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.
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48 results for “Super-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.
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.
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.
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.
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.
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.
In this paper we consider compact, Riemannian manifolds each equipped with a one-parameter family of metrics satisfying the Ricci flow equation. Motivated by a characterization of the super Ricci flow developed by McCann-Topping, we introduce the notion of a super Ricci flow for a family of …
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.
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.
Study heat flows on time-dependent metric measure spaces, proving properties related to super-Ricci flows.
problem Characterize heat flows and their properties on time-dependent metric measure spaces.
method Prove existence, uniqueness, and regularity of heat equations and their duals on time-dependent metric measure spaces.
result Equivalence of dynamic convexity of Boltzmann entropy, monotonicity of Wasserstein distances, gradient estimates, and Bochner inequality.
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.
Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.
problem Defining scalar curvature for non-smooth spaces and flows.
method Gaussian integral approach to scalar curvature, applied to manifolds and flows.
result Characterizes Ricci flows as minimal super-Ricci flows.
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.
We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed…
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.
In this paper we will give a new proof of the monotonicity of Wasserstein distances of two diffusions under super Ricci flow. Our proof is based on the coupling method of B.Andrew and J.Clutterbuck. The same method can also be applied to the contractivity of normalized L-Wasserstein distance under backward Ricci flow.
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 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 …
Hamilton differential Harnack inequality and -entropy for Witten Laplacian on Riemannian manifoldsmath.DG
The paper proves inequalities and entropy formulas for Witten Laplacian on manifolds.
problem Analyzing heat equations and entropy on Riemannian manifolds.
method Proving Hamilton Harnack inequalities and entropy formulas for Witten Laplacian.
result Established Hamilton Harnack inequalities and -entropy formulas for Witten Laplacian.
Paper bounds local curvature for Ricci-harmonic flow under Ricci bounded condition.
problem Local curvature estimates for Ricci-harmonic flow.
method Explicit bound of Δg(t)u(t) and local curvature estimates.
result Local curvature estimates for generalized Ricci flow and conjecture for Einstein's scalar field equations.
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.
We prove that a Kahler supermetric on a supermanifold with one complex fermionic dimension admits a super Ricci-flat supermetric if and only if the bosonic metric has vanishing scalar curvature. As a corollary, it follows that Yau's theorem does not hold for supermanifolds.
The paper establishes bounds for Ricci flows using entropy and heat kernel methods.
problem Analyzing geometric and analytic properties of Ricci flows.
method Entropy and heat kernel bounds, monotonicity formula for variance of conjugate heat kernels.
result Optimal bounds for Ricci flows, including volume, heat kernel, and entropy estimates.
The paper proves stability properties for quotients of spaces with synthetic Ricci curvature bounds.
problem Stability properties of quotients of spaces with synthetic Ricci curvature bounds.
method Analyzes quotients of Riemannian manifolds with isometric group actions and metric-measure foliations/submersions.
result Quotients of Riemannian manifolds with synthetic Ricci curvature bounds are also Riemannian manifolds with synthetic Ricci curvature bounds.
Symplectic geometric flowsmath.SG
New flows introduced for symplectic geometry.
problem No specific problem stated; focuses on new flows.
method Introduces several geometric flows on symplectic manifolds.
result Examples include the Hitchin gradient flow and dual Ricci flow.
Sylvester flows improve variational inference by making transformations more flexible.
problem Flexible approximate posterior distributions for variational inference.
method Introduce Sylvester normalizing flows as a generalization of planar flows.
result Sylvester flows perform favorably compared to planar and inverse autoregressive flows on various datasets.
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.
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
Investigate scalar curvature under geometric flows
problem Behavior of scalar curvature under geometric flows
method Three specific cases: Ricci flow, Kähler-Ricci flow, Laplacian flow
result Long-time existence of flows
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…
Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.
i-flow uses normalizing flows for high-dimensional integration and sampling.
problem High-dimensional integration in science and statistics.
method Normalizing flows for bijective mappings between distributions.
result i-flow outperforms other algorithms for high-dimensional correlated integrals.
The article calculates the -convergence rate for Ricci flows with closed and smooth tangent flows.
problem Analyzing the convergence rate of Ricci flows with specific tangent flows.
method Calculating the -convergence rate for Ricci flows with closed and smooth tangent flows.
result A Ricci flow with closed and smooth tangent flow is close to its tangent flow in the -sense.
Paper introduces Tensor Gauge Flow Models for better data encoding.
problem Lack of expressive flow dynamics in existing Generative Flow Models.
method Incorporates higher-order Tensor Gauge Fields into the Flow Equation.
result Tensor Gauge Flow Models achieve improved generative performance.
Ancient mean curvature flows get codimension bounds from their tangent flow.
problem Understanding the limiting behavior of ancient mean curvature flows.
method Proving codimension bounds using the tangent flow at .
result Ancient mean curvature flows are rigid to their tangent flow at .
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.
The study disproves rotating ancient flows in 4D space.
problem The existence of rotating ancient flows in .
method Analysis of ancient noncollapsed flows in .
result Nonexistence of rotating ancient flows among ancient noncollapsed flows in .
Ancient curve shortening flows have entropy and curvature bounds equivalent.
problem Bounding entropy and total curvature for ancient curve shortening flows.
method Equivalence of entropy and total curvature conditions for ancient curve shortening flows.
result Entropy and total curvature bounds are equivalent for ancient curve shortening flows.
Simplifies residual flows to make flow-based modeling more practical.
problem Extremely high computational cost of residual flows limits their applicability.
method Introduces Quasi-Autoregressive (QuAR) approach to residual flows.
result Significantly reduces compute time and memory requirements for flow-based modeling.
Almost Hermitian Ricci flowmath.DG
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.
Existence of translating solutions shown for curve diffusion flow.
problem Existence of translating solutions for curve diffusion flow.
method Higher order curve shortening flow approach.
result Properly immersed translating solutions exist.
Modeling bone microarchitecture adaptation using geometric flows.
problem Bone microarchitecture adaptation modeling.
method Advection and mean curvature flow model with a sphere as a test case.
result Closed-form solution for sphere under advection and mean curvature flow.
The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flo…
The study examines mass drop and multiplicity in mean curvature flow.
problem Analyzing mass drop and multiplicity in mean curvature flow.
method Defined Brakke flow with variational inequality, proved mass drop conditions.
result Mass drop and multiplicity one conjecture are equivalent for Brakke flows.
Ancient flows of elliptic functionals classified in various dimensions.
problem Classifying ancient solutions to gradient flows of elliptic functionals.
method Analyzing closed ancient solutions in Riemannian manifolds.
result Ancient solutions classified in multiple dimensions and cases.