On uniqueness of heat flow of harmonic maps

Tao Huang; Changyou Wang

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arXiv · 1208.1470

On uniqueness of heat flow of harmonic maps

Tao Huang·Changyou Wang

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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 $S^{k-1}$ 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.

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Tao Huang, Changyou Wang. 2012-08-07. On uniqueness of heat flow of harmonic maps. https://arxiv.org/abs/1208.1470

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