Ancient solutions to the Ricci flow with isotropic curvature conditions

Abstract

We show that every n-dimensional, κ-noncollapsed, noncompact, complete ancient solution to the Ricci flow with uniformly PIC for n = 4 or n ≥ 12 has weakly PIC 2 and bounded curvature. Combining this with the results in [12], we prove that any such solution is isometric to either a family of shrinking cylinders (or a quotient thereof) or the Bryant soliton. Also, we classify all complex 2-dimensional, κ-noncollapsed, complete ancient solutions to the Kähler Ricci flow with weakly PIC.

Ancient solutions to the Kähler Ricci flow

“…Theorem 1.1 extends our earlier result regarding the classification of κ-solutions on Kähler surfaces [23,Theorem 1.3]. In [23], a key observation is that any complete, complex 2-dimensional ancient solution to the Kähler Ricci flow with nonnegative bisectional curvature automatically has nonnegative curvature operator [23,Lemma 4.6]. Here, we obtain a similar curvature improvement (see Theorem 3.3), which states that any complete ancient solution to the Kähler Ricci flow with nonnegative bisectional curvature has weakly PIC 2 .…”

“…Theorem 1.1 extends our earlier result regarding the classification of κ-solutions on Kähler surfaces [23,Theorem 1.3]. In [23], a key observation is that any complete, complex 2-dimensional ancient solution to the Kähler Ricci flow with nonnegative bisectional curvature automatically has nonnegative curvature operator [23,Lemma 4.6].…”

“…The classification of higher dimensional κ-solutions remains open. For recent developments in higher dimensions, see, for example, [10] [36] [11] [23].…”

“…Another important observation is that any κ-solution to the Kähler Ricci flow must be of Type-I, which was initially proved by Deng-Zhu [27,Lemma 2.5] in the compact case and generalized by Cho-Li [23,Lemma 6.2] to the general case. Essentially, this follows from the classical pointpicking argument and the nonexistence of κ-noncollapsed, nonflat, steady Kähler Ricci soliton with nonnegative bisectional curvature [26].…”

“…To complete the proof of Theorem 1.1, we next show that any solution in Theorem 1.1 must have bounded curvature. Here, we follow the strategy in [23] to prove first a canonical neighborhood theorem, which states that any high curvature region in the flow is modeled on a κ-solution to the Kähler Ricci flow (Theorem 3.18). The key point is to show that any Kähler manifold, whose high curvature regions are modeled on κ-solutions, must have uniformly bounded curvature, see Proposition 3.15.…”