A remark on the finite time singularity of the heat flow for harmonic maps (Q1403388)

The paper deals with the study of the asymptotic behaviour of the solution to the heat equation associated to harmonic maps from surfaces into a Riemann manifold. The author first establishes a gradient estimate for the solution across a finite time singularity. This implies in particular that the solution is asymptotically radial around the isolated singular point in space at a finite singular time. An open question raised in the present paper is to understand whether the solution is continuous in space at a finite singular time. The proofs are based on gradient estimates for the solution combined with refined local energy and Harnack estimates.