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We consider the energy critical harmonic heat flow from $\Bbb R^2$ into asmooth compact revolution surface of $\Bbb R^3$. For initial data withcorotational symmetry, the evolution reduces to the semilinear radiallysymmetric parabolic problem $$\partial_t u -\pa^2_{r} u-\frac{\pa_r u}{r} +\frac{f(u)}{r^2}=0$$ for a suitable class of functions $f$ . Given an integer$L\in \Bbb N^*$, we exhibit a set of initial data arbitrarily close to theleast energy harmonic map $Q$ in the energy critical topology such that thecorresponding solution blows up in finite time by concentrating its energy$$\nabla u(t,r)-\nabla Q(\frac{r}{\l(t)})\to u^* in L^2$$ at a speed given bythe {\it quantized} rates: $$\l(t)=c(u_0)(1+o(1))\frac{(T-t)^L}{|\log(T-t)|^{\frac{2L}{2L-1}}},$$ in accordance with the formal predictions [3]. Thecase L=1 corresponds to the stable regime exhibited in [37], and the data for$L\ge 2$ leave on a manifold of codimension $(L-1)$ in some weak sense. Ouranalysis lies in the continuation of [36,32,37] by further exhibiting themechanism for the existence of the excited slow blow up rates and theassociated instability of these threshold dynamics.

Quantized slow blow up dynamics for the corotational energy critical harmonic heat flow