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arXiv · 2607.27072

Existence of a stable shrinker for the corotational harmonic map heat flow in higher space dimensions

Johannes Angerer·Sarah Kistner·Birgit Schörkhuber
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Abstract

We study singularity formation for the heat flow of harmonic maps from $\R^d$ into $\mathbb{S}^d$ in supercritical dimensions $d \in \{3,4,5,6\}$. It is well known that in each of these dimensions there exist infinitely many self-similar solutions that provide examples of loss of regularity in finite time. In this paper, we extend the results of \cite{BieDon18}, \cite{BieDonSch17} for $d=3$ to higher space dimensions $d \in \{4,5,6\}$ and prove the existence of a monotonically increasing self-similar profile $f_0$, which is asymptotically stable under small corotational perturbations. To construct the solution and resolve the spectral problem, we use rigorous computer assistance. As a byproduct of our stability analysis, we also obtain finite-codimension stability of arbitrary self-similar profiles within the corotational class.

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Johannes Angerer, Sarah Kistner, Birgit Schörkhuber. 2026-07-29. Existence of a stable shrinker for the corotational harmonic map heat flow in higher space dimensions. https://arxiv.org/abs/2607.27072

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