Paper · Nov 10, 2021
A local Sobolev inequality on Ricci flow and its applications
As on arXiv
Abstract
In this article, we prove a local Sobolev inequality for complete Ricci flows. Our main result is that the local -functional of a disk on a Ricci flow depends only on the Nash entropy based at the center of the disk, and consequently depends only on the volume of the disk. Furthermore, we introduce some applications of this local Sobolev inequality. These applications reveal the way in which the local geometry evolves along Ricci flow. In particular, we show that several classical theorems related to Perelman's monotonicity formula can be derived from our results.
Comments: New results added in the new version
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Local smooth convergence of -limit flows
Topics Perelman entropy · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “Nash entropy”
-
An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons
Topics Perelman entropy · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “Nash entropy”
-
On noncollapsed -limit metric solitons
Topics Ricci flow · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “Sobolev inequality”
-
Rigidity and gap theorems for Ricci shrinkers
Topics Ricci flow · Same authors Pak-Yeung Chan, Yongjia Zhang
-
Bochner formulas, functional inequalities and generalized Ricci flow
Topics Perelman entropy · Shared terms “Sobolev inequality”
-
A local gap theorem for Ricci shrinkers
Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “local geometry”
-
Lower bounds for the scalar curvatures of Ricci flow singularity models
Topics Ricci flow · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “monotonicity formula”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.12Entropy, Information and Diffusion
Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
- 6B.16Eigenvalue Problems and Perelman’s λ
Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.