# 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.

## Key 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}
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\begin{document}$$\mathbb {R}^d$$\end{document}Rd. For each \documentclass[12pt]{minimal}
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\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.