Evolution of harmonic maps on manifolds flat at infinity

The aim of this article is to prove a global existence result with small data for the heat flow for harmonic maps from a manifold flat at infinity into a compact manifold. By flat at infinity we mean that the growth rate of the volumes of the balls on the manifold is the same as in the flat space. This is true for any manifold for small enough radius, but is in general not true when the radius of the ball grows. So prescribing such a growth rate also at infinity selects a class of manifolds on which our result holds. In this setting estimates are available for the heat kernel and its gradient on the base manifold. From such estimates it is easy to get L ^p −L ^q bounds for the heat kernel. A contraction principle argument then yields a local existence result in a suitable Sobolev space and a global existence result for small data.