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

Abstract

We study singularity formation for the heat flow of harmonic maps from $\R^d$. For each $d \geq 4$, we construct a compact, $d$-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly \emph{shrinker}) that represents a stable blowup mechanism for the corresponding Cauchy problem.