Geometric rigidity for analytic estimates of Müller-Šverák
type: Item
instance of: scholarly article
title: Geometric rigidity for analytic estimates of Müller-Šverák
zbMATH Open document ID: 1282.53008
author: Huy The Nguyen
published in: Mathematische Zeitschrift
publication date: 2012-12-19T00:00:00Z
review text: The author improves results by \textit{S. Müller} and \textit{V. Šverák} [J. Differ. Geom. 42, No. 2, 229--258 (1995; Zbl 0853.53003)] on complete, connected, non-compact surfaces \(\Sigma\) immersed into \(\mathbb R^n\). He presents two new main results: Firstly, in dimension \(n=3\) for surfaces with total curvature \(\int_\Sigma \left|A\right|^2 < 8 \pi\) the author shows that the surface is either conformal to a plane and embedded, isometric to a catenoid and embedded or is isometric to Enneper's minimal surface and not embedded. In dimension \(n=4\) a surface with total curvature \(\int_\Sigma\left|A\right|^2 \leq 4 \pi\) is embedded and either conformal to the plane or isometric to a minimal surface. The second main result is devoted to surfaces with \(\int_\Sigma K =0\) and \(\int_\Sigma \left|A\right|^2 \leq 16 \pi - 4 \delta\) with some \(\delta > 0\). These surfaces are embedded and conformal to a plane. For \(\delta = 0\) the surface is either embedded or \(\Sigma\) is isometric to the Hoffmann-Ossermann cylinder inverted with respect to a point on that cylinder.
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 6116863
zbMATH Keywords: surfaces with finite total curvature
zbMATH Keywords: surfaces conformal to the plane
zbMATH Keywords: Willmore energy
zbMATH Keywords: Hoffmann-Ossermann cylinder
reviewed by: Otto Röschel
MaRDI profile type: Publication
full work available at URL: s00209-011-0974-8
OpenAlex ID: W1921801191
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DOI: 10.1007/S00209-011-0974-8
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description: scientific article; zbMATH DE number 6116863