Tuesday, September 15, 2026 — 4:40 pm — aula M1.7, edificio Matematica
On the Threshold Dynamics of Solutions for H-System Heat Flow
Abstract: We study the threshold dynamics between global existence and finite-time blow-up for the constant mean curvature (CMC) heat flow on the whole plane. After briefly describing its variational origin from the area functional under a volume constraint, we construct a maximal mild solution and establish the corresponding energy identity.
Our main result gives a sharp dichotomy below a critical energy level: the sign of the Nehari functional determines whether the solution exists globally or blows up in finite time. On the stable side, the potential well estimate places the solution strictly below the energy of a nontrivial stationary CMC bubble. A parabolic blow-up argument shows that any finite-time singularity would produce such a bubble, yielding a contradiction with the sharp Wente-type energy threshold. On the unstable side, finite-time blow-up follows from a Levine-type concavity argument. These results provide a unified description of the low-energy dynamics of the CMC heat flow on the whole plane.