Harmonic and Quasi-Harmonic Spheres, Part III Recti ablity of the Parabolic Defect Measure and Generalized Varifold Flows

This is the third part of our project initiated in [LW1] on the study of the weakly convergent sequence of smooth (or certain classes of weak) solutions to the heat equation of harmonic maps or approximated harmonic maps (i.e. the negative gradient ow of the generalized Ginzburg-Landau functionals). The general situation for the heat ow of harmonic maps is as follows. Let un(x; t) : M R+ ! N be a sequence of smooth solutions to the heat ow of harmonic maps from a m-dimensional compact smooth Riemannian manifold M (with possibly nonempty smooth boundary @M) into a compact smooth Riemannian manifold N R without boundary, namely,