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We construct the first example of finite time blow-up solutions for the heatflow of the $H$-system, describing the evolution of surfaces with constant meancurvature \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 thatthe linearized operators around $W$ projected onto $W^\perp$ and in the$W$-direction are in fact decoupled. On $W^\perp$, the linearization is thelinearized harmonic map heat flow, while in the $W$-direction, it is thelinearized Liouville-type flow. Based on this, we also prove the non-degeneracyof the $H$-bubbles with any degree.

Finite-time singularity formation for the heat flow of the $H$-system