Smoothness of conformal heat flow of harmonic maps

Abstract

The conformal heat flow of harmonic maps is a system of evolution equations combined with harmonic map flow with metric evolution in conformal direction. It is known that global weak solution of the flow exists and smooth except at mostly finitely many singular points. In this paper, we show that no finite time singularity occurs, unlike the usual harmonic map flow. And if the initial energy is small, we can obtain the uniform convergence of the map to a point and the conformal factor of the metric under some time sequence $t_n \to \infty$. Also, under the assumption that energy concentration is uniform in time, we show that there exists a sequence of time $t_n \to \infty$ such that $f(\cdot,t_n)$ converges to a harmonic map in $W^{1,2}$ on any compact set away from at most finitely many points.