Heat equation for harmonic maps of the compactification of complete manifolds

Abstract

for i = 1 ..... n, where f : M -+ N is a given initial map which is C 1 with bounded energy density. (This assumption on the initial value f is a standing one throughout this paper.) This harmonic map heat equation is a nonlinear parabolic system which proved to be useful in the study of harmonic maps. The nonlinear terms, which are due to the curvature of the target manifold, give distinct geometric flavor to this problem. The natural question is then how to understand the interplay between these terms and the target geometry. It is universally accepted that a harmonic map behaves much nicer when the target manifold has negative curvature. It is usually due to the geometric inequalities that come from the negativity of curvature of the target manifold. We like to go one step further in this direction. Namely, we would like to propose the thesis that a harmonic map heat equation behaves like a bunch of uncoupled linear equations when the target manifold has nonpositive sectional curvature, even in the presence of topology in the target manifold. In particular, we will show that each component function u i of the solution u = (u 1 ..... u n) of (1.1) can be well