Blow-up for the 3-dimensional axially symmetric harmonic map flow into S2

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, begin{align*} u_t & = Delta u + | abla u|^2 u quad text{in } Omegatimes(0,T) u &= u_b quad text{on } partial Omegatimes(0,T) u(cdot,0) &= u_0 quad text{in } Omega , end{align*} with $u(x,t): bar Omegatimes [0,T) to S^2$. Here $Omega$ is a bounded, smooth axially symmetric domain in $mathbb{R}^3$. We prove that for any circle $Gamma subset Omega$ with the same axial symmetry, and any sufficiently small $T>0$ there exist initial and boundary conditions such that $u(x,t)$ blows-up exactly at time $T$ and precisely on the curve $Gamma$, in fact $$ | abla u(cdot ,t)|^2 rightharpoonup | abla u_*|^2 + 8pi delta_Gamma text{ as } tto T . $$ for a regular function $u_*(x)$, where $delta_Gamma$ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng.