Conformal variational problems (Q1202744)

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 \(E^ m\).