Ancient solutions of the homogeneous Ricci flow on flag manifolds

Abstract

For any flag manifold M=G/K of a compact simple Lie group G we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions pass through an invariant Einstein metric on M, and by [13] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold M=G/K with second Betti number b2(M) = 1, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose α-limit set consists of fixed points at infinity of MG. Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system’s phase space.

Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

“…(b) On 𝖲𝖴(𝑛)∕𝖳 𝑛−1−𝑘 , with 𝑛 ⩾ 3 and 1 ⩽ 𝑘 ⩽ 𝑛 − 1, there exists a ( 𝑘(𝑘+1) 2 + 𝑛 − 2)-parameter family of ancient solutions to the Ricci flow collapsing to the normal Einstein metric on 𝖲𝖴(𝑛)∕𝖳 𝑛−1 . (c) On 𝖲𝖮(4) and 𝖲𝖮(4)∕𝖲 1 there exists a 3, respectively, 1-parameter family of ancient solutions to the Ricci flow collapsing to the normal Einstein metric on 𝖲𝖮(4)∕𝖳 2 .…”

“…, such ancient solutions are known to exist whenever 𝑀 admits a 𝖦-unstable, 𝖦-invariant Einstein metric (see, for example, [2,10]). The second possibility is that scal 𝑀 ( P(𝑡)) → 0 as 𝑡 → −∞.…”

“…(a) On 𝖲𝖴(3), 𝖲𝖴(3)∕𝖲 1 , 𝖲𝖴(4), 𝖲𝖴(4)∕𝖲 1 , 𝖲𝖴(4)∕𝖳 2 , 𝖦 2 and 𝖦 2 ∕𝖲 1 there exists a 3, respectively, 1, 7, 4, 2, 3 and 1-parameter family of ancient solutions to the Ricci flow collapsing, under a suitable rescaling, to a Kähler-Einstein metric on a full flag manifold. (b) On 𝖲𝖴(𝑛)∕𝖳 𝑛−1−𝑘 , with 𝑛 ⩾ 3 and 1 ⩽ 𝑘 ⩽ 𝑛 − 1, there exists a ( 𝑘(𝑘+1) 2 + 𝑛 − 2)-parameter family of ancient solutions to the Ricci flow collapsing to the normal Einstein metric on 𝖲𝖴(𝑛)∕𝖳 𝑛−1 .…”

“…The first possibility is that there exists an $$\varepsilon >0$$ such that $$\operatorname{scal}_{M}(\tilde{P}(t))>\varepsilon$$ for any $$t\leqslant 0$$, in which case $$\tilde{P}(t)$$ (and hence $$P(t)$$) is non‐collapsed and, by [10, Theorem 5.2], $$\tilde{P}(t)$$ converges to an Einstein metric as $$t\rightarrow -\infty$$. Since the traceless Ricci tensor is the negative $$L^2$$‐gradient of the functional $$\operatorname{scal}_{M}|_{\mathcal {M}^{\mathsf {G}}_{{M},1}}$$, such ancient solutions are known to exist whenever $$M$$ admits a $$\mathsf {G}$$ ‐unstable , $$\mathsf {G}$$‐invariant Einstein metric (see, for example, [2, 10]). The second possibility is that $$\operatorname{scal}_{M}(\tilde{P}(t)) \rightarrow 0$$ a...…”

Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

“…Since the traceless Ricci tensor is the negative L 2 -gradient of the functional scal M | M G M ,1 , such ancient solutions are known to exist whenever M admits a G-unstable, G-invariant Einstein metric (see e.g. [2,10]). The second possibility is that scal M ( P (t)) → 0 as t → −∞.…”