1993/01/01 by Richard S. Hamilton · 122 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Cosmology and Gravitation Theories #Flow (mathematics) #Geometric Analysis and Curvature Flows #Geometry #Mathematics #Ricci curvature #Ricci flow
paper · pdf · doi:10.4310/jdg/1214454093
published in Journal of Differential Geometry 38(1) (Lehigh University)
openalex publication_date 1993/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The resultWe consider solutions to the Ricci flow equation on a manifold X of dimension n .We say the solution is eternal if it is defined for all time -oo < t < oo.We are interested in solutions which are complete (which is a way of saying they are also defined for "all" of space) and which have their Riemannian curvature uniformly bounded for all space and time.This is a serious restriction; by the work of W. X.Shi [2] we know then that all the covariant derivatives of the curvature are bounded.Examples of eternal solutions which are complete with bounded curvature are provided by solitons.These are solutions which move under a one-parameter family of diffeomorphisms.If this comes from exponentiating a vector field V , then we have a soliton when