How to produce a Ricci flow via Cheeger–Gromoll exhaustion

Cabezas-Rivas, Esther; Wilking, Burkhard

doi:10.4171/jems/582

Cited by 43 publications

(68 citation statements)

References 39 publications

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“…One interesting thing of such lower bound of existence time is that it only depends on the local property of manifolds in a neighborhood of a point, not globally on the whole manifold. This kind of lower bound on existence time for Ricci flow had been obtained in [4,Corollary 5] for any complete manifolds with stronger curvature assumptions. Note for any point x 0 ∈ M n , we can always find the corresponding r 0 satisfying the assumptions in the above theorem, although such r 0 may be very small.…”

Section: Introductionmentioning

confidence: 63%

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Lower bound of Ricci flow's existence time

2015

Bull. London Math. Soc.

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Let (M n , g) be a compact n-dimensional (n 2) manifold with nonnegative Ricci curvature, and if n 3, then we assume that (M n , g) × R has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on (M n , g) is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows' maximal existence times, which was first proved by E. Cabezas-Rivas and B. Wilking. We also get an interior curvature estimate for n = 3 under Rc 0 assumption among others. Combining these results, we proved the shorttime existence of the Ricci flow on a large class of three-dimensional open manifolds, which admit some suitable exhaustion covering and have nonnegative Ricci curvature.

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Section: Introductionmentioning

confidence: 63%

“…The existence time's lower bound part of the following theorem is not totally new (see [4,Corollary 5]), however in three dimensions the result seems to be new. And our method to obtain such lower bound is different from the method in [4].…”

Section: Introductionmentioning

confidence: 99%

Lower bound of Ricci flow's existence time

2015

Bull. London Math. Soc.

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“…However, one can hope to prove the existence of Ricci flows starting with unbounded-curvature manifolds with certain positivity-of-curvature conditions, and Cabezas-Rivas and Wilking [CRW11] have done this for positive complex sectional curvature. The Ricci flows in our second result, Theorem 1.4 have similar properties to four-dimensional Ricci flows constructed by the same authors [CRW11]. Of course, by taking the product of our examples with Euclidean space, our work immediately yields examples also in all dimensions larger than two.…”

Section: If T < ∞ Then We Havementioning

confidence: 99%

“…This was originally proved in [GT13] on general surfaces, with a somewhat simpler construction. Specific higher-dimensional examples of Ricci flows with this type of behaviour were constructed by Cabezas-Rivas and Wilking [CRW11].…”

Section: If T < ∞ Then We Havementioning

confidence: 99%

Ricci flows with bursts of unbounded curvature

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2016

Communications in Partial Differential Equations

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Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metric, this Ricci flow always exists for all time and converges (after scaling by a factor 1 2t ) to this hyperbolic metric [GT11], i.e. our Ricci flow geometrises the surface. In this paper we show that there exist complete, bounded curvature initial metrics, including those conformal to a hyperbolic metric, which have subsequent Ricci flows developing unbounded curvature at certain intermediate times. In particular, when coupled with the uniqueness from [Top13], we find that any complete Ricci flow starting with such initial metrics must develop unbounded curvature over some intermediate time interval, but that nevertheless, the curvature must later become bounded and the flow must achieve geometrisation as t → ∞, even though there are other conformal deformations to hyperbolic metrics that do not involve unbounded curvature.Another consequence of our constructions is that while our Ricci flow from [GT11] must agree initially with the classical flow of Hamilton and Shi in the special case that the initial surface is complete and of bounded curvature, by uniqueness, it is now clear that our flow lasts for a longer time interval in general, with Shi's flow stopping when the curvature blows up, but our flow continuing strictly beyond in these situations.All our constructions of unbounded curvature developing and then disappearing are in two dimensions. Generalisations to higher dimensions are then immediate. IntroductionHamilton [Ham82] and Shi [Shi89] proved that given a complete Riemannian manifold (M, g 0 ) with bounded curvature, there exists a complete Ricci flow g(t) on M for a short time, with g(0) = g 0 (see [Top06] for an introduction to this topic). The curvature of this Ricci flow is initially bounded, and the flow can be extended until such time that the curvature becomes unbounded.Ricci flows with possibly unbounded curvature in their initial condition and/or during the flow itself, were studied by the second author in [Top10] in the special case of surfaces, and in [GT11] we proved that one can always find an instantaneously complete Ricci flow starting at a completely arbitrary initial surface, whether of unbounded curvature or not, or indeed whether complete or not, which exists for a specific amount of time, and in [Top13] this solution was shown to be unique. More precisely, we proved: (1.1)Then there exists a smooth Ricci flow g(t) t∈[0,T ) such thatis complete for all t ∈ (0, T ));and this flow is unique in the sense that if g 2 (t) t∈[0,T2) is any other Ricci flow on M satisfying (i) and (ii), then T 2 ≤ T and g 2 (t) = g(t) for all t ∈ [0, T 2 ). If T < ∞, then we haveand in particular, T is the maximal existence time.It has been understood since the work of Hamilton and C...

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“…One other special case is that it is possible to find an immortal Ricci flow on any Riemann surface which has bounded curvature for t 2 OE0; 1/ but unbounded curvature for sufficiently large t . Examples of Ricci flows with unbounded curvature in three dimensions, and with unbounded curvature for t 1 in four dimensions, were constructed by Cabezas-Rivas and Wilking [1]. Ricci flows on surfaces with unbounded curvature for all t 0 were first constructed in [7].…”

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confidence: 99%

Remarks on Hamilton's Compactness Theorem for Ricci flow

Topping¹

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. A fundamental tool in the analysis of Ricci flow is a compactness result of Hamilton in the spirit of the work of Cheeger, Gromov and others. Roughly speaking it allows one to take a sequence of Ricci flows with uniformly bounded curvature and uniformly controlled injectivity radius, and extract a subsequence that converges to a complete limiting Ricci flow. A widely quoted extension of this result allows the curvature to be bounded uniformly only in a local sense. However, in this note we give a counterexample.

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How to produce a Ricci flow via Cheeger–Gromoll exhaustion

Cited by 43 publications

References 39 publications

Lower bound of Ricci flow's existence time

Lower bound of Ricci flow's existence time

Ricci flows with bursts of unbounded curvature

Remarks on Hamilton's Compactness Theorem for Ricci flow

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