Relationship between volume and energy of vector fields.
type: Item
instance of: scholarly article
title: Relationship between volume and energy of vector fields.
zbMATH Open document ID: 1066.53068
DOI: 10.1016/S0926-2245(01)00053-5
author: O. Gil-Medrano
published in: Differential Geometry and its Applications
publication date: 2003-08-06T00:00:00Z
review text: The energy and volume on the space of smooth maps \(C^\infty(M,N)\) and smooth immersions \(\text{Imm}(M,N)\) give rise to the concepts of harmonic maps and minimal immersions provided that suitable Riemannian metrics are given on the ambient manifolds. By varying the Riemannian metric on \(M\) and considering the energy and volume as functions on \(C^\infty(M,N)\times{\mathcal M}\), where \({\mathcal M}\) is the space of Riemannian metrics on \(M\), the author gives a unified treatment of these functionals. This approach is then applied to the submanifold \({\mathcal X}(M)\times{\mathcal M}\) with \({\mathcal X}(M)\) the space of vector fields on \(M\) (viewed as maps \(M\to T(M)\)). Various known and new results are obtained for the critical points of the energy and volume functionals in different settings.
reviewed by: Gábor Tóth
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 1959883
zbMATH Keywords: energy functional
zbMATH Keywords: volume of an immersion critical points
zbMATH Keywords: variational problems
Wikidata QID: Q115338266
MaRDI profile type: Publication
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full work available at URL: s0926-2245(01)00053-5
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description: scientific article; zbMATH DE number 1959883