  We construct the first example of finite time blow-up solutions for the heat
flow of the $H$-system, describing the evolution of surfaces with constant mean
curvature \begin{equation*} \left\{ \begin{aligned} &u_t = \Delta u -
2u_{x_1}\wedge u_{x_2}~\quad\text{ in }~\mathbb{R}^2\times\mathbb{R}_+,\\
&u(\cdot, 0) = u_0~\qquad\qquad~\text{ in }~\mathbb{R}^2, \end{aligned} \right.
\end{equation*} where $u$: $\mathbb{R}^2\times\mathbb{R}_+\to \mathbb{R}^3$.
The singularity at finite time forms as a scaled least energy $H$-bubble,
denoted as $W$, exhibiting type II blow-up speed. One key observation is that
the linearized operators around $W$ projected onto $W^\perp$ and in the
$W$-direction are in fact decoupled. On $W^\perp$, the linearization is the
linearized harmonic map heat flow, while in the $W$-direction, it is the
linearized Liouville-type flow. Based on this, we also prove the non-degeneracy
of the $H$-bubbles with any degree.
