Corrects a mistake in Ricci Flow's curve-shrinking flow equations.
problem Incorrect upper bound for total curvature evolution.
method Shows the multiplicative constant depends on initial total curvature and length.
result Finite-time extinction result remains unaffected.
Study shows how embryo wounds heal through a mathematical model.
problem Understanding wound closure in embryonic epidermal healing.
method Developed a curvature flow model linked to physical wound closure.
result Closed, initially convex curves shrink to a round point in finite time under the flow.
We consider two types of p-centro affine flows on smooth, centrally symmetric, closed convex planar curves, p-contracting, respectively, p-expanding. Here p is an arbitrary real number greater than 1. We show that, under any p-contracting flow, the evolving curves shrink to a point in finite time and the only…
The paper proves the existence of a continuous family of translating surfaces under a specific curvature flow.
problem Existence of convex translating surfaces under flow by α-th power of Gauss curvature. method Constructing a family of translating surfaces using Jacobi fields and analyzing their effective growth rates.
result The family of translating surfaces is a topological manifold with quantitative convergence rates.
Positive curvature forces shrinking Ricci solitons to be compact.
problem Characterizing shrinking Ricci solitons with positive curvature.
method Analyzing sectional curvature properties of shrinking Ricci solitons.
result Positive curvature ensures compactness of shrinking Ricci solitons.
Researchers classify 4D Ricci solitons with positive curvature.
problem Classifying 4D shrinking Ricci solitons with positive sectional curvature.
method Analyzing sectional and scalar curvatures to classify solitons.
result Four-dimensional shrinking Ricci solitons are classified.
Study on evolving spoon-shaped networks in convex domains.
problem Evolution of spoon-shaped networks composed of two curves in convex domains.
method Curvature-driven evolution of the network, focusing on the enclosed area and length constraints.
result The network evolves to a Brakke spoon, with maximal existence time dependent on the enclosed area.
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 F-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 F-convergence rate for Ricci flows with closed and smooth tangent flows. result A Ricci flow with closed and smooth tangent flow is ∣logλ∣−θ close to its tangent flow in the F-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 R4. method Analysis of ancient noncollapsed flows in R4. result Nonexistence of rotating ancient flows among ancient noncollapsed flows in R4. 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.
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…
Study shows flows with maximal entropy are either Bernoulli or a product of a Bernoulli and rotational flow.
problem Characterizing ergodic measures of maximal entropy for smooth flows.
method Analyzes the structure of ergodic measures of maximal entropy for smooth three-dimensional flows.
result Smooth flows with maximal entropy are either Bernoulli or a product of a Bernoulli and rotational flow.
Paper studies geometric flows generalizing Ricci flow.
problem Geometric flows generalizing Ricci flow.
method Prove short time existence and provide curvature estimates for suitable scalar parameters.
result Proves short time existence and curvature estimates for geometric flows.
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.
RG-2 flow preserves area monotonicity in asymptotically flat spacetimes.
problem Mathematical and physical properties of RG-2 flow compared to Ricci flow.
method Study of RG-2 flow in asymptotically flat spacetimes with time-symmetric foliation.
result Area of a closed surface is monotonous under RG-2 flow, similar to Ricci flow.
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.
Two graphs show mean curvature flow differs from heat flow in dimensions n≥2.
problem Comparing mean curvature flow and heat flow on entire graphs.
method Analyzing two specific graphs in dimensions n≥2.
result Mean curvature flow and heat flow behave differently, with oscillation vs. stabilization.
Support Vector Machines predict gas-liquid flow patterns with 97% accuracy.
problem Predicting gas-liquid flow patterns in multiphase flow systems.
method Support Vector Machine (SVM) applied to a dataset of two-phase flow patterns.
result Achieved 97% correct classification of flow patterns.
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 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.
Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.
problem Whether mean curvature flow is a gradient flow on nondegenerate metric spaces of simple closed plane curves.
method Examined two nondegenerate metric spaces: uniformness-preserving and curvature-weighted structures.
result Mean curvature flow is not a gradient flow on either metric space.
Using the conformally invariant Cotton tensor, we define a geometric flow, the "Cotton flow", which is exclusive to three dimensions. This flow tends to evolve the initial metrics into conformally flat ones, and is somewhat orthogonal to the Yamabe flow, the latter being a flow within a conformal class. We define an en…
SurVAE Flows combine VAEs and flows using surjective transformations.
problem Combining the strengths of VAEs and flows to model complex densities.
method Modular framework of composable deterministic and stochastic transformations.
result Exact likelihood computation and lower bound on likelihood.
Survey of geometric flows from unified string theories.
problem None explicitly stated, but related to understanding geometric flows in string theories.
method Survey of geometric flows in various geometries (complex, almost-complex, symplectic) motivated by string theories.
result Intermediate flows between Ricci and Kähler-Ricci flows, often coupled to additional fields.
New derivation of Type IIA flow metrics.
problem Flow of metrics in Type IIA theory.
method Adapted to Laplacian flow, uses projected Levi-Civita connection.
result New derivation of flow equations.
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.
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.
Gauge Flow Models use a learnable Gauge Field in Generative Flow Models.
problem Improving generative model performance.
method Integrates a learnable Gauge Field into Flow ODEs.
result Gauge Flow Models outperform traditional Flow Models in Flow Matching experiments.
Discrete flows extend normalizing flows to discrete data, improving various applications.
problem Applying normalizing flows to discrete data distributions.
method Developed discrete autoregressive and bipartite flows, showing their effectiveness on various discrete data tasks.
result Discrete autoregressive flows outperform autoregressive baselines on synthetic discrete distributions and Potts models.
Higher Gauge Flow Models integrate higher geometry and symmetries into Generative Flow Models.
problem Improving generative models' performance.
method Integrates L∞-algebra into Generative Flow Models, leveraging higher geometry and symmetries. result Substantial performance improvements on Gaussian Mixture Model datasets.
By the method of discrete Morse flows, we construct an energy reducing multiple-valued function flow. The flow we get is Holder continuous with respect to the L-2 norm. We also give another way of constructing flows in some special cases, where the flow we get behaves like ordinary heat flow.
A new geometric flow K-flow on 3-manifolds shrinks or preserves homogeneous spheres.
problem Analyzing the behavior of Thurston's model geometries under the K-flow. method Defining and studying the K-flow on 3-dimensional Riemannian manifolds, using a DeTurck-type argument for short-time existence. result The K-flow shrinks or preserves homogeneous spheres, showing short-time existence.