Abstract
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 Sk−1 or a Riemannian homogeneous manifold. For such a class of solutions, we also establish the convexity property of the Dirichlet energy for t ≥ t0 > 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 whose gradients belong to LqtL l x, for q > 2 and l > n satisfying the Serrin’s condition.
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