Conformal variational problems
type: Item
instance of: scholarly article
title: Conformal variational problems
zbMATH Open document ID: 0766.53035
DOI: 10.1007/BF02844662
author: Aristide Sanini
publication date: 1993-02-03T00:00:00Z
review text: In an earlier paper [Rend. Mat. Appl., III. Ser. 3, 53-63 (1983; Zbl 0514.53037)] the author studied the conditions for a map \(f:(M^ m,g)\to (\overline{M},\overline{g})\) (\(M\) is compact) to have critical energy with respect to deformations of the metric on \(M\). On the other hand, \textit{K. Uhlenbeck} [Semin. on minimal submanifolds, Ann. Math. Stud. 103, 169-176 (1983; Zbl 0535.53050)] introduced the functional \(E^ m(f)\) generalizing the energy and depending only on the conformal class chosen on \(M\). Using similar methods to those in the first quoted paper the work under review studies the dependence of \(E^ m\) on the metric \(g\) and on the map \(f\). It is shown that \(E^ m\) is stationary with respect to any deformation of \(g\) iff \(f\) is weakly conformal. Further, \(E^ m\) is stationary with respect to deformations of \(f\) iff the \(m\)-tension field \(\tau^ m(f)\) (associated to \(E^ m(f)\) in the same way as the usual tension field \(\tau(f)\) to the energy \(E(f)\)) is identically zero. A special attention is paid in the last section to real valued maps which are critical for the functional \(E^ m\).
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 109281
zbMATH Keywords: energy functional
zbMATH Keywords: weakly conformal maps
zbMATH Keywords: critical energy
zbMATH Keywords: tension field
reviewed by: Liviu Ornea
MaRDI profile type: Publication
cites work: Q3662315
cites work: Q3945267
cites work: Quelques formules de variation pour une structure riemannienne
cites work: A Report on Harmonic Maps
cites work: Q4080368
cites work: Q3660479
cites work: Q3319388
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 109281