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

2023/11/24 by Yannick Sire, Juncheng Wei, Sire, Yannick +5
Mathematics · #53A10 #53E10 (Primary) 35B44 (Secondary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2311.14336

openalex publication_date 2023/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \ \beginaligned amp;u_t = Δu - 2u_x1\wedge u_x2~ in ~ℝ²×ℝ₊,
amp;u(⋅, 0) = u₀~ ~ in ~ℝ², \endaligned . where u: ℝ²×ℝ₊→ ℝ³. 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^⊥ and in the W-direction are in fact decoupled. On W^⊥, 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.

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