  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.
