In this paper, we establish the uniqueness of heat flow of harmonic maps into (N, h) that have sufficiently small renormalized energies, provided that N is either a unit sphere $S^{ k-1}$ or a compact Riemannian homogeneous manifold without boundary. For such a class of solutions, we also establish the convexity property of the Dirichlet energy for $t\ge t_0>0$ and the unique limit property at time infinity. As a corollary, the uniqueness is shown for heat flow of harmonic maps into any compact Riemannian manifold N without boundary whose gradients belong to $L^q_t L^l_x$ for $q>2$ and $l>n$ satisfying the Serrin's condition.
Categories: math.AP math.DG