Nonlinear asymptotic stability of homothetically shrinking Yang–Mills solitons in the equivariant case

We study the heat flow for Yang-Mills connections on $\mathbb{R}^d\times SO(d)$ for $5\leq d\leq 9$. It is well-known that for this model homothetically shrinking solitons exist and an explicit example was found by Weinkove [21]. In this paper, we prove the nonlinear asymptotic stability of this solution under small $SO(d)$−equivariant perturbations and extend the results of [8] for $d=5$ to higher space dimensions. Also, we substantially simplify proof and provide new techniques to rigorously solve the spectral problem for the linearization, which turns out to be more involved in higher space dimensions.