Perelman's entropy on ancient Ricci flows

Ma, Zilu

doi:10.1016/j.jfa.2021.109195

Cited by 12 publications

(11 citation statements)

References 24 publications

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“…On the other hand, the Nash entropy on an ancient solution always converges to the entropy of its asymptotic (metric) soliton (c.f. [4,46]). So if the ancient solution in question is the canonical form (M, g t ), then the Nash entropy converges to the quantity μ g defined in formula (2.2).…”

Section: The Triviality Of the Entropy On A Ricci Shrinkermentioning

confidence: 99%

Volume Growth Estimates of Gradient Ricci Solitons

Chan

2022

J Geom Anal

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Section: The Triviality Of the Entropy On A Ricci Shrinkermentioning

confidence: 99%

Volume Growth Estimates of Gradient Ricci Solitons

Chan

2022

J Geom Anal

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“…The main results of [Bam20b] were generalized to the non-compact setting in [MZ21,CMZ21a]; we include a brief overview for completeness:…”

Section: Appendix a Justification Of The Non-compact Casementioning

confidence: 99%

On the fundamental group of non-collapsed ancient Ricci flows

Bamler¹

2021

Preprint

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“…The Ricci soliton is an important field of study, since they arise naturally as rescaled limits of Ricci flows near singularities. The blow-up limit at a Type I finite-time singularity, or the backward scaled limit of a Type I ancient solution, is the canonical form of a Ricci shrinker (see [Per02,Nab10,EMT11,MZ21] and the references therein), whereas Type II and Type III scaled limits of Ricci flows are closely related to steady and expanding solitons, respectively (see [Ham93,Ham95,Cao97,CZ00,Lot07,GZ08]). Moreover, a…”

Section: Introductionmentioning

confidence: 99%

“…See also [Der17,Cha20,Che20,Zhl21] for other estimates on the expanding solitons. Ancient Ricci flows and steady solitons satisfying the estimate (1.4) have been recently studied in [MZ21,CMZ21a].…”

Section: Introductionmentioning

confidence: 99%

Hamilton-Ivey estimates for gradient Ricci solitons

Chan¹,

Ma²

2021

Preprint

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We first show that any 4-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy |Rm| ≤ cR for some positive constant c. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for 4-dimensional steady gradient solitons with linear scalar curvatrue decay and proper potential function. The technique is also used to establish a sufficient condition for a 3-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. As an application, we remove the positive curvature condition in a classification result by [Cho14] in dimension three and show that any 3-dimensional gradient Ricci expander C 2 asymptotic to C(S 2 ), dt 2 + αt 2 g S 2 is rotationally symmetric, where α ∈ (0, 1] is a constant and g S 2 is the standard metric on S 2 with constant curvature 1.

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Perelman's entropy on ancient Ricci flows

Cited by 12 publications

References 24 publications

Volume Growth Estimates of Gradient Ricci Solitons

Volume Growth Estimates of Gradient Ricci Solitons

On the fundamental group of non-collapsed ancient Ricci flows

Hamilton-Ivey estimates for gradient Ricci solitons

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