Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$

We study m-corotational solutions to the Harmonic Map Heat Flow from $\mathbb{R}^2$ to $\mathbb{S}^2$. We first consider maps of zero topological degree, with initial energy below the threshold given by twice the energy of the harmonic map solutions. For $m \geq 2$, we establish the smooth global existence and decay of such solutions via the {\it concentration-compactness} approach of Kenig-Merle, recovering classical results of Struwe by this alternate method. The proof relies on a profile decomposition, and the energy dissipation relation. We then consider maps of degree $m$ and initial energy above the harmonic map threshold energy, but below three times this energy. For $m \geq 4$, we establish the smooth global existence of such solutions, and their decay to a harmonic map (stability), extending results of Gustafson-Nakanishi-Tsai to higher energies. The proof rests on a stability-type argument used to rule out finite-time bubbling.