On the fundamental group of non-collapsed ancient Ricci flows

Bamler, Richard

doi:10.48550/arxiv.2110.02254

Cited by 3 publications

(6 citation statements)

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“…On the other hand, one can extend Theorem 6.2 for complete non-compact Ricci flow (M, g(t)) t∈[−T,0) with bounded curvature on compact time-intervals and bounded Nash entropy, or the Ricci flow induced by a Ricci shrinker (cf. [5] and [46]). Remark 6.4.…”

Section: Uniqueness Of the Tangent Flowmentioning

confidence: 99%

The rigidity of Ricci shrinkers of dimension four

Wang

2019

Trans. Amer. Math. Soc.

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In dimension 4, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinkers satisfying some natural geometric properties.Ricci shrinker is also called as gradient shrinking Ricci soliton. Direct calculation shows that ∇(R + |∇ f | 2 − f ) = 0. Adding f by a constant if necessary, we assume throughout that3) where µ = µ(g, 1) is the entropy functional of Perelman(c.f. [35]). The Ricci shrinker (1.1) was introduced by Hamilton [22] in mid 1980's. As critical points of Perelman's µ-entropy, the Ricci shrinkers play important roles in the singularity analysis of the Ricci flow. For example, it is proved by Enders-Müller-Topping [20] that the proper rescaling limit of a type-I singularity is always a nontrivial Ricci shrinker. More information and references can be found in Chapter 30 of the book [18] by Chow and his coauthors. In dimension 2, the only Ricci shrinkers are R 2 , S 2 and RP 2 with standard metrics, due to the classification of Hamilton [23]. In dimension 3, based on the breakthrough of Perelman([35], [36]), through the efforts of Naber [33], Ni-Wallach [34] and Cao-Chen-Zhu [6], etc, we know that R 3 , S 2 × R, S 3 and their quotients are all the possible Ricci shrinkers. In dimension 4 and higher, much fewer is known about Ricci shrinkers. Typically, some extra conditions of the curvature operator(e.g. [3], [9], [7], [17], etc), or geometric properties at infinity(c.f. [27], [32]) are required to draw definite geometry conclusion. We refer the readers the surveys [4], [5] for more detailed picture. Without such extra conditions, it is still not clear how to classify Ricci shrinkers. However, one can study the moduli of Ricci shrinkers. In [8], Cao-Sesum showed the weak compactness of the moduli of Kähler Ricci solitons with uniformly bounded diameter and uniformly lower bound of Ricci curvature and µ-functional. The extra conditions were gradually weakened or removed by X. Zhang [39], Weber [38], Chen-Wang [15], Z.L. Zhang [40] and Haslhofer-Müller [24], etc.In this article, we focus on the study of the moduli M * (A, H), which consists of 4d Ricci shrinkers which have uniform entropy lower bound and Harnack inequality of scalar curvature on unit ball(c.f. Definition 2.3 for precise definition). Moreover, we require each Ricci shrinker has bounded(not uniformly) scalar curvature. In light of the results of Haslhofer-Müller(c.f. Theorem 2.6), it is known that such moduli has weak compactness. In other words, any sequence of Ricci shrinkers in M * (A, H) sub-converges to an orbifold Ricci shrinker with locally finite singularities, in the pointed-Gromov-Hausdorff topology. Now we reverse the process and ask the following question:What kind of orbifolds can be approximated by a sequence of Ricci shrinkers in M * (A, H)?Note that flat cones R 4 /Γ are naturally the s...

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Section: Uniqueness Of the Tangent Flowmentioning

confidence: 99%

The rigidity of Ricci shrinkers of dimension four

Wang

2019

Trans. Amer. Math. Soc.

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“…Though Bamler [2] proved the above result for Ricci flows on closed manifolds, yet one may check the proof of Theorem 6.1 in [2] and easily verify its validity for Ricci flows with bounded geometry on each time-slice; one may need to apply Theorem 4.4 of [21] in this verification. Indeed, Bamler's recent preprint [3] also pointed out the fact that his theory in [2] can be generalized to this more general case. Fortunately, all the Ricci flows we work with in this paper satisfy this condition.…”

Section: The Asymptotic Shrinkermentioning

confidence: 99%

On the noncollapsedness of positively curved Type I ancient Ricci flows

Cheng

2023

Mathematische Nachrichten

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In this paper, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such ancient solutions must be noncollapsed on all scales. This furthermore gives a complete classification for three‐dimensional noncompact Type I ancient solutions without assuming the noncollapsing condition.

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“…Notice that the results in [3] and [4] are already generalized by Bamler to Ricci flows with complete time-slices and bounded curvature on compact time-intervals (cf. [5]). In [5, Appendix A], some issues in the non-compact case are addressed and can be resolved similarly in the setting of Ricci shrinkers by the results and techniques developed in previous sections.…”

Section: Metric Flows and F-convergencementioning

confidence: 99%

“…Proof. The conditions (1)- (5) in the definition of the metric flow can be easily checked. Condition (6) follows from (3.15) and ( 7) from the semigroup property (3.1).…”

Section: Appendix a Integral Estimates For The Conjugate Heat Kernel ...mentioning

confidence: 99%