A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

2006/01/01 by Huai-Dong Cao, Xi-Ping Zhu · 7 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research #Ricci flow #Mathematics #Poincaré conjecture #Flow (mathematics) #Pure mathematics #Geometry #Ricci curvature

paper · pdf · doi:10.4310/ajm.2006.v10.n2.a2

openalex publication_date 2006/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/18

Abstract

In this paper, we give a complete proof of the Poincar and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.

Citations

• Four-manifolds with positive isotropic curvature
• Metric Structures for Riemannian and Non-Riemannian Spaces
• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• Three-manifolds with positive Ricci curvature
• Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
• Gaussian densities and stability for some Ricci solitons
• The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature
• Deforming the metric on complete Riemannian manifolds
• Limiting behaviour of the Ricci flow
• Elliptic Partial Differential Equations of Second Order
• On the parabolic kernel of the Schrödinger operator
• A theorem on holonomy
• An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures
• Projective Manifolds with Ample Tangent Bundles
• Manifolds with positive curvature
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Ricci flow with surgery on three-manifolds
• Strong rigidity ofQ-rank 1 lattices
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

Cited by

• Off-diagonal solutions in Einsteingravity modeling f(R) gravity and dynamical darkenergy vs Lambda CDM cosmology
• Collapsing irreducible 3-manifolds with nontrivial fundamental group
• A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds
• Notes on Perelman’s papers
• Geometrization conjecture [wikipedia]
• Grigori Perelman [wikipedia]
• Scalar curvature [wikipedia]