On surfaces of finite total curvature
type: Item
instance of: scholarly article
title: On surfaces of finite total curvature
zbMATH Open document ID: 0853.53003
published in: Journal of Differential Geometry
publication date: 1997-01-09T00:00:00Z
review text: The authors study two-dimensional manifolds \(M\) immersed in \(\mathbb{R}^n\). \(M\) shall be complete, connected, noncompact and oriented. The second fundamental form is denoted by \(A\). Following a result of Huber for the case \(\int_M |A|^2<+ \infty\) there exists a conformal parametrization \(f: S\setminus \{a_1, \dots, a_q \}\to M\to \mathbb{R}^n\) with a compact Riemann surface \(S\). The authors study \(f\) in the neighbourhood of the ends \(a_i\). They show that these immersions are proper and, if \(\int_M |A|^2\leq 4\pi\) or, for \(n=3\), \(\int_M |A|^2< 8\pi\), \(M\) is embedded.
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 894384
zbMATH Keywords: surfaces
zbMATH Keywords: finite total curvature
zbMATH Keywords: embeddings
zbMATH Keywords: proper immersions
zbMATH Keywords: conformal parametrization
author: Stefan Müller
author: Vladimír Šverák
reviewed by: Otto Röschel
MaRDI profile type: Publication
Wikidata QID: Q115177911
full work available at URL: https://doi.org/10.4310/jdg/1214457233
OpenAlex ID: W2103395362
DOI: 10.4310/JDG/1214457233
description: scientific article; zbMATH DE number 894384