Regularity theory for type I Ricci flows

Gianniotis, Panagiotis

doi:10.1007/s00526-019-1644-7

Cited by 4 publications

(4 citation statements)

References 39 publications

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“…A similar stratification was studied for Riemannian manifolds satisfying (1.1),(1.2) in [CN13], where it was used to prove L p estimates for the Riemannian curvature tensor of Einstein manifolds. The Ricci flow version was again first studied for Type-I Ricci flows [Gia19], while two different but related definitions for general Ricci flows were used in [Bam20c]. These are the quantitative strata S ǫ,k r 1 ,r 2 and the weak quantitative strata S ǫ,k r 1 ,r 2 .…”

Section: Suppose (M 2nmentioning

confidence: 99%

Tangent Flows of Kähler Metric Flows

Hallgren¹,

Jian²

2022

Preprint

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We improve the description of F-limits of noncollapsed Ricci flows in the Kähler setting. In particular, the singular strata S k of such metric flows satisfy S 2j = S 2j+1 . We also prove an analogous result for quantitative strata, and show that any tangent flow admits a nontrivial oneparameter action by isometries, which is locally free on the cone link in the static case. The main results are established using parabolic regularizations of conjugate heat kernel potential functions based at almost-selfsimilar points, which may be of independent interest.

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Section: Suppose (M 2nmentioning

confidence: 99%

Tangent Flows of Kähler Metric Flows

Hallgren¹,

Jian²

2022

Preprint

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“…In fact, the dimension of the singular set is related to the rate of convergence of its volume to zero as t approaches the singular time. This approach revealed useful in the study of Type I Ricci flows by Gianniotis (see [18,19]) and we briefly recall the heuristic behind it. An actual estimate on the (intrinsic) Minkowski content cannot be available in general.…”

Section: The Ricci Singular Sets and A Localised Version Of Sesum's R...mentioning

confidence: 99%

“…Given Theorem 1.1, it is natural to ask whether in general for an n-dimensional Ricci flow whose scalar curvature remains bounded up to a finite singular time the singular set has codimension 8. In this context, the codimension has to be understood in a sense similar to the one defined in the works of Gianniotis [18,19], namely as a decay in time of the volume of the singular set. We give a partial answer to this question in Theorem 1.12 below.…”

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confidence: 99%

A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature

Buzano¹,

Matteo²

2020

Preprint

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We prove that Ricci flows with bounded scalar curvature cannot develop finite time singularities in dimensions less than eight. In order to study such flows in higher dimensions, we then develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points. In particular, near any singular point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a result of Enders, Topping and the first author that relied on a global Type I assumption. We also prove analogous results for the Ricci tensor, as well as a localised version of Sesum's result, namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at least at a Type I rate. Combining these and other results, we then show that for Ricci flows with bounded scalar curvature in higher dimensions the singular set has codimension eight in a suitable sense.

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“…In fact, the dimension of the singular set is related to the rate of convergence of its volume to zero as t approaches the singular time. This approach revealed useful in the study of Type I Ricci flows by Gianniotis (see [ 21 , 22 ]) and we briefly recall the heuristic behind it. An actual estimate on the (intrinsic) Minkowski content cannot be available in general.…”

Section: Introductionmentioning

confidence: 99%

A local singularity analysis for the Ricci flow and its applications to Ricci flows with bounded scalar curvature

Buzano

Matteo

2022

Calc. Var.

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We develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points. In particular, near any singular point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a result of Enders, Topping and the first author that relied on a global Type I assumption. We also prove analogous results for the Ricci tensor, as well as a localised version of Sesum’s result, namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at least at a Type I rate. Finally, we show some applications of the theory to Ricci flows with bounded scalar curvature.

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Regularity theory for type I Ricci flows

Cited by 4 publications

References 39 publications

Tangent Flows of Kähler Metric Flows

Tangent Flows of Kähler Metric Flows

A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature

A local singularity analysis for the Ricci flow and its applications to Ricci flows with bounded scalar curvature

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