Ricci flows with unbounded curvature

Giesen, Gregor; Topping, Peter M.

doi:10.1007/s00209-012-1014-z

Cited by 26 publications

(32 citation statements)

References 12 publications

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“…l c = r −2 c . Combined with the techniques in [GT13] (in particular using pseudolocality in the cylinder region, or alternatively barrier arguments), we could modify Lemma 5.3 such that the upper barrier (5.9) holds for longer times. An inspection of the proof of Proposition 6.1 shows that we also have the lower curvature bound (6.2) for longer and longer times, which is enough to conclude the result in this case.…”

Section: Burst Of Unbounded Curvaturementioning

confidence: 99%

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Ricci flows with bursts of unbounded curvature

Giesen

Topping

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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Section: Burst Of Unbounded Curvaturementioning

confidence: 99%

“…This can be made precise using slight variants of the estimates we proved in Sections 4 and 5 (without requiring Section 5.2) and in Proposition 6.1. As in the C case of this sketch proof, the variant of Lemma 6.2 will have a slightly adjusted proof using the techniques from [GT13] or barrier arguments.…”

Section: Burst Of Unbounded Curvaturementioning

confidence: 99%

Ricci flows with bursts of unbounded curvature

Giesen

Topping

2016

Communications in Partial Differential Equations

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“…In fact, for each k we will be constructing two cigar Ricci soliton flows C lower k .s; t/ and C upper k .s; t / which will be lower and upper barriers respectively for u k .s; t/ over the range .s; t / 2 OEk; 1/ OE0; 1 2 , and so that u k .s; 0/ will coincide with C lower k .s; 0/ for large enough s. The idea of using upper and lower cigar barriers in this way was introduced in [7].…”

Section: 2mentioning

confidence: 99%

“…for t 2 OE0; we will be able to apply the comparison principle for s k (strictly speaking we should return to working on the disc D e k and apply it there, as in [7]) to deduce that…”

Section: 2mentioning

confidence: 99%

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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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Ricci Flow and Ricci Limit Spaces

Topping

2020

Geometric Analysis

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Ricci flows with unbounded curvature

Abstract: We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t ∈ [0, ∞), which has unbounded curvature for all t ∈ [0, ∞).

Cited by 26 publications

References 12 publications

Ricci flows with bursts of unbounded curvature

Ricci flows with bursts of unbounded curvature

Remarks on Hamilton's Compactness Theorem for Ricci flow

Ricci Flow and Ricci Limit Spaces

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