  The conformal heat flow for 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 and the
conformal factor. 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 $C^1$ on any compact
set away from at most finitely many points.
