Kähler-Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature

Abstract

In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC 1 , without any curvature bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are also obtained. The main new feature is that no curvature upper bound is assumed.2010 Mathematics Subject Classification. 53C44, 53C55.

Ancient solutions to the Ricci flow with isotropic curvature conditions

“…By taking b → (b max + bmax ), we can conclude that (M, g(t)) t∈(−∞,0] has weakly PIC 1 . Then by [1, Lemma 4.2](see also [39,Proposition 6.2]), we conclude that (M, g(t)) t∈(−∞,0] has weakly PIC 2 .…”

“…Since the curvature cones pinch toward weakly PIC 1 , the ancient solution has weakly PIC 1 . Therefore, by [1,Lemma 4.2] (see also [39,Proposition 6.2]), the ancient solution has weakly PIC 2 . Moreover, we obtain a general theorem for curvature improvement, see Theorem 3.3.…”

“…Khler manifold with complex dimension 2) has weakly PIC if and only if it has nonnegative orthogonal bisectional curvature, which is preserved under Kähler Ricci flow [33]. In general, an ancient solution to the Kähler Ricci flow with nonnegative orthogonal bisectional curvature has nonnegative bisectional curvature [39]. For the κ-solutions (see Definition 6.1) to the Kähler Ricci flow, there is a parallel theory to Perelman's theory on κ-solutions to the 3-dimensional Ricci flows.…”

Ancient solutions to the Kähler Ricci flow

“…Therefore, to prove Theorem 1.1, it is essential to understand all Type-I κ-solutions to the Ricci flow. One motivation is the similar classification result of Ricci shrinkers [38] [34]. More precisely, it was proved in [34,Theorem 3.1(ii)] that any Ricci shrinker with weakly PIC 2 is locally symmetric.…”

Rigidity of the round cylinders in Ricci shrinkers

“…In higher dimensions, there exist many nontrivial, non-product Ricci shrinkers (e.g., [37] [11] [27]), and the classification of Ricci shrinkers is only achieved when extra assumptions are assumed. Such assumptions include non-negativity of curvatures (e.g., [51] [46][54] [45] [48]), restriction of the Weyl curvatures (e.g., [58][49] [12] [23]), restriction of asymptotic behavior at infinity (e.g., [40] [41]), Kähler conditions (e.g., [55] [22] [20]), and others. In general, much less is known if no extra assumptions are assumed.…”