The Ricci flow under almost non-negative curvature conditions

Abstract

We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time.As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than −1 can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than −C. Here the time of existence and the constant C only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kähler case. We also get a local version of the main theorem.As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds). arXiv:1707.03002v2 [math.DG] 25 Jul 2017 for some ε ∈ [0, 1]. Then the Ricci flow g(t) with initial metric g exists until time τ , is Kähler if (M, g) is Kähler, and we have the curvature bounds Rm g(t) + Cε I ∈ C and |Rm g(t) | ≤ C t for all t ∈ (0, τ ].(1.2)

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

“…Note that the above result generalizes Theorem 2 of [24]. The same argument shows that Lemma 4.2 in [2] holds without the bounded curvature assumption. Proposition 6.2.…”

“…For any n ≥ 2, = 3 and ν 0 , there exist positive constants C = C(n, ν 0 ) and τ = τ (n, ν 0 ) such that if (M, g) is Kähler manifold with bounded curvature, dim C (M ) = n, V ol g (B g (p, 1)) ≥ ν 0 , ∀p ∈ M, and Rm +ǫ id has NOB for some ǫ ∈ [0, 1], then the Kähler-Ricci flow exists on [0, τ ] with Rm g(t) +Cǫ id has NOB and | Rm | ≤ C t for all t ∈ (0, τ ]. As a consequence of the above one can have a similar result as Corollary 3 of [2]. Namely for given D > 0, v 0 > 0, there exists an ǫ = ǫ(D, v 0 , n) such that if a Kähler manifold (M n , g) satisfies that V ol(M ) ≥ v 0 , Diam(M ) ≤ D, and Rm +ǫ id has NOB, then M admits a Kähler metric with NOB.…”

“…Note that the part (i) was known for compact manifolds [32], and the part (ii) of the above was proved under additional assumption of bounded curvature recently in [2]. The main feature of our results is that no upper curvature bound is assumed.…”

“…First by combining the argument of the proof of Proposition 5.1 and a modification of the argument in [2], we strengthen Proposition 5.1 to show that in fact the ancient solution with B ⊥ ≥ 0 has nonnegative bisectional curvature. This result is needed in extending a recent result of [2] to Kähler manifolds with negative lower bound of B ⊥ . We start with a lemma.…”

“…Hence in this paper, for the discussion of shrinkers of weakly PIC n = dim R (M ) ≥ 5 is assumed. Note that in a recent work [2], it was shown that for n ≥ 7, R being of weakly PIC implies that the Ricci curvature is 3-nonnegative. There have been many works on gradient shrinking solitons since [31].…”