Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

Abstract

Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. We show that for any initial riemannian metric on M the solution to the Ricci flow with surgery, defined in our previous paper math.DG/0303109, becomes extinct in finite time. The proof uses a version of the minimal disk argument from 1999 paper by Richard Hamilton, and a regularization of the curve shortening flow, worked out by Altschuler and Grayson.

7 pages

Cited by in corpus (31)

• Completion of the Proof of the Geometrization Conjecture
• Remarks on the cohomology of finite fundamental groups of 3-manifolds
• Nonholonomic Ricci Flows, Exact Solutions in Gravity, and Symmetric and Nonsymmetric Metrics
• Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective
• Weak collapsing and geometrisation of aspherical 3-manifolds
• Triangulated Manifolds with Few Vertices: Geometric 3-Manifolds
• The Normalized Ricci Flow on Four-Manifolds and Exotic Smooth Structures
• Geometry, Heegaard splittings and rank of the fundamental group of hyperbolic 3-manifolds
• Multivariance as a crucial property of microcosm
• Stable Cohomotopy Seiberg-Witten Invariants of Connected Sums of Four-Manifolds with Positive First Betti Number
• Width and finite extinction time of Ricci flow
• Subgroups of the mapping class group from the geometrical viewpoint
• Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds
• Curve Shortening Flow in a Riemannian Manifold
• The Existence of Soliton Metrics for Nilpotent Lie Groups
• On the Farrell-Jones and related Conjectures
• Minimally invasive surgery for Ricci flow singularities
• Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds
• The maximal number of exceptional Dehn surgeries
• The Ricci Flow for Nilmanifolds
• Nonsingular Ricci flow on a noncompact manifold in dimension three
• Some applications of Ricci flow to 3-manifolds
• Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere
• Width and mean curvature flow
• Structures of Three-Manifilds
• Existence of Good Sweepouts on Closed Manifolds
• Mean Curvature Driven Ricci Flow
• Surface subgroups of Kleinian groups with torsion
• Surgical distance between lens spaces
• Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman
• Connection between continuous and digital n-manifolds and the Poincare conjecture