Total bending of flows with mean curvature correction
type: Item
instance of: scholarly article
title: Total bending of flows with mean curvature correction
zbMATH Open document ID: 0995.53023
DOI: 10.1016/S0926-2245(00)00007-3
published in: Differential Geometry and its Applications
publication date: 2002-10-17T00:00:00Z
review text: The author introduces the functional \[ D(\vec{v}) = \int_M \left( \|\nabla \vec{v} \|^2 +\;(n-1) (n-3) \|\vec{H}_{\vec{v}_{\bot}}\|^2 \right) dM, \] on the unit vector fields \(\vec v\) of a closed Riemannian manifold \(M\), where \(\vec{H}_{\vec{v}_{\bot}}\) is the mean curvature vector of the distribution \(\vec{v}_{\bot}\), orthogonal to \(\vec{v}\). He shows that Hopf flows uniquely minimize \(D\) on odd-dimensional unit spheres. On \(S^3\) the functional \(D\) coincides, up to a constant, with the total bending of a unit vector field as defined by G. Wiegmink. Therefore it follows that Hopf flows minimize \(B\) on \(S^3\) and, moreover, these are the unique absolute minima of \(B\). The paper also extends a result of H. Gluck and W. Ziller on the lower bound for the volume of vector fields on \(S^3.\)
reviewed by: Alina Stancu
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1485492
zbMATH Keywords: Hopf flows
zbMATH Keywords: mean curvature of a distribution
zbMATH Keywords: total bending
Wikidata QID: Q115338447
author: Fabiano G. B. Brito
MaRDI profile type: Publication
full work available at URL: s0926-2245(00)00007-3
OpenAlex ID: W2069746143
description: scientific article; zbMATH DE number 1485492