A uniform Sobolev inequality under Ricci flow

Abstract

Let $\mathbf{M}$ 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 $(\mathbf{M},g(t))$ 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.

An application to 3-d Ricci flow with surgery is added

References in corpus (1)

• Uniform Sobolev inequalities for manifolds evolving by Ricci flow

Cited by in corpus (8)

• The logarithmic Sobolev inequality along the Ricci flow
• Space of Ricci flows (II)
• The Calabi flow on Kähler surface with bounded Sobolev constant
• Stability of Kähler-Ricci flow
• First variation of the Log Entropy functional along the Ricci flow
• The (logarithmic) Sobolev inequalities along geometric flow and applications
• Volume bounds of the Ricci flow on closed manifolds
• Multiplier ideal sheaves and geometric problems