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

TL;DR

This paper constructs the first finite-time blow-up solutions for the heat flow of the H-system, showing singularity formation as a scaled H-bubble with decoupled linearized operators, advancing understanding of surface evolution with constant mean curvature.

Contribution

It provides the first explicit example of finite-time blow-up solutions for the H-system heat flow, revealing the decoupled structure of linearized operators around the blow-up profile.

Findings

Finite-time blow-up solutions are constructed.

Singularity forms as a scaled H-bubble with type II blow-up speed.

Linearized operators around the bubble are decoupled, with one resembling harmonic map flow and the other Liouville flow.

Abstract

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.