The paper characterizes -Ricci solitons and flows using -entropy.
problem Characterizing -Ricci solitons and flows.
method Using -entropy and super Perelman Ricci flows.
result Characterizes -Ricci solitons and flows.
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48 results for “super Perelman Ricci flows”
On Perelman's -entropy and Shannon entropy power for super Ricci flows on metric measure spacesmath.DG
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's -entropy and Shannon entropy power to super Ricci flows.
result Equivalence between volume non-local collapsing property and lower boundedness of -entropy on RCD spaces.
Article proves Liouville theorem for heat equation in super Ricci flow.
problem Proving Liouville theorem for heat equation in super Ricci flow.
method Formulated under a growth condition concerning Perelman's reduced distance.
result Established Liouville theorem for heat equation in ancient super Ricci flow.
Paper proves Harnack inequalities for Witten Laplacian on manifolds with specific flows.
problem Proving Harnack inequalities for Witten Laplacian on Riemannian manifolds.
method Using Li-Yau and Hamilton type inequalities for heat equation associated with time-dependent Witten Laplacian on manifolds with specific flows.
result Proves Li-Yau and Hamilton type Harnack inequalities for Witten Laplacian.
In this paper, we prove logarithmic Sobolev inequalities and derive the Hamilton Harnack inequality for the heat semigroup of the Witten Laplacian on complete Riemannian manifolds equipped with -super Perelman Ricci flow. We establish the -entropy formula for the heat equation of the Witten Laplacian and prove a …
Unified approach to analysis on Ricci nonnegative manifolds and flows using optimal transport.
problem Generalizing Perelman's functionals to super Ricci flows.
method Optimal transport, Bochner inequality, gradient estimates, EVI.
result Unified condition equivalent to Ricci nonnegativity for smooth evolutions of Riemannian manifolds.
Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometrymath.DG
Study Liouville theorems for harmonic maps along ancient super Ricci flows.
problem Proving Liouville theorems for harmonic maps under specific geometric conditions.
method Using Perelman's reduced geometric viewpoint, derive Liouville theorems with controlled growth.
result Sharp growth conditions and new Liouville theorems for both non-positively and positively curved target spaces.
Derives gradient estimation for a specific heat equation on evolving manifolds.
problem Gradient estimation for a generalized heat equation on evolving weighted Riemannian manifolds.
method Derives gradient estimation for a specific heat equation on evolving weighted Riemannian manifolds.
result Derives a Harnack type inequality and a Liouville type theorem as applications of gradient estimation.
Survey on -entropy formulas for heat equations and Langevin deformation on Riemannian manifolds.
problem Entropy formulas for heat equations and Langevin deformation on Riemannian manifolds.
method Proving -entropy formulas for heat equations and Langevin deformation.
result Proved -entropy formulas for heat equations and Langevin deformation.
The paper analyzes Perelman's entropies on manifolds with conical singularities.
problem Analyzing manifolds with conical singularities using Perelman's entropies.
method Employing singular Ricci de Turck flow and Perelman's entropies to study manifolds with conical singularities.
result Entropy is monotone along the singular Ricci de Turck flow and helps in proving properties of Ricci solitons.
The paper proves entropy power properties on Riemannian manifolds and Ricci flows.
problem Entropy power on Riemannian manifolds and Ricci flows.
method Proving concavity and convexity of Shannon entropy power for heat and conjugate heat equations on Riemannian manifolds and Ricci flows.
result Entropy power rigidity models on Einstein or quasi Einstein manifolds and shrinking Ricci solitons.
In recent years, there has seen much interest and increased research activities on Perelman's paper. Section one and two of this paper aim to establish Perelman's local non-collapsing result for the Ricci flow. This will provide a positive lower bound on the injectivity radius for the Ricci flow under blow-up analysis.…
The paper studies conical structures and Perelman's functionals on Ricci flows.
problem Characterizing conical structures and Perelman's functionals on manifolds.
method Analysis of Perelman's functionals on cones and adaptation of the pseudolocality theorem.
result Cone structures can be smoothed out by type III immortal solutions on Ricci flows.
Perelman's Ricci flow emerges in quantum gravity, linking math and physics.
problem Understanding Perelman's Ricci flow equations in quantum gravity.
method Mapping Perelman's Ricci flow equations to localization equations in topological quantum gravity.
result Perelman's dilaton and fixed volume condition emerge dynamically.
Ancient ovals are key blowup limits in 3D Ricci flow near singularities.
problem Understanding blowup limits in 3D Ricci flow near singularities.
method Proving ancient ovals are blowup limits if and only if spherical singularities accumulate.
result Ancient ovals are necessary and sufficient for blowup limits in 3D Ricci flow.
Ancient Ricci flows with nonnegative curvature operator have bounded entropy.
problem Conditions for bounded entropy in ancient Ricci flows.
method Used Perelman's entropy and Hamilton's trace Harnack inequality.
result Curvature operator nonnegativity is not necessary for bounded entropy.
We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by c…
Study Brownian motion on Perelman's almost Ricci-flat manifold, proving convergence to Ricci flow limits.
problem Characterize Brownian motion and stochastic transport on Perelman's manifold.
method Construct sequences of projected Brownian motions and stochastic parallel transports, analyze Laplace and horizontal Laplacian martingale problems.
result Convergence of projected Brownian motions and stochastic parallel transports to Ricci flow limits as .
Proves curvature bounds for close to 1 Perelman's reduced volume.
problem Curvature bounds for Ricci flow with close to 1 reduced volume.
method ε-regularity theorem for Perelman's reduced volume.
result Curvature radius cannot be too small if reduced volume is close to 1.
Survey classifies singularity models in 3D Ricci flow.
problem Understanding singularity formation in 3D Ricci flow.
method Analysis of ancient κ-solutions and steady gradient Ricci solitons.
result Complete classification of singularity models in 3D.
Notes on Perelman's papersmath.DG
These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds".
Two new proofs show Ricci flow breathers are special solutions.
problem Characterize solutions to Ricci flow on closed manifolds.
method Use singularity models and ancient solutions to show they are gradient Ricci solitons.
result Ricci flow breathers are gradient Ricci solitons.
We show a quite simple second variation formula for Perelman's -functional along the modified Kähler-Ricci flow over Fano manifolds.
Ancient Ricci flows with asymptotic solitons have uniform bounds and inequalities.
problem Bounding and understanding ancient Ricci flows with asymptotic solitons.
method Analyzing asymptotic solitons, proving uniform bounds on Perelman's ν-functional, and showing Nash entropy bounds.
result Uniform bounds on Perelman's ν-functional and logarithmic/Sobolev inequalities for ancient solutions.
The study improves Perelman's theorems on Ricci flow.
problem No local collapse in Ricci flow on minimal projective manifolds.
method Localization of entropy functionals and further development of Li-Yau estimate.
result Generalization of no-local-collapsing theorem and pseudo-locality theorem.
Extends heat kernel estimates for super Ricci flow.
problem Heat kernel estimates for super Ricci flow.
method Generalizes Bamler-Zhang's geometric analysis to super Ricci flow.
result Obtains Gaussian heat kernel estimates for super Ricci flow.
In this expository note, we study the second variation of Perelman's entropy on the space of Kahler metrics at a Kähler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelman's entropy is stable along the Kähler-Ricci flow. The Chinese version of this note has appeared in…
In his study of Ricci flow, Perelman introduced a smooth-manifold invariant called lambda-bar. We show here that, for completely elementary reasons, this invariant simply equals the Yamabe invariant, alias the sigma constant, whenever the latter is non-positive. On the other hand, the Perelman invariant just equals + i…
In this thesis we give a review on Ricci flow, an overview on Poincare conjecture, maximum principle, Li-Yau-Perelman estimate, Two functional F and W of Perelman, Reduced volume and reduced length and k-non collapsing estimate
The paper analyzes Kähler-Ricci flow near Kähler-Ricci solitons under complex structure deformation.
problem Behavior of Kähler-Ricci flow near Kähler-Ricci solitons under complex structure deformation.
method Established Lojasiewicz's type inequality for Perelman's entropy and proved convergence of Kähler-Ricci flow.
result Solved Yau-Tian-Donaldson conjecture and showed the kernel corresponds to local moduli space of modified -semistable Fano manifolds.
The article proves a new entropy formula for surfaces with boundaries.
problem Entropy formula for surfaces with boundaries.
method Established a monotonicity formula of Hamilton type entropy.
result Entropy functional and -functional relation studied.
Non linear sigma models are quantum field theories describing, in the large deviations sense, random fluctuations of harmonic maps between a Riemann surface and a Riemannian manifold. Via their formal renormalization group analysis, they provide a framework for possible generalizations of the Hamilton-Perelman Ricci fl…
Study weak super Ricci flow through neckpinch in metric measure spaces.
problem Understanding Ricci flow behavior at singularities.
method Introduce weak super Ricci flow and show conditions for continuation.
result Weak super Ricci flow properties at singularities.
Introduces new flows for metric measure spaces, proving stability and compactness.
problem Stability and compactness of metric measure spaces.
method Introduces super-Ricci flows and Ricci flows, proving stability and compactness under mGH-convergence.
result Uniformly bounded families of super-Ricci flows are compact.
Survey on Ricci flow on spaces with conical singularities.
problem Analyzing Ricci flow on spaces with isolated conical singularities.
method Ricci de Turck flow preserving conical singularities, stability of Ricci flat metrics, preservation of positive scalar curvature under certain conditions.
result Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow.
Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.
problem Stability and instability of ALE Ricci-flat metrics.
method Use of a modified Perelman's λ-functional and Lojasiewicz inequality.
result Demonstrates dynamical instability of ALE Ricci-flat metrics.
The Nonexistence of Noncompact Type-I Ancient 3-d -Solutions of Ricci Flow with Positive Curvaturemath.DG
Paper disproves certain 3D Ricci flow solutions with positive curvature.
problem Existence of noncompact Type-I ancient 3-d κ-solutions with positive curvature.
method Analyzes Ricci flow constraints and curvature conditions.
result No noncompact Type-I ancient 3-d κ-solutions with positive curvature exist.
New method confirms Ricci iteration converges to Kähler-Einstein metrics.
problem Confirming the conjecture that the Ricci iteration converges to Kähler-Einstein metrics.
method Using Perelman's convergence theory for the Ricci flow, the article confirms the conjecture for the Ricci iteration.
result The Ricci iteration converges to Kähler-Einstein metrics, providing a new method of uniformization of the Riemann sphere.
Study almost rigidity of super Ricci flow with non-negative Muller quantity.
problem Almost rigidity properties of super Ricci flow with non-negative Muller quantity.
method Almost splitting and quantitative stratification theorems established by Bamler for Ricci flow.
result Obtained almost constancy for a certain integral quantity concerning scalar curvature at an almost self-similar point.
This is the first of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper we first fix a notion of Ricci flows with surgery, which will be used in this and the following three papers. Then we review Perelman's long-time estimates and generalize them to the case in which …
The paper disproves a conjecture about Ricci flow solutions on noncompact 3-manifolds.
problem Existence of -solutions of Ricci flow on noncompact 3-manifolds with positive curvature.
method Analyzing the blow-up behavior of solutions and integrating curvature over time.
result Proves the nonexistence of -solutions under certain conditions, partially confirming a Perelman conjecture.
Compactness theory for super Ricci flows provides convergence results.
problem Understanding convergence of super Ricci flows.
method Developed a compactness theory for super Ricci flows.
result Subsequential convergence to a metric flow under certain conditions.
Local Sobolev inequality on Ricci flows with applications.
problem Understanding the local geometry of Ricci flows.
method Proving a local Sobolev inequality for Ricci flows.
result The local ν-functional depends only on the Nash entropy at the center of a disk.
New spinorial functional connects Perelman's W- and F-functionals.
problem Unifying Perelman's functionals for spin manifolds.
method Introduced a new energy functional on spin manifolds, computed its first variation, and established a gradient flow.
result Critical points of the functional are twisted Ricci solitons and eigen-spinsors.
New method for Ricci flows on graphs that can handle changing structures.
problem Challenges of Ricci flows on graphs with evolving structures.
method Introducing super Ricci flows and studying heat flow on singular graphs.
result Consistency with classical Ricci flows in a discrete to continuum limit.
New proof for curvature and diameter estimates on Fano manifolds.
problem Curvature and diameter estimates for Kähler-Ricci flow on Fano manifolds.
method New Harnack estimate for special functions in space-time.
result Established new estimates for scalar curvature and diameter.
We show that Perelman's W-functional can be generalized to Sasaki-Ricci flow. When the basic first Chern class is positive, we prove a uniform bound on the scalar curvature, the diameter and a uniform bound for the transverse Ricci potential along the Sasaki-Ricci flow, which generalizes Perelman's results Kahler…
The paper extends Perelman's theorems on Ricci flow entropy.
problem Understanding the behavior of Ricci flow under various conditions.
method Localization of entropy functionals and development of Li-Yau estimates.
result Generalization of Perelman's no-local-collapsing and pseudo-locality theorems.