Constructs closed aspherical 4-manifolds that are homeomorphic but not diffeomorphic, providing counterexamples to the smooth Borel conjecture in dimension 4.
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Ricci flow with surgery on three-manifolds
18 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print; the exceptions are (1) the statement that manifolds that can collapse with local lower bound on sectional curvature are graph manifolds - this is deferred to a separate paper, since the proof has nothing to do with the Ricci flow, and (2) the claim on the lower bound for the volume of maximal horns and the smoothness of solutions from some time on, which turned out to be unjustified and, on the other hand, irrelevant for the other conclusions.
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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.
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.
Proves Lojasiewicz inequality for W-entropy near generalized cylinders in Ricci flow, yielding strong uniqueness of tangent flows and horizontal parabolic k-rectifiability of the corresponding singularity set.
κ-solutions with round cylinder asymptotic shrinker are uniformly PIC, implying classification as the round shrinking cylinder, Bryant steady soliton, or Perelman's ancient solution.
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.
The paper extends Perelman's entropy functionals to 4D Lorentzian spacetimes and proves long-time well-posedness of Ricci flow using gradient flow properties of the coupled system.
Inflationary models on Thurston geometries admit a stable anisotropic fixed point triggered by eccentricity-induced vector field coupling to the inflaton.
Proves global well-posedness and smooth convergence of renormalized metrics to constant negative scalar curvature for large-data vacuum Einstein-Λ flow on negative Yamabe type 3-manifolds, confirming Ringström conjecture via integrable damping from Λ.
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.
Shrinking gradient Ricci solitons with constant scalar curvature k/2, nonnegative Ricci curvature and sectional curvature bounded by 1/(2(k-1)) are finite quotients of R^{n-k} x S^k; those with R=(n-2)/2 and vanishing Weyl curvature on level sets of f are finite quotients of R^2 x S^{n-2}.
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.
An overview of the geometrisation theorem for 3-manifolds that explains its content and effects in various situations.
citing papers explorer
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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.
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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.
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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.
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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.
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Strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow
Proves Lojasiewicz inequality for W-entropy near generalized cylinders in Ricci flow, yielding strong uniqueness of tangent flows and horizontal parabolic k-rectifiability of the corresponding singularity set.
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$\kappa$-solutions with the round cylinder as an asymptotic shrinker
κ-solutions with round cylinder asymptotic shrinker are uniformly PIC, implying classification as the round shrinking cylinder, Bryant steady soliton, or Perelman's ancient solution.
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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.
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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.
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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.
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Well-posedness of Ricci Flow in Lorentzian Spacetime and its Entropy Formula
The paper extends Perelman's entropy functionals to 4D Lorentzian spacetimes and proves long-time well-posedness of Ricci flow using gradient flow properties of the coupled system.
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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.
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A large data result for vacuum Einstein's equations
Proves global well-posedness and smooth convergence of renormalized metrics to constant negative scalar curvature for large-data vacuum Einstein-Λ flow on negative Yamabe type 3-manifolds, confirming Ringström conjecture via integrable damping from Λ.
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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.
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A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature
Shrinking gradient Ricci solitons with constant scalar curvature k/2, nonnegative Ricci curvature and sectional curvature bounded by 1/(2(k-1)) are finite quotients of R^{n-k} x S^k; those with R=(n-2)/2 and vanishing Weyl curvature on level sets of f are finite quotients of R^2 x S^{n-2}.
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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.
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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.
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Notes on harmonic-Ricci flow on surface
Establishes several evolution formulas for functionals along the harmonic-Ricci flow on surfaces with boundary.
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Geometrisation of 3-manifolds
An overview of the geometrisation theorem for 3-manifolds that explains its content and effects in various situations.