Nonexistence results for extremals of curvature functionals
type: Item
instance of: scholarly article
title: Nonexistence results for extremals of curvature functionals
zbMATH Open document ID: 0907.49022
author: Heiko von der Mosel
published in: Archiv der Mathematik
publication date: 1999-02-08T00:00:00Z
review text: The author considers variational integrals of the form \[ F(X)= \int_X f(H, K)dA, \] where \(H\) and \(K\) denote mean and Gaussian curvature of the immersed surface \(X:\overline B\to \mathbb{R}^3\) defined on the closed unit disc \(\overline B\) in \(\mathbb{R}^2\). The integrand \(f\) is given by \(f(H, K)= \alpha+ \beta(H- H_0)^2- \gamma K\) with constants \(\alpha, \beta, \gamma\geq 0\) and \(H_0\in\mathbb{R}\). It is shown that any tubular surface solving the Euler equation for \(F\) is a circular cylinder, a torus or a surface consisting of pieces of cylinders and tori, and a smooth ruled surface \(S\) cannot satisfy the Euler equation for \(\beta H_0\neq 0\) unless \(S\) is locally cylindrical.
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1145773
zbMATH Keywords: curvature functionals
zbMATH Keywords: variational integrals
reviewed by: M. Fuchs
MaRDI profile type: Publication
full work available at URL: s000130050141
OpenAlex ID: W2027075884
DOI: 10.1007/S000130050141
description: scientific article; zbMATH DE number 1145773