Study stability and instability of Ricci-flat metrics under generalized Ricci flow.
problem Stability and instability of Ricci-flat metrics under Ricci flow.
method Analysis of generalized Ricci flow for Ricci-flat metrics and vanishing 3-forms.
result Dynamical stability and instability results for Ricci-flat metrics and vanishing 3-forms.
Study geometric bounds on generalized Ricci flow.
problem No specific problem stated; focuses on bounds.
method Analogous geometric quantities and bounds proven.
result Geometric and analytic bounds established.
Study on shrinking solitons of generalized Ricci flow.
problem Characterizing shrinking solitons in generalized Ricci flow.
method Analyzing gradient shrinking solitons and pluriclosed solitons on compact manifolds.
result First non-trivial shrinking generalized soliton constructed.
Smooth 3D flows from non-smooth starting points.
problem Creating smooth Ricci flows from non-smooth initial conditions.
method Generalized singular Ricci flow applied to 3D complete manifolds.
result Existence of smooth Ricci flows starting from non-smooth initial conditions.
Book introduces generalized Ricci flow for constructing canonical metrics.
problem Constructing canonical metrics in generalized Riemannian and complex geometry.
method Introduces generalized Ricci flow as a tool for constructing canonical metrics.
result Global convergence results and applications to complex geometry.
We introduce a flow of Kähler structures over Fano manifolds with formal limit at infinite time a Kähler-Ricci soliton. This flow correspond to a Perelman's modified backward Kähler-Ricci type flow that we call Soliton-Kähler-Ricci flow. It can be generated by the Soliton-Ricci flow. We assume that the Soliton-Ricci fl…
Theory of Ricci flow for Courant algebroids, preserving isometry group.
problem Analytic theory of geometric flows for Courant algebroids.
method Develops Ricci flow theory for Courant algebroids, proving existence and uniqueness.
result Shows scalar curvature monotonicity and convergence for certain solutions.
We consider a generalized Ricci flow with a given (not necessarily closed) three-form and establish the higher derivatives estimates for compact manifolds. As an application, we prove the compactness theorem for this generalized Ricci flow. The similar results still hold for a more generalized Ricci flow.
Develops a new framework for generalized Ricci flow on Lie groups.
problem Global existence and geometric properties of Ricci flow on Lie groups.
method Inspired by Lauret's bracket flow, studies generalized Ricci flow on discrete quotients of Lie groups.
result Establishes global existence on solvmanifolds in arbitrary dimensions.
This paper proves certain quasi-Einstein manifolds are rigid under Ricci flow.
problem Understanding the behavior of quasi-Einstein metrics under Ricci flow.
method Employing a curvature evolution identity associated with Ricci flow.
result Certain closed quasi-Einstein manifolds are rigid under Ricci flow.
Proves uniqueness of Ricci flow with scaling invariant estimates.
problem Proving uniqueness of Ricci flow with scaling invariant curvature bound.
method Solving Ricci-harmonic map heat flow in unbounded curvature background.
result Complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique in dimension three.
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.
The (α,β)-Ricci-Yamabe flow exists on closed manifolds.
problem Existence of solutions to the (α,β)-Ricci-Yamabe flow. method Showed short time existence and established long time existence theorems.
result Existence of smooth solutions to the (α,β)-Ricci-Yamabe flow on closed manifolds. Illustrates Ricci flow for a general audience.
problem Explains Ricci flow to non-experts.
method Illustrative introduction without prerequisites.
result Provides a working definition and intuition for Ricci flow.
We define solitons for the generalized Ricci flow on an exact Courant algebroid, building on the definitions of M. Garcia-Fernandez and J. Streets. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flo…
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.
Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize this result for a Ricci-mean curvature flow moving along a Ricci flow constructe…
Study describes global existence and convergence of flows on surfaces and fibrations.
problem Global existence and convergence of flows on surfaces and fibrations.
method Complete description of Ricci-Yang-Mills flow and pluriclosed flow on Tk bundles over Riemann surfaces. result Equivalence of solutions to generalized Ricci flow and pluriclosed flow with symmetry.
The paper improves heat equation estimates under weaker Ricci curvature conditions.
problem Improving heat equation estimates under weaker Ricci curvature conditions.
method Establishing Li-Yau-type and Hamilton-type estimates for positive solutions of the heat equation under generalized Ricci flow.
result Deriving Harnack-type inequalities and monotonicity of parabolic frequency.
New formulations for Ricci flows without smoothness.
problem Characterize Ricci flows without smooth solutions.
method Weak formulations of super Ricci flows with saturation condition.
result Generalized formulations for singular settings.
We show that on a smooth Hermitian minimal model of general type the Chern-Ricci flow converges to a closed positive current on M. Moreover, the flow converges smoothly to a Kahler-Einstein metric on compact sets away from the null locus of K_M. This generalizes work of Tsuji and Tian-Zhang to Hermitian manifolds, prov…
New formulas derived for scalar curvature in generalized Ricci flow.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estim…
Study of t-Gauduchon Ricci-flat condition under Chern-Ricci flow on non-Kähler manifolds.
problem Investigating the t-Gauduchon Ricci-flat condition on non-Kähler manifolds. method Chern-Ricci flow approach, examples of non-Kähler Calabi-Yau manifolds, and geometric flow analysis.
result Examples of Chern-Ricci flow on non-Kähler Calabi-Yau manifolds that do not preserve the t-Gauduchon Ricci-flat condition. This project serves to analyze the behavior of Ricci Flow in five dimensional manifolds. Ricci Flow was introduced by Richard Hamilton in 1982 and was an essential tool in proving the Geometrization and Poincare Conjectures. In general, Ricci Flow is a nonlinear PDE whose solutions are rather difficult to calculate; ho…
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 models Ricci flow on complex surfaces, showing mixed behavior.
problem Understanding long-time behavior of Ricci flow on complex surfaces.
method BiLipschitz models for 4-manifolds (minimal surfaces of general type).
result Exhibits a combination of expanding and static behavior.
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 C1 bound for the transverse Ricci potential along the Sasaki-Ricci flow, which generalizes Perelman's results Kahler…
In this note, we provide some general discussion on the Ricci lower bound along Kähler-Ricci flow with singularity over closed manifold.
In this paper we study a generalization of the Kahler-Ricci flow, in which the Ricci form is twisted by a closed, non-negative (1,1)-form. We show that when a twisted Kahler-Einstein metric exists, then this twisted flow converges exponentially. This generalizes a result of Perelman on the convergence of the Kahler-Ric…
We explore the harmonic-Ricci flow---that is, Ricci flow coupled with harmonic map flow---both as it arises naturally in certain principal bundle constructions related to Ricci flow and as a geometric flow in its own right. We demonstrate that one natural geometric context for the flow is a special case of the locally …
The paper studies Ricci curvature on Kähler-Ricci flow.
problem Analyzing Ricci curvature on Kähler-Ricci flow.
method Examining n-dimensional compact Kähler manifolds with semi-ample canonical line bundles under Kähler Ricci Flow.
result Ricci curvature converges to negative of generalized Kähler Einstein metric ωB locally away from singular set. In this paper, we continue to study the generalized Ricci flow. We give a criterion on steady gradient Ricci soliton on complete and noncompact Riemannian manifolds that is Ricci-flat, and then introduce a natural flow whose stable points are Ricci-flat metrics. Modifying the argument used by Shi and List, we prove the…
Ancient Ricci flows on non-collapsed manifolds have finite fundamental groups.
problem Understanding the fundamental groups of ancient Ricci flows.
method Analyzing the structure of ancient Ricci flows and their tangent flows.
result The fundamental group of non-collapsed ancient Ricci flows is finite and a quotient of the regular part's fundamental group.
For some class of geometric flows, we obtain the (logarithmic) Sobolev inequalities and their equivalence up to different factors directly and also obtain the long time non-collapsing and non-inflated properties, which generalize the results in the case of Ricci flow or List-Ricci flow or harmonic-Ricci flow. As applic…
Ricci flow simulations show unstable Fubini-Study metrics develop singularities.
problem Understanding the behavior of unstable perturbations in Ricci flow.
method Numerical simulations of Ricci flow starting from unstable Fubini-Study metrics.
result Ricci flow solutions from unstable Fubini-Study metrics develop local singularities.
The study preserves Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.
problem Preserving Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.
method Investigation of infinitely many generalized Wallach spaces (GWS) where Ricci curvature positivity is preserved.
result Infinite number of GWS where Ricci curvature positivity is preserved under the normalized Ricci flow.
Paper defines parabolic frequency for Ricci flow solutions, proving monotonicity and uniqueness.
problem Backwards uniqueness for solutions of parabolic equations on Ricci flows.
method Defines and proves monotonicity of parabolic frequency for Ricci flow solutions.
result Backwards uniqueness for solutions of parabolic equations on Ricci flows.
We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local Ricci flow, using which we prove the short-time existence of Ricci flow solutions o…
The study examines stability of specific geometric flows.
problem Stability of Pluriclosed and Generalized Ricci solitons.
method Analyzes the second variation of generalized Einstein--Hilbert functional and infinitesimal deformations.
result Stability of the flows and solitons under specific conditions.
Study of 4D flows with nilpotent symmetry, showing immortal solutions and blowdown limits.
problem Understanding 4D generalized Ricci flows with nilpotent symmetry.
method Immortal solutions, type III curvature and diameter estimates, new energy monotonicity.
result Blowdown limits lie in a finite-dimensional family of solutions.
New proof of energy functional monotonicity via geodesics in measure space.
problem Proving monotonicity of energy functional in generalized Ricci flow.
method Defining adapted cost functional, geodesics, and entropy functional.
result Monotonicity of cost along backwards heat flow and energy functional along generalized Ricci flow.
In this paper, we derive a relative volume comparison estimate along Ricci flow and apply it to studying the Gromov-Hausdorff convergence of Kähler-Ricci flow on a minimal manifold. This new estimate generalizes Perelman's no local collapsing estimate and can be regarded as an analogue of the Bishop-Gromov volume compa…
Paper studies Kähler-Ricci flow convergence on Fano manifolds.
problem Uniform convergence of Kähler-Ricci flow on Fano manifolds.
method Analyzes flow behavior with varied initial metrics and complex structures.
result Proves uniqueness of Kähler-Ricci solitons in diffeomorphism orbits.
The paper derives inequalities and formulas for generalized Ricci flow.
problem Understanding and characterizing generalized Ricci flow.
method Using Bochner formula and adapted Malliavin gradient, the paper derives inequalities and characterizes generalized Ricci flow.
result Characterizations of generalized Ricci flow via inequalities for the associated Malliavin gradient.
Ricci-Filtration enhances retrieval-augmented generation rerankers for query-answer tasks by using discrete Ricci flow on graphs.
problem Improving retrieval-augmented generation rerankers for query-answer tasks.
method Discrete Ricci flow on graphs to evaluate structural importance of chunks.
result Ricci-Filtration outperforms baseline methods in accuracy, precision, recall, and F1 scores.
The paper applies combinatorial Ricci flows to prove hyperbolic structures on 3-manifolds.
problem Proving the existence of hyperbolic structures on 3-manifolds with cusps.
method Combinatorial Ricci curvature flow methods to study pseudo 3-manifolds and ideal triangulations.
result The extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a zero Ricci curvature metric.
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.