Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.
problem Understanding the asymptotic geometry of ancient Ricci flows with nonnegative Ricci curvature.
method Analyze tangent flows at infinity and use estimates for noncollapsed F-limit metric solitons.
result Two dichotomy theorems for ancient Ricci flows: either the asymptotic volume ratio is zero or every tangent flow is a Ricci flat cone.
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.
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.
Paper introduces ∗−Ricci flow and its properties.
problem Exploring geometric curvature tensors under ∗−Ricci flow. method Introducing ∗−Ricci flow and analyzing its properties. result Found deformation of geometric curvature tensors under ∗−Ricci flow. The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.
problem Understanding the behavior of parabolic frequency under Ricci flow and Ricci-harmonic flow.
method Investigates the monotonicity of parabolic frequency for solutions of linear and heat equations with bounded curvatures.
result Establishes monotonicity results for parabolic frequency under specific curvature conditions.
Paper compares Ricci flow volume changes and applies to Kähler-Ricci flow.
problem Volume comparison in Ricci flow and convergence of Kähler-Ricci flow.
method Derives a new relative volume comparison estimate and applies it to Kähler-Ricci flow.
result Generalizes Perelman's no local collapsing estimate and provides an analogue of Bishop-Gromov volume comparison for Ricci flow.
The paper examines conditions for Ricci solitons to be Ricci flat or Einstein.
problem Conditions for Ricci solitons to be Ricci flat or Einstein.
method Study of evolution equations of scalar and Ricci curvatures under Ricci flow.
result Conditions for Ricci solitons to be Ricci flat or Einstein.
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…
We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped pr…
Modified flow smooths conical Kähler-Ricci problem.
problem Smooth approximation of modified conical Kähler-Ricci flow.
method Constructing a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.
result Long-time solution of the modified conical Kähler-Ricci flow.
Study Ricci flow on torus bundles and related manifolds.
problem Compute Ricci flow formulas for torus bundles.
method Use invariant metrics compatible with connections on principal G-bundles.
result Solutions to Ricci flows on Heisenberg groups and Berger 3-spheres.
In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In add…
Study the Ricci flow on Finsler surfaces and find solutions.
problem Existence and uniqueness of solutions to the Ricci flow on Finsler surfaces.
method First, study the Finslerian Ricci-DeTurck flow and find a unique short time solution. Then, use this to find a solution to the original Ricci flow.
result Find solutions to the Ricci flow on Finsler surfaces.
The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.
problem Characterizing and understanding Ricci flows with closed and smooth tangent flows.
method Analyzing ancient and finite-time singularity Ricci flows to prove uniqueness and characterizations.
result The tangent flow is unique and characterizes ancient and finite-time singularity flows.
Survey on new Ricci flow techniques.
problem None explicitly stated.
method None explicitly stated.
result None explicitly stated.
We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. We consider the behavior of Ricci flow with surgery starting from a fixed initial compact Riemannian 3-manifold, as the surgery parameter varies. We prove that the flow with surgery subconverges to a singular Ri…
Analyzing Ricci Flow on 5D homogeneous manifolds.
problem Understanding Ricci Flow behavior in 5D homogeneous manifolds.
method Applying Ricci Flow to a specific class of 5D homogeneous manifolds.
result Calculations and analysis of Ricci Flow in 5D homogeneous manifolds.
New Ricci flows found with Einstein orbifolds at infinity.
problem Ancient and immortal Ricci flows with Einstein orbifolds at infinity.
method Continuous families of non-isometric ancient Ricci flows and half-PIC ancient flows constructed.
result Found continuous families of non-isometric ancient Ricci flows and half-PIC ancient flows on specific manifolds.
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.
Proves analogous result for harmonic Ricci flow, agreeing with gradient Ricci solitons.
problem Consistency of singular points in Ricci flow.
method Developed refined compactness theorems, pseudolocality theorem, and reduced length/volume concept.
result Analogous result for harmonic Ricci flow, agreeing with gradient Ricci solitons.
New flow preserves almost Hermitian metrics for manifold study.
problem Curvature flow for almost Hermitian manifolds.
method Introducing a new curvature flow matching Ricci flow and preserving almost Hermitian condition.
result Ricci flow can be used to study almost Hermitian manifolds.
The paper studies mean curvature flow in a Ricci flow background with extended Ricci flow.
problem Analyzing mean curvature flow in a Ricci flow background.
method Computing variational properties and deriving evolution equations for mean curvature and second fundamental form.
result Established a Huisken's monotonicity-type formula for mean curvature solitons in an extended Ricci flow.
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.
New energy functional bounds Ricci flows on ancient spaces.
problem Bounding Ricci flows on ancient spaces.
method Introducing a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows.
result Provides an upper bound for the ordinary λ-functional.
Survey on Chern-Ricci flow for complex manifolds.
problem Understanding and solving open problems in Chern-Ricci flow.
method Parabolic flow of Hermitian metrics on complex manifolds.
result Open problems and new directions in Chern-Ricci flow highlighted.
Study on stability of harmonic maps under Ricci flow.
problem Stability of harmonic maps under Ricci flow.
method Analysis of stability and non-stability properties.
result Examples of preserved and non-preserved stability under Ricci flow.
We characterize the conjugate linearized Ricci flow and the associated backward heat kernel on closed three--manifolds of bounded geometry. We discuss their properties, and introduce the notion of Ricci flow conjugated constraint sets which characterizes a way of Ricci flow averaging metric dependent geometrical data. …
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.
Existence and uniqueness theorem for Ricci flow on weighted graphs proved.
problem Existence and uniqueness of solutions to Ricci flow equations on weighted graphs.
method Continuous time normalized Ricci flow approach.
result Existence and uniqueness theorem for solutions to Ricci flow on weighted graphs.
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
Study investigates non-Kahler Ricci flow singularities converging to Kahler-Ricci solitons.
problem Investigating finite-time Type-I singularities in non-Kahler Ricci flow solutions.
method Examining the convergence of parabolic rescalings at singularities to shrinking Kahler-Ricci solitons.
result Supports the conjecture that the blowdown soliton is stable under Ricci flow and the subspace of Kahler metrics is stable under Ricci flow.
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. Proves an ε-regularity theorem for Ricci flows, leading to new singularity estimates.
problem Analyzing singularity models in Fano Kähler-Ricci flows.
method Proves ε-regularity theorem and uses it to derive new estimates.
result Establishes new estimates for singularity models of Fano Kähler-Ricci flows.
Study shows Ricci flow's convergence and harmonic map heat flow's long-time existence.
problem Analyzing convergence of Ricci flow and harmonic map heat flow.
method Established long-time existence of harmonic map heat flow between Ricci flow and shrinker.
result Ricci flow converges exponentially to compact integrable shrinkers and at singularities modelled on the shrinker.
New curvature defined via graph resistances leads to Ricci flow.
problem Defining curvature on graph edges for analysis.
method Introducing Ricci--Foster curvature based on effective resistances and studying Ricci flow.
result Existence of solutions to Ricci flow on short time intervals, preservation of nonnegative curvature.
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. The paper studies Gauss maps of a Ricci-mean curvature flow.
problem Investigating the Gauss maps of a Ricci-mean curvature flow.
method Deduced the evolution equation for the Gauss maps of a Ricci-mean curvature flow and proved they satisfy the vertically harmonic map heat flow equation.
result The Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation.
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.
In this note we study conformal Ricci flow introduced by Arthur Fischer. We use DeTurck's trick to rewrite conformal Ricci flow as a strong parabolic-elliptic partial differential equations. Then we prove short time existences for conformal Ricci flow on compact manifolds as well as on asymptotically flat manifolds. We…
Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.
problem Finite time singularities in Kähler-Ricci flow on Hirzebruch surfaces.
method Analyze tangent flows based at singular points.
result Tangent flows are K-R flows with orbifold singularities.
Enhances understanding of Kähler-Ricci flow singularities.
problem Understanding singularities in Kähler-Ricci flow.
method Relates to classic Kähler-Ricci flow and degenerate complex Monge-Ampère equation.
result Improves understanding of finite and infinite time singularities.
Ancient Ricci flows are identified without curvature sign condition.
problem Identifying type II ancient Ricci flows and their backward limits.
method Using a size condition of the sharp log Sobolev functional near infinity.
result Rigidity result for ancient Ricci flows without sign condition on curvatures.
ALE Ricci-flat manifolds stable under Ricci flow if integrable.
problem Stability of ALE Ricci-flat manifolds under Ricci flow.
method Adapting Tian's approach for closed manifolds, proving integrability for ALE Calabi-Yau manifolds.
result ALE Ricci-flat manifolds are dynamically stable if integrable.
Ricci flow smooths locally collapsing manifolds with controlled curvature.
problem Locally collapsing manifolds with controlled Ricci curvature.
method Ricci flow for a definite period of time, detecting collapsing infranil fiber bundles.
result Topological conditions detect collapsing infranil fiber bundles.
Sharp eigenvalue bounds and splitting for modified Ricci flow.
problem Eigenvalue bounds and splitting in modified Ricci flow.
method Sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow.
result Splitting theorem in the case of equality.
Simple Ricci flow proof for Riemann surfaces.
problem Uniformization theorem of Riemann surfaces
method Ricci flow and Hamilton's isoperimetric estimate
result Simple proof of uniformization theorem
New nontrivial breathers found for Ricci flow on noncompact manifolds.
problem Perelman's theorem that nontrivial breathers don't exist for closed manifolds.
method Constructing nontrivial expanding breathers in all dimensions for noncompact manifolds.
result Nontrivial breathers found for Ricci flow on noncompact manifolds.
Local existence and uniqueness of Bakry-Émery Ricci flow solutions on finite graphs.
problem Analyzing the behavior of Ricci flow on finite graphs.
method Local existence and uniqueness proof for solutions of the Bakry-Émery Ricci flow.
result Local existence and uniqueness of solutions to the Ricci flow on finite graphs.