An obstruction to the existence of immersed curves of prescribed curvature
type: Item
instance of: scholarly article
title: An obstruction to the existence of immersed curves of prescribed curvature
zbMATH Open document ID: 1188.53004
author: Stephane Kirsch
author: Paul Laurain
published in: Potential Analysis
publication date: 2010-01-29T00:00:00Z
review text: The main result is the following: Theorem 1.2. Let \(H:\mathbb{R}^2\to\mathbb{R}\) be a positive Lipschitz function such that there exists \(\overrightarrow{e}\in\mathbb{R}^2\) verifying \[ \langle\nabla H(x),\overrightarrow{e}\rangle>0,\quad \text{for almost every}\;\;x\in\mathbb{R}^2. \] Then there exists no \(C^1\)-closed curve immersed in \(\mathbb{R}^2\) whose curvature at every point \(x\) is \(H(x)\). The related result for \(C^1\)-closed hypersurfaces embedded in \(\mathbb{R}^n\) of mean curvature \(H\), namely the theorem 1.1 in this paper, is proved by the first author in his thesis (PIMS, UBC, Canada). The authors want to investigate a generalization of the result of the first author in the immersed case.
reviewed by: Weihuan Chen
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 5664591
zbMATH Keywords: immersed curve
zbMATH Keywords: prescribed curvature
MaRDI profile type: Publication
full work available at URL: s11118-009-9142-8
OpenAlex ID: W1972197120
cites work: Q3283395
cites work: An obstruction for the mean curvature of a conformal immersion 𝑆ⁿ→ℝⁿ⁺¹
cites work: Q4039783
cites work: Scalar Curvature Functions in a Conformal Class of Metrics and Conformal Transformations
cites work: Some Integral Formulas for Hyper-Surfaces in Euclidean Spaces
cites work: Q4838489
cites work: Differential geometry in the large. Seminar lectures New York University 1946 and Stanford University 1956. With a preface by S. S. Chern.
cites work: Curvature functions for compact 2-manifolds
cites work: The Jordan-Brouwer Separation Theorem for Smooth Hypersurfaces
DOI: 10.1007/S11118-009-9142-8
Recommended article: Immersions with bounded curvature
Recommended article: Q6058497
Recommended article: An obstruction for the mean curvature of a conformal immersion 𝑆ⁿ→ℝⁿ⁺¹
Recommended article: Q3802203
Recommended article: Obstruction to prescribed positive Ricci curvature
Recommended article: Intrinsic obstructions to the existence of isometric minimal immersions.
Recommended article: An obstruction to removing intersection points in immersed surfaces
Recommended article: The total absolute curvature of an equiaffine immersion
Recommended article: Obstruction to the existence of metric whose curvature has umbilical Hessian in a \(K\)-surface
Recommended article: Bounding immersed curves
description: scientific article; zbMATH DE number 5664591