Ricci flow under local almost non-negative curvature conditions

Lai, Yi

doi:10.1016/j.aim.2018.11.006

Cited by 30 publications

(29 citation statements)

References 17 publications

(20 reference statements)

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“…Moreover, for any 0 ≤ t ≤ γT , again using that γ − 1 2 ≤ ε, [Lai18]. Doing so gives us constants C 0 = C 0 (n, v 0 ) ≥ 1 and S 0 = S 0 (n, v 0 ) > 0 such that for all 0 < t ≤ min {γT, S 0 } the conclusion (3.2) hold for g γ (t) instead of g(t).…”

Section: Local Flows and Curvature Estimatesmentioning

confidence: 99%

“…We conclude this section by recording that it is possible to find a local solution to the Ricci flow, assuming a lower K IC 1 bound. This is the content of Theorem 1.1 in [Lai18]; we state a minor variant that is more convenient for our purposes. In particular, we reduce the initial noncollapsedness hypothesis to a lower volume bound for a single unit ball, rescale the result to apply to any ball of radius strictly larger than one, and add a lower injectivity radius bound to the conclusion.…”

Section: Local Flows and Curvature Estimatesmentioning

confidence: 99%

“…In particular, we reduce the initial noncollapsedness hypothesis to a lower volume bound for a single unit ball, rescale the result to apply to any ball of radius strictly larger than one, and add a lower injectivity radius bound to the conclusion. The injectivity radii bounds are implicitly obtained within the proof of Theorem 1.1 in [Lai18], and the following result simply makes these explicit.…”

Section: Local Flows and Curvature Estimatesmentioning

confidence: 99%

“…Theorem 3.3 (Local Existence; Variant of Theorem 1.1 in [Lai18]). Given n ∈ N, R ≥ 1 and ε, α 0 , v 0 > 0 there exist positive constants C, τ > 0, both depending only on n, α 0 , v 0 , ε and R, for which the following is true.…”

Section: Local Flows and Curvature Estimatesmentioning

confidence: 99%

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Pyramid Ricci flow in higher dimensions

McLeod

Topping

2020

Math. Z.

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In this paper, we construct a pyramid Ricci flow starting with a complete Riemannian manifold (M n , g 0 ) that is PIC1, or more generally satisfies a lower curvature bound K IC1 ≥ −α 0 . That is, instead of constructing a flow on M ×[0, T ], we construct it on a subset of space-time that is a union of parabolic cylinders B g0 (x 0 , k) × [0, T k ] for each k ∈ N, where T k ↓ 0, and prove estimates on the curvature and Riemannian distance. More generally, we construct a pyramid Ricci flow starting with any noncollapsed IC 1 -limit space, and use it to establish that such limit spaces are globally homeomorphic to smooth manifolds via homeomorphisms that are locally bi-Hölder. IntroductionA central issue in differential geometry is to understand Riemannian manifolds with lower curvature bounds. One of the important tasks in this direction is to understand the topological implications of such geometric bounds. Another, which is the main focus of this paper, is to understand the structure of Gromov-Hausdorff limits of sequences of manifolds satisfying a uniform lower curvature bound.There is some choice as to the precise notion of curvature bound to consider. Imposing a uniform lower bound on the sectional curvatures gives limits that are Alexandrov spaces, studied since the middle of the twentieth century, and about which we now have a great deal of information, e.g. [BGP92]. In practice, we often know a uniform lower bound not for each sectional curvature, but for a suitable average of sectional curvatures, and the instance that has received the most attention is the case of limits of manifolds with a uniform lower Ricci bound. Such Ricci limit spaces have been studied extensively since the work of Cheeger-Colding, starting in the 1990s, and have been widely applied, for example in the study of Einstein manifolds, [Che01]. One result that is particularly relevant to the present paper is the topological regularity of (non-collapsed) three-dimensional Ricci limit spaces in the sense that they are globally homeomorphic to smooth manifolds via homeomorphisms that are locally bi-Hölder [ST17, MT18, Hoc16]. This paper is concerned principally with a way of averaging sectional curvatures that is less familiar than Ricci curvature. Positivity of this average is generally referred to as PIC1, with this concept first appearing in the seminal work of Micallef and Moore [MM88] that was principally concerned with the weaker notion of positive isotropic curvature, itself now abbreviated as PIC. We will give the definition and basic properties of PIC1, and its nonnegative version WPIC1 (sometimes called NIC1) in Section 2.The PIC1 condition is natural for multiple reasons. To begin with, it can be naturally compared with other curvature conditions. For example, it is implied by 1 4 -pinching, positive curvature operator, 2-positive curvature operator, and positive complex sectional curvature separately [MM88, §5]. As we recall in Section 2, the PIC1 condition implies that Ric > 0, so volume comparison and compactness are at our d...

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Section: Local Flows and Curvature Estimatesmentioning

confidence: 99%

Section: Local Flows and Curvature Estimatesmentioning

confidence: 99%

Section: Local Flows and Curvature Estimatesmentioning

confidence: 99%

Section: Local Flows and Curvature Estimatesmentioning

confidence: 99%

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Pyramid Ricci flow in higher dimensions

McLeod

Topping

2020

Math. Z.

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“…When the initial metric is Kähler with non-negative bisectional curvature, the author and Tam [21] were able to construct a short-time solution to the Kähler-Ricci flow. When the initial metric is only Riemannian and has almost weakly PIC 1 , Yi [20] showed that a short-time solution also exists. We also refer the earlier work by Bamler, Cabezas-Rivas and Wilking [2] for the case when g 0 has complex sectional curvature bounded from below or equivalently g 0 has almost weakly PIC 2 .…”

Section: Introductionmentioning

confidence: 99%