Four-dimensional steady gradient Ricci solitons with 3-cylindrical tangent flows at infinity

Bamler, Richard; Chow, Bennett; Deng, Yue; Ma, Zilu

doi:10.1016/j.aim.2022.108285

Cited by 8 publications

(5 citation statements)

References 19 publications

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“…Remark 4.3. A similar idea to the above argument for showing that Γ does not change along the subsequence was used in Proposition 2.2 of [4].…”

Section: Orbifold Structure In Dimensionmentioning

confidence: 88%

“…For p ∈ [2,4), we are unable to get a bound which is uniform in the parameter A, even if we only take the supremum over t ∈ [ T 2 , T ) and include an upper bound on diameter or only integrate over geodesic balls of fixed radius. It is unclear whether or not the estimate of Theorem 1.5 can be made uniform in the parameters A, T, p. Another interesting question is whether this estimate holds when p = 4.…”

Section: Introductionmentioning

confidence: 99%

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Ricci Flow with Ricci Curvature and Volume Bounded Below

Hallgren¹

2021

Preprint

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We show that a simply-connected closed four-dimensional Ricci flow whose Ricci curvature is uniformly bounded below and whose volume does not approach zero must converge to a C 0 orbifold at any finite-time singularity, so has an extension through the singularity via orbifold Ricci flow. Moreover, a Type-I blowup of the flow based at any orbifold point converges to a flat cone in the Gromov-Hausdorff sense, without passing to a subsequence. In addition, we prove L p bounds for the curvature tensor on time-slices for any p < 2. In higher dimensions, we show that every singular point of the flow is a Type-II point in the sense of [13], and that any tangent flow (in the sense of [6,8]) at a singular point is a static flow corresponding to a Ricci flat cone.

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“…Remark 4.3. A similar idea to the above argument for showing that Γ does not change along the subsequence was used in Proposition 2.2 of [4].…”

Section: Orbifold Structure In Dimensionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Ricci Flow with Ricci Curvature and Volume Bounded Below

Hallgren¹

2021

Preprint

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“…To conclude the introduction section, we remark that in an upcoming work of the second author with Bamler, Chow, Deng, and Ma [23], it is proved that the canonical form of a noncollapsed steady gradient Ricci soliton with nonnegative Ricci curvature always admits an asymptotic shrinking gradient Ricci soliton.…”

Section: Introductionmentioning

confidence: 97%

Perelman-type no breather theorem for noncompact Ricci flows

Cheng

2020

Preprint

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In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete 3-dimensional shrinking breathers. Second, we can show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient-a generalization of Naber's result in [11]. Furthermore, we develop a general condition for the existence of the asymptotic shrinking gradient Ricci soliton, which hopefully will contribute to the study of ancient solutions.

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“…Also, Brendle [11] extended his work [10] to dimension n ≥ 4 under the extra assumption that the steady soliton is asymptotically cylindrical; see also related recent works by Deng and Zhu [41,42]. For some of the recent progress on 4D gradient steady solitons, see, e.g., [3,61] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On Curvature Estimates for four-dimensional gradient Ricci solitons

Cao¹

2022

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This is a sequel to our paper [24], in which we investigated the geometry of 4-dimensional gradient shrinking Ricci solitons with half positive (nonnegative) isotropic curvature. In this paper, we mainly focus on 4-dimensional gradient steady Ricci solitons with nonnegative isotropic curvature (WPIC) or half nonnegative isotropic curvature (half WPIC). In particular, for 4D complete ancient solutions with WPIC, we are able to prove the nonnegativity of the Ricci curvature Rc ≥ 0 and bound the curvature tensor Rm by |Rm| ≤ R. For 4D gradient steady solitons with WPIC, we obtain a classification result. We also give a partial classification of 4D gradient steady Ricci solitons with half WPIC. Moreover, we obtain a preliminary classification result for 4D complete gradient expanding Ricci solitons with WPIC. Finally, motivated by the recent work [60], we improve our earlier results in [24] on 4D gradient shrinking Ricci solitons with half PIC or half WPIC, and also provide a characterization of complete gradient Kähler-Ricci shrinkers in complex dimension two among 4-dimensional gradient Ricci shrinkers.

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Four-dimensional steady gradient Ricci solitons with 3-cylindrical tangent flows at infinity

Cited by 8 publications

References 19 publications

Ricci Flow with Ricci Curvature and Volume Bounded Below

Ricci Flow with Ricci Curvature and Volume Bounded Below

Perelman-type no breather theorem for noncompact Ricci flows

On Curvature Estimates for four-dimensional gradient Ricci solitons

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