A variational problem for pullback metrics

Abstract

Let (M, g) and (N, h) be Riemannian manifolds without boundary and let f : M → N be a smooth map. Let ${\|f^*h\|}$ denote the norm of the pullback metric of h by f. In this paper, we consider the functional ${{\Phi (f) = \int_M \|f^*h\|^2 dv_g}}$. We prove the existence of minimizers of the functional Φ in each 3-homotopy class of maps, where maps f ₁ and f ₂ are 3-homotopic if they are homotopic on the three dimensional skeltons of a triangulation of M. Furthermore, we give a monotonicity formula and a Bochner type formula.