The Whitham flow for hyperelliptic curves has singularities that can be smoothly extended.
problem Singularities in the Whitham flow for hyperelliptic spectral curves.
method Analysis of deformations preserving periods of a meromorphic differential.
result Stable and unstable manifolds are non-empty, and the flow can be extended through the singularity.
We map out the moduli space of Lawson symmetric constant mean curvature surfaces in the 3-sphere of genus g>1 by flowing numerically from Delaunay tori with even lobe count via the generalized Whitham flow.
Study of harmonic maps from 2-torus to S^3 using spectral curves and Whitham deformations.
problem Investigating the space of harmonic maps from a 2-torus to S^3.
method Using spectral curve correspondence and Whitham deformations.
result The space of harmonic maps is smooth and has dimension two in an open and dense subset of the parameter space.
We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space which corresponds to the pairing between the space of quadratic differentials an…
Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.
problem Parameterize spectral data of constant mean curvature tori in R^3.
method Use Whitham deformations, blowups, and spectral data analysis.
result Prove the Wente family is parameterized by the bisector of the right angle.
Paper constructs solutions to WDVV equations for Frobenius manifolds.
problem Solving open WDVV equations for Frobenius manifolds.
method Explicit construction of flat F-manifolds and principal hierarchies.
result Recovery of polynomial solutions for A- and D-type singularities.
A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is em…
Paper constructs counterexamples of higher solutions to self-duality equations.
problem Whether all real sections give rise to global solutions of self-duality equations.
method Using Willmore surfaces and generalized Whitham flow.
result Higher solutions exist on an open dense subset of the Riemann surface, not globally.
In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, by using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surf…
We propose multidimensional versions of the Painlevé VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role o…
This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. In this theory, solutions of a PDE are sections of a fiber bundle Y over a base manifold X of dimension n+$1, typically taken to be spacetime. Given a conn…
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…
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.
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.
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.
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…
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.
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.