Ricci Flow and the Poincare Conjecture

Abstract

This manuscript contains a detailed proof of the Poincare Conjecture. The arguments we present here are expanded versions of the ones given by Perelman in his three preprints posted in 2002 and 2003. This is a revised version taking in account the comments of the referees and others. It has been reformatted in the AMS book style.

493 pages with over 30 figures and 3 pages of front material

References in corpus (2)

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

Cited by in corpus (102)

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• Completion of the Proof of the Geometrization Conjecture
• Mean curvature flow with surgery
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• Nonholonomic Ricci Flows: II. Evolution Equations and Dynamics
• Ricci Solitons and Einstein-Scalar Field Theory
• The minimal volume orientable hyperbolic 2-cusped 3-manifolds
• Short-time behaviour of a modified Laplacian coflow of G2-structures
• Towards physically motivated proofs of the Poincare' and geometrization conjectures
• Symmetric links and Conway sums: volume and Jones polynomial
• Fractional Nonholonomic Ricci Flows
• The Ricci flow on noncommutative two-tori
• On a classification of the gradient shrinking solitons
• Remarks on the cohomology of finite fundamental groups of 3-manifolds
• Global existence and convergence of solutions to gradient systems and applications to Yang-Mills gradient flow
• Nonholonomic Ricci Flows, Exact Solutions in Gravity, and Symmetric and Nonsymmetric Metrics
• Smooth approximations of the Conical Kahler-Ricci flows
• The Entropy of Lagrange-Finsler Spaces and Ricci Flows
• Ricci flow on open 3-manifolds and positive scalar curvature
• On the conditions to extend Ricci flow(II)
• Some Geometry and Analysis on Ricci Solitons
• Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids
• Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective
• Orientation reversal of manifolds
• Gravity localization in a string-cigar braneworld
• Marden's Tameness Conjecture: history and applications
• Some global, analytical and topological properties of regular black holes
• Which 3-manifold groups are Kähler groups?
• Bordered Floer homology and existence of incompressible tori in homology spheres
• A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows
• Numerical solutions of Einstein's equations for cosmological spacetimes with spatial topology S3 and symmetry group U(1)
• An operad for splicing
• Long-time behavior of 3 dimensional Ricci flow -- Introduction
• Quasi-hyperbolic planes in relatively hyperbolic groups
• Nonholonomic Ricci Flows and Parametric Deformations of the Solitonic pp--Waves and Schwarzschild Solutions
• Characterization of finite groups generated by reflections and rotations
• The Ricci flow of left invariant metrics on full flag manifold SU(3)/T from a dynamical systems point of view
• Swampland geometry and the gauge couplings
• The global existence and convergence of the Calabi flow on ${\mathbb{C}}^n/{\mathbb{Z}}^n+i{\mathbb{Z}}^n$
• Entropy Functionals, Sobolev Inequalities and kappa-Noncollapsing Estimates along the Ricci Flow
• Entropy functionals and thermodynamics of relativistic geometric flows, stationary quasi-periodic Ricci solitons, and gravity
• Classical and quantum geometric information flows and entanglement of relativistic mechanical systems
• Nonholonomic Ricci Flows: Exact Solutions and Gravity
• Local min-max surfaces and strongly irreducible minimal Heegaard splittings
• Nonholonomic Ricci Flows: III. Curve Flows and Solitonic Hierarchies
• Constantin Carathéodory axiomatic approach and Grigory Perelman thermodynamics for geometric flows and cosmological solitonic solutions
• Geometric simple connectivity and finitely presented groups
• Systoles and Dehn surgery for hyperbolic 3-manifolds
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• Kaluza--Klein gravity & cosmology emerging from G. Perelman's entropy functionals and quantum geometric information flows
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• Diffusion and Self-Organized Criticality in Ricci Flow Evolution of Einstein and Finsler Spaces
• Producing 3d Ricci flows with non-negative Ricci curvature via singular Ricci flows
• Finite group actions and G-monopole classes on smooth 4-manifolds
• The compression body graph has infinite diameter
• Asymptotic behavior of positively curved steady Ricci Solitons, II
• Long-time analysis of 3 dimensional Ricci flow I
• Extremal metrics for spectral functions of Dirac operators in even and odd dimensions
• Topological spherical space forms
• Asymptotic behavior of a matter filled universe with exotic topology
• The maximal number of exceptional Dehn surgeries
• A lower bound for the scalar curvature of the standard solution of the Ricci flow
• The Geometry and Arithmetic of Kleinian Groups
• Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions
• All finitely presented groups are QSF
• On long-time existence for the flow of static metrics with rotational symmetry
• Nonsingular Ricci flow on a noncompact manifold in dimension three
• Two-dimensional Ricci flow as a stochastic process
• Minimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds
• Ancient solutions to the Ricci flow with isotropic curvature conditions
• Einstein flow with matter sources: stability and convergence
• Compact manifolds of dimension $n\ge 12$ with positive isotropic curvature
• An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions
• Introduction to Lie groups, isometric and adjoint actions and some generalizations
• A harmonic map flow associated with the standard solution of Ricci flow
• Local derivative estimates for the heat equation coupled to the Ricci flow
• Ricci flow with bounded curvature integrals
• Perelman's collapsing theorem for 3-manifolds
• Monotonic invariants under blowups
• Homogeneous metric ANR-compacta
• Perelman's functional and reduced volume
• Characteristics of conformal Ricci soliton on warped product spaces
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• Ricci flow for homogeneous compact models of the universe
• Harnack type inequalities for matrices in majorization
• A topological equivalence relation for finitely presented groups
• Wave Approximation of Backward Heat Equation with Ricci Flow
• On the Alexandroff-Borsuk problem
• A Simple Proof On Poincaré Conjecture
• Classification of Ricci solitons on Euclidean hypersurfaces
• On quotient orbifolds of hyperbolic 3-manifolds of genus two
• The Ricci flow of the `RP3 geon' and noncompact manifolds with essential minimal spheres
• Coupling of Brownian motions and Perelman's L-functional
• Connection between continuous and digital n-manifolds and the Poincare conjecture
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• An example of how the Ricci Flow can increase topological entropy
• Geometric Structure of Higher-Dimensional Spheres
• The Kähler-Ricci flow on compact Kähler manifolds
• Two dimensional axisymmetric smooth lattice Ricci flow
• Ricci flow on Poincare' and Thurston's Geometrization Conjecture
• Renormalization group flow, Entropy and Eigenvalues