Harmonic Map Heat Flow Coupled to a Green-Potential
Abstract
We introduce a Green--potential heat flow for maps from a closed Riemannian surface. Inspired by Parker--Wolfson renormalization~\cite{ParkerWolfson1993,Parker1996}, the conformal factor is reconstructed at each time from an energy-based source determined by the current map. The resulting parabolic--elliptic system incorporates renormalization into the evolution while leaving the Dirichlet energy functional and its harmonic critical points unchanged. We establish local well-posedness for H3 initial data and a continuation criterion in terms of energy concentration. For sufficiently large coupling, we exclude finite-time bubbling and obtain global solutions. At infinite time, the energy identity provides only weighted L2 control of the tension field. Localizing to regular regions of concentration charts yields harmonic profiles along subsequences of prescribed times. We represent these profiles in the flow's packet charts, organize them into a finite bubble tree, and decompose the limiting energy into harmonic component energies and neck energies. A critically normalized radial example illustrates how the modified flow can prevent finite-time blow-up and produce harmonic bubbles with no-neck property.
Related
- Rigidity of Lipschitz map using harmonic map heat flow
- Numerical approximation of probabilistically strong solutions to the stochastic harmonic map heat flow
- Partial regularity of heat flow of fractional harmonic maps to spheres
- Controlled harmonic map heat flow and blow-up from the disk to the sphere
- Classification for the $2$-equivariant harmonic map heat flow and the radial energy-critical nonlinear heat equation in dimension $6$
BibTeX
@misc{zhang2026harmonic,
title = {{Harmonic Map Heat Flow Coupled to a Green-Potential}},
author = {Zhe Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25893},
eprint = {2609.25893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25893},
}
RIS
TY - UNPB
TI - Harmonic Map Heat Flow Coupled to a Green-Potential
AU - Zhe Zhang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25893
UR - https://arxiv.org/abs/2609.25893
AB - We introduce a Green--potential heat flow for maps from a closed Riemannian surface. Inspired by Parker--Wolfson renormalization~\cite{ParkerWolfson1993,Parker1996}, the conformal factor is reconstructed at each time from an energy-based source determined by the current map. The resulting parabolic--elliptic system incorporates renormalization into the evolution while leaving the Dirichlet energy functional and its harmonic critical points unchanged. We establish local well-posedness for H^3 initial data and a continuation criterion in terms of energy concentration. For sufficiently large coupling, we exclude finite-time bubbling and obtain global solutions. At infinite time, the energy identity provides only weighted L^2 control of the tension field. Localizing to regular regions of concentration charts yields harmonic profiles along subsequences of prescribed times. We represent these profiles in the flow's packet charts, organize them into a finite bubble tree, and decompose the limiting energy into harmonic component energies and neck energies. A critically normalized radial example illustrates how the modified flow can prevent finite-time blow-up and produce harmonic bubbles with no-neck property.
ER -