  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 + |\nabla u|^2 u
\quad \text{in } \Omega\times(0,T) \\ u &= u_b \quad \text{on } \partial
\Omega\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } \Omega , \end{align*}
with $u(x,t): \bar \Omega\times [0,T) \to S^2$. 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 $$
|\nabla u(\cdot ,t)|^2 \rightharpoonup |\nabla u_*|^2 + 8\pi \delta_\Gamma
\text{ as } t\to T . $$ 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.
