On blow-up trees for the harmonic map heat flow from B² to S²

2025/07/31 by Dylan Samuelian, Samuelian, Dylan
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Black Holes and Theoretical Physics

paper · pdf · doi:10.1007/s00526-026-03396-0

Abstract

Abstract We consider finite-time and k -equivariant solutions to the harmonic map heat flow from B² <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> to S² <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> under general time-dependent boundary data and prove that the bubble tree decomposition contains only one bubble. The method relies on the Maximum and Comparison Principle. We also exhibit solutions blowing up in infinite time for any k ≥ 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> .

Citations

• Harmonic Mappings of Riemannian Manifolds

Related

• Bubble decomposition for the harmonic map heat flow in the equivariant case
• Finite time blowup and type II rate for harmonic heat flow from Riemannian manifolds
• Finite-time singularity formation for the heat flow of the H-system
• Type II blow-up mechanism for supercritical harmonic map heat flow
• Singularity formation for the two-dimensional harmonic map flow into S²