Partial Regularity for Weak Heat Flows into a General Compact Riemannian Manifold

In this paper the partial regularity of the weak heat flow of harmonic maps from a Riemannian manifold M into a general compact Riemannian manifold N without boundary is considered. Partial results have been obtained for target manifolds that are spheres [12, 4] or homogeneous spaces [6]. The proofs in these special cases relied heavily on the geometry of these manifolds, and cannot be applied to the general case. We prove in this article that the singular set Sing(u) of the stationary weak heat flow satisfies H ⁿ _ρ (Sing(u))=0, with n=dimension M, where H ⁿ _ρ is the Hausdorff measure with respect to parabolic metric .