The space of embedded minimal surfaces of fixed genus in a 3-manifold. V. Fixed genus
type: Item
instance of: scholarly article
title: The space of embedded minimal surfaces of fixed genus in a 3-manifold. V. Fixed genus
zbMATH Open document ID: 1322.53059
published in: Annals of Mathematics. Second Series
publication date: 2015-01-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0509647
review text: The following compactness questions naturally arise when considering a class of submanifolds of a fixed complete Riemannian manifold \((M, g)\): (1) Given a sequence \(\Sigma_i\) of smoothly immersed submanifolds, to what extent can we say that, up to passing to a subsequence, the \(\Sigma_i\) converge to a limit \(\Sigma\)? (2) Moreover, assuming that all the \(\Sigma_i\) have a property \(\mathcal{P}\), to what extent does this affect the answer to the previous question and to what extent does \(\Sigma\) have the property \(\mathcal{P}\)? Answering this type of questions has important consequences because, often, when studying a class of submanifolds useful information can be obtained about the class by considering to what extent the class is compact and by analyzing how failure of compactness may happen. The paper under review is the fifth of a series of papers containing Colding and Minicozzi's recent work regarding compactness properties of embedded minimal surfaces in three-manifolds. They mainly study the failure of smooth convergence for sequences of embedded minimal surfaces in \(\mathbb{R}^3\) with bounded genus because, as they see, the key is to understand the structure of an embedded minimal planar domain in a ball in \(\mathbb{R}^3\), that is, the case when the surfaces have genus zero, since the general case of fixed genus requires only minor changes. In the first four papers, they considered the case of minimal disks. Roughly speaking, they characterized the failure of smooth compactness for embedded minimal disks in \(\mathbb{R}^3\), with the failure being always modelled on the helicoid. Their proof utilized and combined many properties of embedded minimal surfaces in \(\mathbb{R}^3\) and, in particular, they proved the so-called one-sided curvature estimate, which gives a uniform curvature bound for an embedded minimal disk that is close to, and on one side of, a plane. The focus in this paper is on non-simply connected planar domains. Sequences o…
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58D10
zbMATH DE Number: 6383661
zbMATH Keywords: lamination
zbMATH Keywords: minimal surfaces
zbMATH Keywords: multi-valued graphs
zbMATH Keywords: no-mixing theorem
zbMATH Keywords: Riemann examples
author: Tobias Holck Colding
author: William P. II Minicozzi
reviewed by: Oscar J. Garay
MaRDI profile type: Publication
OpenAlex ID: W2140037202
arXiv ID: math/0509647
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DOI: 10.4007/ANNALS.2015.181.1.1
description: scientific article