Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions

TL;DR

This paper investigates how singularities form in harmonic map heat flow in higher dimensions and identifies stable shrinking solutions.

Contribution

The paper constructs stable, self-similar shrinking solutions for harmonic map heat flow in dimensions four and above.

Findings

For each dimension d ≥ 4, a compact, rotationally symmetric target manifold is constructed.

A corotational self-similar shrinking solution exists and represents a stable blowup mechanism.

Abstract

We study singularity formation for the heat flow of harmonic maps from \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbb {R}^d$$\end{document}Rd. For each \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d \ge 4$$\end{document}d≥4, we construct a compact, d-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly shrinker) that represents a stable blowup mechanism for the corresponding Cauchy problem.