Perelman's functional and reduced volume

2010/04/11 by Hassan Jolany, Jolany, Hassan
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1004.1785

openalex publication_date 2010/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In recent years, there has seen much interest and increased research activities on Perelman's paper. Section one and two of this paper aim to establish Perelman's local non-collapsing result for the Ricci flow. This will provide a positive lower bound on the injectivity radius for the Ricci flow under blow-up analysis. We also discuss the gradient flow formalism of the Ricci flow and Perelman's motivation from physics. In this paper, we survey some of the recent progress on Perelman's functional, reduced volume and reduced length.

Citations

• Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective
• Ricci flow with surgery on three-manifolds

Related

• Simple proofs of some results of Perelman on Ricci flow
• Ricci flow on Poincare' and Thurston's Geometrization Conjecture
• On the asymptotic reduced volume of the Ricci flow
• An ε-regularity theorem for Perelman's reduced volume
• On the l-Function and the Reduced volume of Perelman II