Constructs closed aspherical 4-manifolds that are homeomorphic but not diffeomorphic, providing counterexamples to the smooth Borel conjecture in dimension 4.
hub
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
16 Pith papers cite this work. Polarity classification is still indexing.
16
Pith papers citing it
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.
hub tools
citation-role summary
background 2
citation-polarity summary
verdicts
UNVERDICTED 16roles
background 2polarities
background 2representative citing papers
A perturbative Ricci-flow formulation in gravity yields a renormalization scheme for Newton's constant that exhibits a non-Gaussian fixed point at two-loop order.
A discrete Ricci flow on graphs converges exponentially to prescribed Lin-Lu-Yau curvatures iff attainable, with an explicit max-edge-density condition for constant curvature on girth-at-least-6 graphs.
Riemannian manifolds with a closed parallel torsion 3-form are locally N × G (G semisimple), enabling simplified proofs and explicit classification of strong G2, Spin(7), and certain 8D HKT manifolds.
Constructs O(2)×O(n-1)-invariant ancient Ricci flows with positive curvature operator and bounded girth for n≥3 and determines their backward asymptotic limits.
Renyi differential privacy for manifold-valued data is characterized via dimension-free Harnack inequalities and governed by Ricci curvature, with heat diffusion and Langevin mechanisms plus application to private Frechet mean estimation.
The Calabi flow on finite graphs converges globally if and only if a weight function exists realizing the prescribed curvature, with convergence for constant curvature under topological conditions.
In a Thurston-geometry-dependent gravity theory, non-tilted BKS cosmologies admit shear-free perfect-fluid and static vacuum solutions for all topologies, isotropize under positive Lambda except for some Bianchi II cases, and never recollapse when the weak energy condition holds.
Establishes a Lojasiewicz inequality for pointed W-entropy near cylindrical singularities in Ricci flow and applies it to prove strong uniqueness of the cylindrical tangent flow at the first singular time under a fixed gauge.
Inflationary models on Thurston geometries admit a stable anisotropic fixed point triggered by eccentricity-induced vector field coupling to the inflaton.
An explicit formula is given for the local connection 1-form α on the anti-canonical bundle of a twisted almost Kähler structure, yielding the Chern-Ricci form as ρ = -dα.
Smooth compact Ricci flows are characterized weakly solely via metrics and measures by defining super Ricci flows and adding a saturation condition to recover equality.
The authors introduce Thurston spacetimes as cosmological backgrounds, solve transfer equations for temperature and polarization patterns, and analyze symmetries in Stokes parameters to attempt isolation of individual geometries.
Establishes several evolution formulas for functionals along the harmonic-Ricci flow on surfaces with boundary.
The monograph organizes and derives classical Riemannian geometry structures explicitly in coordinate and matrix form for direct use in optimization algorithms on nonlinear manifolds.
An overview of the geometrisation theorem for 3-manifolds that explains its content and effects in various situations.
citing papers explorer
-
Exotic aspherical 4-manifolds
Constructs closed aspherical 4-manifolds that are homeomorphic but not diffeomorphic, providing counterexamples to the smooth Borel conjecture in dimension 4.
-
The perturbative Ricci flow in gravity
A perturbative Ricci-flow formulation in gravity yields a renormalization scheme for Newton's constant that exhibits a non-Gaussian fixed point at two-loop order.
-
The Ricci flow with prescribed curvature on graphs
A discrete Ricci flow on graphs converges exponentially to prescribed Lin-Lu-Yau curvatures iff attainable, with an explicit max-edge-density condition for constant curvature on girth-at-least-6 graphs.
-
On the rigidity of special and exceptional geometries with torsion a closed $3$-form
Riemannian manifolds with a closed parallel torsion 3-form are locally N × G (G semisimple), enabling simplified proofs and explicit classification of strong G2, Spin(7), and certain 8D HKT manifolds.
-
Ancient Ricci flows of bounded girth
Constructs O(2)×O(n-1)-invariant ancient Ricci flows with positive curvature operator and bounded girth for n≥3 and determines their backward asymptotic limits.
-
Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms
Renyi differential privacy for manifold-valued data is characterized via dimension-free Harnack inequalities and governed by Ricci curvature, with heat diffusion and Langevin mechanisms plus application to private Frechet mean estimation.
-
The Calabi flow with prescribed curvature on finite graphs
The Calabi flow on finite graphs converges globally if and only if a weight function exists realizing the prescribed curvature, with convergence for constant curvature under topological conditions.
-
Bianchi cosmologies in a Thurston-based theory of gravity
In a Thurston-geometry-dependent gravity theory, non-tilted BKS cosmologies admit shear-free perfect-fluid and static vacuum solutions for all topologies, isotropize under positive Lambda except for some Bianchi II cases, and never recollapse when the weak energy condition holds.
-
Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow
Establishes a Lojasiewicz inequality for pointed W-entropy near cylindrical singularities in Ricci flow and applies it to prove strong uniqueness of the cylindrical tangent flow at the first singular time under a fixed gauge.
-
Cosmological viability of anisotropic inflation in Thurston spacetimes
Inflationary models on Thurston geometries admit a stable anisotropic fixed point triggered by eccentricity-induced vector field coupling to the inflaton.
-
On the Chern-Ricci form of a twisted almost K\"{a}hler structure
An explicit formula is given for the local connection 1-form α on the anti-canonical bundle of a twisted almost Kähler structure, yielding the Chern-Ricci form as ρ = -dα.
-
On weak formulations of (super) Ricci flows
Smooth compact Ricci flows are characterized weakly solely via metrics and measures by defining super Ricci flows and adding a saturation condition to recover equality.
-
Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes
The authors introduce Thurston spacetimes as cosmological backgrounds, solve transfer equations for temperature and polarization patterns, and analyze symmetries in Stokes parameters to attempt isolation of individual geometries.
-
Notes on harmonic-Ricci flow on surface
Establishes several evolution formulas for functionals along the harmonic-Ricci flow on surfaces with boundary.
-
Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations
The monograph organizes and derives classical Riemannian geometry structures explicitly in coordinate and matrix form for direct use in optimization algorithms on nonlinear manifolds.
-
Geometrisation of 3-manifolds
An overview of the geometrisation theorem for 3-manifolds that explains its content and effects in various situations.