Remarks on Hamilton's Compactness Theorem for Ricci flow

Topping, Peter M.

doi:10.1515/crelle-2012-0079

Cited by 10 publications

(14 citation statements)

References 15 publications

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“…In order to show that the cusp will start collapsing at a uniformly controlled rate before some uniformly bounded time we require some more involved a priori estimates. In [32], we prove these estimates in a slightly different situation to the one outlined above, notably replacing the bulb metric by long thin cigars with k-dependent geometry, but with area uniformly bounded above and below.…”

Section: Intuition Behind the Constructionmentioning

confidence: 80%

“…For the two-dimensional theory of Ricci flow from the viewpoint of these lectures, one can see particularly [13,31,15,33], and we will also draw on elements of [12] and [14] in order to explain some subtleties of Hamilton's compactness theorem [32].…”

Section: Background Readingmentioning

confidence: 99%

“…In this section we sketch an intuitive construction that indicates that a limiting Ricci flow arising in Theorem 7.1 can be complete over an open time interval containing t = 0, but incomplete beyond a certain time. The precise construction we make in [32] is a little different to reduce the amount of technology required to make it rigorous. (That technology was later developed in [15].…”

Section: Intuition Behind the Constructionmentioning

confidence: 99%

“…The core principle of this lecture is: Such flows h(t) can arise as Hamilton-Cheeger-Gromov limits of complete Ricci flows within the extension of Hamilton's compactness theorem given in Theorem 7.1. A precise statement can be found in [32]; here we sketch how one could hope to construct a sequence of complete, bounded-curvature Ricci flows satisfying the hypotheses of Theorem 7.1, with a limit flow as given above.…”

Section: Intuition Behind the Constructionmentioning

confidence: 99%

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Applications of Hamilton’s compactness theorem for Ricci flow

Topping¹

2016

Geometric Analysis

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Section: Intuition Behind the Constructionmentioning

confidence: 80%

Section: Background Readingmentioning

confidence: 99%

Section: Intuition Behind the Constructionmentioning

confidence: 99%

Section: Intuition Behind the Constructionmentioning

confidence: 99%

See 2 more Smart Citations

Applications of Hamilton’s compactness theorem for Ricci flow

Topping¹

2016

Geometric Analysis

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“…One of the underlying techniques of this paper was introduced in [Top14] where a sequence of complete Ricci flows with locally-controlled curvature was constructed that converged in the Cheeger-Gromov sense to an incomplete Ricci flow. With a great deal of extra technical effort, the same ideas should allow one to construct a Ricci flow with unbounded curvature precisely on an extremely general subset of time [0, ∞).…”

Section: If T < ∞ Then We Havementioning

confidence: 99%

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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Compactness theory of the space of Super Ricci flows

Bamler

2023

Invent. math.

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Remarks on Hamilton's Compactness Theorem for Ricci flow

Cited by 10 publications

References 15 publications

Applications of Hamilton’s compactness theorem for Ricci flow

Applications of Hamilton’s compactness theorem for Ricci flow

Ricci flows with bursts of unbounded curvature

Compactness theory of the space of Super Ricci flows

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