A uniform Sobolev inequality under Ricci flow

By Qi S. Zhang. 2007

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Let ${\bf M}$ be a compact Riemannian manifold and the metrics $g=g(t)$ evolve by the Ricci flow. We prove the following result. The Sobolev imbedding by Aubin or Hebey, perturbed by a scalar curvature term and modulo sharpness of constants, holds uniformly for $({\bf M}, g(t))$ for all time if the Ricci flow exists for all time; and if the Ricci flow develops a singularity in finite time, then the same Sobolev imbedding holds uniformly after a standard normalization. As a consequence, long time non-collapsing results are derived, which improve Perelman's local non-collapsing results. An application to 3-d Ricci flow with surgery is also presented.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/42897b870e7fb04ebd909522d9e4a8cc5c056cf8

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Kähler–Ricci flow and optimal degenerations (2016, background): “It is well known [33] that ft is smooth, and in addition the uniform control of the Sobolev constant [41, 42] and the scalar curvature [34] along the flow implies that we have a uniform bound |Φt| < C, where Φt = e −ft/2.”
Regularity of Kähler–Ricci flows on Fano manifolds (2013, background): “When the Riemannian manifold is a spatial slice of Kähler-Ricci flow on a Fano manifold, the Sobolev constant has a global estimate by [56] [55].”
New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow (2012, methodology): “In [20], Davies’s method [7] for deriving L∞ heat kernel estimates from log-Sobolev inequalities is applied to the Ricci flow, based on Corollary 2.”
Space of Ricci Flows I (2009, background): “For other important progress in Kähler Ricci flow, we refer interested readers to the following papers(far away from being complete): [84], [122], [120], [111], [123], [91], [111], [90], [80], [113], [102], as well as references listed therein.”
Lectures on Stability and Constant Scalar Curvature (2008, result): “Combined with Perelman’s recent results [100] and subsequent uniform Sobolev estimates [160, 164 ], they readily imply a version of Perelman’s unpublished result, namely, that when Aut 0 (X) = 0, the existence of a K¨ahler-Einstein metric implies the convergence in C ∞ of the K¨ahler-Ricci flow.”