Nonexistence of shrinkers for the harmonic map flow in higher dimensions

TL;DR

This paper proves that in dimensions seven and higher, there are no self-similar shrinking solutions for the harmonic map flow from Euclidean space to spheres, indicating limitations on singularity models in these dimensions.

Contribution

It establishes the nonexistence of equivariant self-similar shrinkers for the harmonic map flow in high dimensions, extending understanding of flow behavior.

Findings

No equivariant self-similar shrinking solutions in dimensions d ≥ 7

Results apply to harmonic map flow from ℝ^d to S^d

Implications for singularity formation in high-dimensional flows

Abstract

We prove that the harmonic map flow from the Euclidean space $\mathbb{R}^d$ into the sphere $S^d$ has no equivariant self-similar shrinking solutions in dimensions $d\geq 7$.