Relationship between volume and energy of vector fields.
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.
scientific article; zbMATH DE number 1959883
scientific article