Long-time behavior of awesome homogeneous Ricci flows (Q6631322)

A Ricci flow \((M, g(t))\) defined on a maximal time interval \([0,T)\) is called immortal if \(T = +\infty\), and otherwise, is said to have finite extinction time. The dynamical Alekseevski conjecture [\textit{A. Naber} et al., Geometrie, Oberwolfach Rep. 19, No. 2, 1551--1601 (2022; Zbl 1519.00025)] states that the universal cover of an immortal homogeneous Ricci flow solution is diffeomorphic to \(\mathbb{R}^n\). In the paper under review, the author proves the dynamical Alekseevski conjecture for a certain class of homogeneous metrics known as awesome metrics.\N\NLet \(G\) be a semisimple Lie group; \(\mathfrak{g},\, \mathfrak{h}\) the Lie algebras of \(G\), \(H\) respectively; \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) a Cartan decomposition of \(\mathfrak{g}\); \(\mathsf{K}\) the integral subgroup corresponding to \(\mathfrak{k}\); and \(\mathfrak{m} = \mathfrak{l} \oplus \mathfrak{p}\) a reductive complement to \(\mathfrak{h}\). Then a \(G\)-invariant metric on the homogeneous space \(G/H\) is said to be awesome [\textit{Y. G. Nikonorov}, Sib. Math. J. 41, No. 2, 349--356 (2000; Zbl 0947.53025)] if \(g(\mathfrak{l}, \mathfrak{p}) = 0\), that is, \(\mathfrak{l}\) and \(\mathfrak{p}\) are orthogonal.\N\NThe paper also contains convergence results for awesome Ricci flows. The author shows that for an awesome Ricci flow with finite extinction time \(T\), any sequence of rescaled metrics \(\{R(g(t_a))\cdot g(t_a)\}\) subconverges in pointed \(C^\infty\)-topology as \(t_a \to T\), to the Riemannian product \(E_\infty \times \mathbb{E}^d\) of a compact homogeneous Einstein manifold with positive scalar curvature, with flat Euclidean space of dimension \(d \geq \dim \mathfrak{p}\).\N\NFor immortal awesome Ricci flows, the author shows that the parabolic rescaling \(t^{-1}g(t)\) converges in pointed \(C^\infty\)-topology to the Riemannian product \(\Sigma_\infty\times \mathbb{E}^{\dim \mathfrak{l}}\), where \(\Sigma_\infty\) is the noncompact Einstein symmetric space defined by the pair \((\mathfrak{g}, \mathfrak{k})\). This generalizes a result of \textit{J. Lott} [Math. Ann. 339, No. 3, 627--666 (2007; Zbl 0156.10701)] which states that the parabolic blow-down of any left-invariant metric on \(\widetilde{SL(2,\mathbb{R})}\) converges to \(\mathbb{H}^2 \times \mathbb{R}\).