minimal submanifold in \(H^n\) with the ideal boundary \(\partial_\infty\Sigma=\Gamma\) in the infinite sphere \(\mathbb{S}^{n-1}=\partial_\infty H^n\). If the Möbius volume of \(\Gamma\) obeys \(\Gamma\widetilde{\text{Vol}}(\Gamma)< 2{\text{Vol}}(\mathbb{S}^{m-1})\), then \(\Sigma\) is embedded, and if the equality \(\widetilde{\text{Vol}}(\Gamma)=2{\text{Vol}}(\mathbb{S}^{m-1})\) holds, then \(\Sigma\) is embedded unless it is a cone.NEWLINENEWLINE (iii)~If a proper minimal surface \(\Sigma\) in \(H^2\times R\) is vertically regular at infinity and has two ends, then \(\Sigma\) is embedded.
description: scientific article; zbMATH DE number 6231016