On the heat equation for harmonic maps from noncompact manifolds

Abstract

Harmonic maps are critical points of the energy functional for maps between Riemannian manifolds. In this paper we study the heat equation for harmonic maps from a non-compact manifold M into N. We show that if the target manifold N is compact and has non-positive sectional curvature, and if the initial map has finite total energy, then there exists a solution u(x,t) : M×[0,∞)→N and a sequence t j →∞, such that u(.,t j ) converges on compact subsets of M to a harmonic from M into N. We also obtain some basic properties of the solution u(x,t). In particular, we prove a uniqueness theorem for the solution and a monotonicity theorem for the energy functional