2006/10/12 by Peter M. Topping · 1 citation
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research #Ricci flow #Mathematics #Gravitational singularity #Pure mathematics #Ricci curvature #Conjecture #Flow (mathematics) #Exposition (narrative) #Compact space #Curvature #Mathematical analysis #Geometry
paper · doi:10.1017/cbo9780511721465
openalex publication_date 2006/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Hamilton's Ricci flow has attracted considerable attention since its introduction in 1982, owing partly to its promise in addressing the Poincaré conjecture and Thurston's geometrization conjecture. This book gives a concise introduction to the subject with the hindsight of Perelman's breakthroughs from 2002/2003. After describing the basic properties of, and intuition behind the Ricci flow, core elements of the theory are discussed such as consequences of various forms of maximum principle, issues related to existence theory, and basic properties of singularities in the flow. A detailed exposition of Perelman's entropy functionals is combined with a description of Cheeger-Gromov-Hamilton compactness of manifolds and flows to show how a 'tangent' flow can be extracted from a singular Ricci flow. Finally, all these threads are pulled together to give a modern proof of Hamilton's theorem that a closed three-dimensional manifold whichcarries a metric of positive Ricci curvature is a spherical space form.