Constructs uncountably many pairwise conformally inequivalent non-rotationally symmetric type II ancient Yamabe flows on S^n (n≥3) via non-radial inner-outer gluing after stereographic projection.
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The entropy formula for the Ricci flow and its geometric applications
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abstract
We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.
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representative citing papers
Normalized Kähler-Ricci flow converges in Gromov-Hausdorff sense to the metric completion of the twisted Kähler-Einstein metric on the canonical model when the canonical bundle is semiample.
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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.
Hypercomplete ANR homology manifolds are cohomologically smooth Poincaré duality complexes whose Spivak fibration destabilizes to a pointed S^d-fibration; conical homotopy manifolds are topological manifolds.
Proves a J-adapted Levi-Malcev decomposition for many 2-step solvable Lie algebras, confirming the Fino-Vezzoni conjecture for unimodular cases and characterizing SKT metrics on completely solvable ones.
Functional renormalization group flow is recast as a potential-modified Ricci flow on the Fisher information metric of coupling space, with an RG-flow entropy serving as the infinite-dimensional analog of Perelman's F-entropy and fixed points appearing as Ricci solitons.
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.
Applies discrete Ricci flow and curvature to networks viewed as geometric objects to identify communities via geometric decomposition.
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.
Chern-Ricci flow on Hermitian minimal models of general type admits uniform estimates yielding subsequential Gromov-Hausdorff convergence under a local Kähler assumption.
For 3D Ricci flows, time-slices converge in the Gromov–Hausdorff sense to an intrinsic terminal time-slice, and the singular set is horizontally parabolic 1-rectifiable with time image of zero 1/2-dimensional measure.
Existence, uniqueness, and convergence of the Ollivier Ricci flow with prescribed curvature are established on infinite graphs with girth at least 6.
Closed orientable 4-manifolds with transversely oriented C² codim-1 foliations by R³-leaves are homeomorphic to T⁴ (and diffeomorphic under an extra smooth 1-form condition).
Complete non-compact gradient generalized m-quasi-Einstein manifolds with R≤0, λ>0, m>1, and μ=1/m are Euclidean under a variety of integrability or volume-growth conditions on the weighted function v=e^{-f/m}λ.
Singular Kähler-Ricci shrinkers from noncollapsed limits are complex analytic varieties with log terminal singularities, yielding geometric consequences including simple connectedness and unique tangent cones.
Proves sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along Ricci flow via monotonicity formula, with consequences for concentration estimates, log-Sobolev inequalities, and related results.
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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.
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Inflationary models on Thurston geometries admit a stable anisotropic fixed point triggered by eccentricity-induced vector field coupling to the inflaton.
Establishes symmetry principles and uniqueness characterizations for steady and expanding gradient Ricci solitons with specified asymptotic links.
citing papers explorer
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Uncountably many non-rotationally symmetric type II ancient Yamabe flows on the sphere
Constructs uncountably many pairwise conformally inequivalent non-rotationally symmetric type II ancient Yamabe flows on S^n (n≥3) via non-radial inner-outer gluing after stereographic projection.
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Gromov-Hausdorff limits of immortal K\"ahler-Ricci flows
Normalized Kähler-Ricci flow converges in Gromov-Hausdorff sense to the metric completion of the twisted Kähler-Einstein metric on the canonical model when the canonical bundle is semiample.
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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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Homology manifolds via six functor formalisms
Hypercomplete ANR homology manifolds are cohomologically smooth Poincaré duality complexes whose Spivak fibration destabilizes to a pointed S^d-fibration; conical homotopy manifolds are topological manifolds.
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A Levi-type decomposition on two-step solvable Lie algebras with a complex structure
Proves a J-adapted Levi-Malcev decomposition for many 2-step solvable Lie algebras, confirming the Fino-Vezzoni conjecture for unimodular cases and characterizing SKT metrics on completely solvable ones.
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Functional Renormalization Group as a Ricci Flow: An \(\mathcal{F}\)-Entropy Perspective on Information Metric Dynamics
Functional renormalization group flow is recast as a potential-modified Ricci flow on the Fisher information metric of coupling space, with an RG-flow entropy serving as the infinite-dimensional analog of Perelman's F-entropy and fixed points appearing as Ricci solitons.
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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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Community Detection on Networks with Ricci Flow
Applies discrete Ricci flow and curvature to networks viewed as geometric objects to identify communities via geometric decomposition.
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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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Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type
Chern-Ricci flow on Hermitian minimal models of general type admits uniform estimates yielding subsequential Gromov-Hausdorff convergence under a local Kähler assumption.
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Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three
For 3D Ricci flows, time-slices converge in the Gromov–Hausdorff sense to an intrinsic terminal time-slice, and the singular set is horizontally parabolic 1-rectifiable with time image of zero 1/2-dimensional measure.
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The Ollivier Ricci flow with prescribed curvature on infinite graphs
Existence, uniqueness, and convergence of the Ollivier Ricci flow with prescribed curvature are established on infinite graphs with girth at least 6.
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Closed $4$--Manifolds Foliated by Hyperplanes
Closed orientable 4-manifolds with transversely oriented C² codim-1 foliations by R³-leaves are homeomorphic to T⁴ (and diffeomorphic under an extra smooth 1-form condition).
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Rigidity of complete non-compact generalized m-quasi-Einstein manifolds
Complete non-compact gradient generalized m-quasi-Einstein manifolds with R≤0, λ>0, m>1, and μ=1/m are Euclidean under a variety of integrability or volume-growth conditions on the weighted function v=e^{-f/m}λ.
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Singular K\"ahler-Ricci Shrinkers are Complex Analytic
Singular Kähler-Ricci shrinkers from noncollapsed limits are complex analytic varieties with log terminal singularities, yielding geometric consequences including simple connectedness and unique tangent cones.
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Sharp Gaussian Isoperimetry along a Ricci Flow
Proves sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along Ricci flow via monotonicity formula, with consequences for concentration estimates, log-Sobolev inequalities, and related results.
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A vector field induced de Rham-Hodge theory on manifolds
Constructs a vector field-induced de Rham-Hodge theory on compact oriented manifolds, proving ellipticity of the induced Laplacian and Hodge decompositions for closed manifolds and with boundary under specific conditions.
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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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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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On steady and expanding Ricci solitons with asymptotic symmetries
Establishes symmetry principles and uniqueness characterizations for steady and expanding gradient Ricci solitons with specified asymptotic links.
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Uniqueness of Ricci flow with scaling invariant estimates
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A large data result for vacuum Einstein's equations
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Morse-Bott inequalities, Topology Change and Cobordisms to Nothing
Morse-Bott inequalities yield homology bounds and topology-change counts for generic cobordisms to nothing in string theory compactifications.
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On Warped Product Gradient Ricci-Harmonic Soliton
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Explicit Laplace Spectra of Homogeneous Principal Bundles
A unified representation-theoretic approach computes the complete Laplace-Beltrami spectra on homogeneous principal bundles and applies the results to classify scalar stability and Yamabe bifurcations on specific manifold families.
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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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Monotone quantities on $3$-manifolds with nonnegative scalar curvature
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On the structure of complete $G_2$-solitons
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Entropy Production and the Gravitational Origin of the Second Law
Entropy production in a stochastic spacetime metric flow vanishes exactly on solutions of the Einstein equations, making classical GR the reversible limit of an underlying stochastic geometro-dynamics.
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First-Principles Prediction of Material Properties from Topological Invariants
A string theory and graph Laplacian model recovers the Jiron-Castellon virtual volumes for nematic liquid crystals and predicts anisotropic thermal expansion and refractive indices to better than 0.06% accuracy with no fitted parameters.
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The K\"ahler-Ricci soliton on bounded pseudoconvex domains
Kähler-Ricci solitons on bounded pseudoconvex domains with C² boundary are Kähler-Einstein under suitable assumptions, with an analogous result for Bergman versions.
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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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Ricci solitons as critical points of quadratic curvature functionals
Ricci solitons are studied as critical points of a quadratic curvature functional with analysis of their rigidity properties.
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Shi-type estimates and finite-time singularities of reasonable flows of Spin(7)-structures
Establishes Shi-type estimates, blow-up criteria for Lambda, lower bounds on singularity rates, and compactness for reasonable flows of Spin(7)-structures.
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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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On an invariant curvature cone along 4-dimensional Ricci flow
Proves gap theorems and regularity results for 4D manifolds with curvature in the invariant cone C_η,μ under Ricci flow with maximal volume growth.
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Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow
Certain closed quasi-Einstein manifolds are rigid, reducing to Einstein metrics, via a curvature identity tied to the Ricci flow.
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Well-posedness of Ricci Flow in Lorentzian Spacetime and its Entropy Formula
A claim that Perelman entropy can be extended to Lorentzian spacetime to prove well-posedness of Ricci flow fails because the key monotonicity step is false.
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The Master-Slave Encoder Model for Improving Patent Text Summarization: A New Approach to Combining Specifications and Claims
MSEA uses a master-slave encoder architecture on patent specifications and claims, enhanced with pointer networks and repetition suppression, to generate better summaries as measured by small ROUGE score gains.
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Geometric Reductions of the $G_2$-Hilbert Functional via Circle Actions
Under S1-invariant G2-structures with constant or varying fiber length, the unnormalized negative L2-gradient flow of the G2-Hilbert functional has only trivial stationary configurations: flat connections, scalar-flat base metrics, and constant fiber lengths.
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A Diagnostics-First Composite Index for Macro-Financial Resilience to Socioeconomic Challenges: The Gondauri Index with Benchmarking and Scenario Evidence
The Gondauri Index is a new composite index that benchmarks macro-financial resilience on a 0-100 scale by integrating inequality resilience, liquidity and systemic resilience, and inflation forecast coherence with percentile normalization and scenario projections.
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Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow
Investigates the relationship between long-time Kähler-Ricci flow behavior on asymptotically conical gradient expanders and initial data asymptotics at spatial infinity.
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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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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.
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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.