The paper defines flows on Z-graded manifolds and proves unique maximal flows for vector fields.
problem Lack of a treatment for flows on Z-graded manifolds. method Definition and proof of maximal flows for vector fields on Z-graded manifolds. result Every vector field admits a unique maximal flow, with conditions for vector fields invariant under flows and commuting flows.
Vector fields on schemes have flows if rings are finitely generated.
problem Understanding vector fields and flows on schemes.
method Analyzing vector fields on affine C∞-schemes with finitely generated rings. result Vector fields on affine C∞-schemes with finitely generated rings have flows and are groupoid internal maps. Study shows limitations of Lie bracket commutation for nonsmooth vector fields.
problem Limitations of Lie bracket commutation for nonsmooth vector fields.
method Analysis of nonsmooth vector fields, focusing on commutation of flows and Lie bracket conditions.
result Lie bracket commutation cannot be extended to general a.e. differentiable vector fields, but holds for certain Sobolev regular fields.
Proofs for flows of linear vector fields and their applications.
problem Existence of flows for linear vector fields and related properties.
method Detailed proofs and flow construction techniques.
result Smooth triviality of vector bundles over contractible bases and isomorphy of fibers of transitive Lie algebroids.
Researchers identify surfaces with special fluid flow fields.
problem Understanding fluid flows on curved surfaces.
method Defined and analyzed hydrodynamic Killing vector fields (HKVF) on surfaces.
result Any connected, orientable surface with HKVF is conformally equivalent to one of 14 canonical Riemann surfaces.
In this paper we introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution of this flow, discuss its convergence and possible applications, and its relation to the …
Study vector fields and flows on singular spaces like submanifolds.
problem Understanding vector fields and flows on singular spaces.
method Integrate derivations of the C∞-ring of global smooth functions into flows. result Derivations integrate to smooth flows on subcartesian spaces.
The geodesic flow on the tangent bundle is the flow of a certain vector field which is called the spray S:TM→TTM. The flow lines of the vector field $\ka_{TM}øTS:TTM\to TTTM$ project to the Jacobi fields on TM. This could be called the Jacobi flow.
Two proofs of Kalman Theorem using flows of vector fields.
problem Classical result of Control Theory (Kalman Theorem).
method Two proofs using flows of vector fields.
result New criteria for local controllability of non-linear systems.
Analyzes vector fields in polytope decompositions, proving curve finiteness.
problem Analyzing vector fields in polytope decompositions.
method Proves integral curves are chopped into finitely many pieces by polytope decompositions.
result Finiteness of edge flips in discrete Yamabe flow.
Flow of curves with curvature and forcing vector field exists.
problem Existence of a curve flow with curvature and forcing.
method Proved existence through Brakke motion law.
result Non-trivial flow of curves exists through singularities.
Study Kähler-Ricci flow on manifolds with singularities.
problem Behavior of Kähler-Ricci flow on manifolds with finite-time singularities.
method Use of holomorphic vector fields to prove estimates.
result Proves estimates related to previous work on the flow.
This paper gives quantitative global estimates between a time dependent flow on a Riemannian manifold (M) and the flow of a vector field constructed by truncating the formal Magnus expansion for the logarithm of the flow. As a corollary, we also find quantitative estimates between the composition of the …
Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
problem Geodesic completeness and flow properties of compact Brinkmann spacetimes.
method Proof of geodesic completeness and flow properties of isotropic parallel vector fields in compact Brinkmann spaces.
result Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
Study geometric flows with varying parameters and prove continuous dependence.
problem Continuous dependence of flows on parameters in geometric settings.
method Derived suitable topologies for vector fields and flows, proved new continuous dependence.
result Proved continuous dependence of flows on parameters in a general topological space.
Paper extends isoperimetric inequalities for non-starshaped hypersurfaces.
problem Isoperimetric inequalities for non-starshaped hypersurfaces.
method Volume preserving and area decreasing mean curvature flow with conformal Killing vector fields.
result Established isoperimetric inequalities for a broader class of hypersurfaces.
Proves properties of Morse vector fields on compact manifolds.
problem Properties of gradient vector fields of Morse functions.
method Analyzes connectedness of critical points and shrinkage of flow.
result Shows connectedness of critical points through orbits and exponential shrinkage.
Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.
problem Computing homology in discrete and smooth dynamical systems.
method Counting flow lines between orbits and critical points.
result Directly recovers Z2 homology from flow lines. This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows o…
Exact universal interpolation property for landmark configurations in Euclidean space.
problem Representing and deforming landmark configurations through flows of vector fields.
method Explicitly describe vector fields for exact universal interpolation property in all dimensions.
result Achieve controllability by combining constant and polynomial vector fields.
Develops a new exponential map for time-varying vector fields.
problem Lack of global flows for general time-varying vector fields.
method Categorical development of spaces of vector fields and flows, allowing for systematic localisation.
result Derives the homeomorphism of the exponential map for vector fields with measurable time-dependence.
DFM simplifies CNF training without interpolants.
problem Efficiently training CNFs with computationally expensive ODE solving.
method DFM optimizes dual vector fields for bijective transformations.
result DFM outperforms CNF trained with FM or ML objectives.
Lyapunov 1-forms on orbifolds help understand flows on compact spaces.
problem Understanding flows on orbifolds using Lyapunov 1-forms.
method Introducing Lyapunov 1-forms, using asymptotic cycles and chain-recurrent sets.
result Existence of a Lyapunov 1-form in a prescribed cohomology class for compact orbifolds.
RFM uses tangent vector fields to match data on manifolds, analyzing TV convergence for Euler discretization.
problem Matching data on curved manifolds using flow-based models.
method Developed a nonasymptotic TV convergence analysis for RFM samplers using Euler discretization.
result Explicit bounds on TV convergence separating numerical discretization and learning errors.
Study the isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.
problem Isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.
method Introduced a mean curvature type flow to study the isoperimetric problem.
result Established the isoperimetric inequality for star-shaped hypersurfaces in such manifolds.
The paper classifies solitons in a curved product space.
problem Classifying solitons in a curved product space.
method Examined vector fields tangent to fibers and rotations, classified solitons under specific symmetries.
result A classification of solitons in s2imesR under certain symmetries. Rescaling expansiveness proven for k*-expansive vector fields.
problem Proving rescaling expansiveness for k*-expansive vector fields.
method Introducing and exploring singular-expansive flows.
result Rescaling expansiveness established for k*-expansive vector fields.
We consider the dynamics of vector fields on three-manifolds which are constrained to lie within a plane field, such as occurs in nonholonomic dynamics. On compact manifolds, such vector fields force dynamics beyond that of a gradient flow, except in cases where the underlying manifold is topologically simple. Furtherm…
In this paper we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In…
The paper examines timelike conformal fields on 3-manifolds and finds they are rigidly tied to specific geometric structures.
problem Investigating timelike conformal vector fields on closed Lorentzian 3-manifolds.
method Performing conformal changes to unit vectors and analyzing the resulting flows through stable Hamiltonian structures and cohomology.
result Timelike conformal vector fields on 3-manifolds are either Reeb vector fields of Sasakian or co-Kähler structures.
Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
A new framework for generative modeling using controlled vector fields.
problem Expressive modeling with limited parameters.
method Continuous-time modeling with modulated fixed vector fields and learned scalar controls.
result Expressive transport achieved with a small number of learned control channels.
New proof confirms periodic orbit conjecture for Eulerisable flows.
problem Periodic orbit conjecture for non-vanishing vector fields on closed manifolds.
method Characterization of Eulerisable flows and use of strongly adapted one-forms.
result Periodic orbit conjecture holds for Eulerisable flows.
Improved flow matching using Gaussian processes for better sample quality.
problem Training continuous normalizing flows with reduced variance and flexibility.
method Extending conditional flow matching to streams modeled with Gaussian processes.
result Improved quality of generated samples with moderate computational cost.
High order discretization schemes of SDEs by using free Lie algebra valued random variables are introduced by Kusuoka, Lyons-Victoir, Ninomiya-Victoir and Ninomiya-Ninomiya. These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical me…
Study classifies harmonic vector fields on 3-manifolds.
problem Classifying harmonic unit vector fields on 3-manifolds.
method Investigates under mild curvature assumptions, classifying vector fields and manifolds.
result Classifies both vector fields and manifolds supporting them.
The paper simplifies proofs and characterizes contact structures in 3D.
problem Contact structures induced by geodesic vector fields in 3D.
method New proofs and characterizations of contact structures.
result Contact structures in 3D are universally tight under certain conditions.
From the paper "Formality Conjecture" (Ascona 1996): "I am aware of only one such a class, it corresponds to simplest good graph, the complete graph with 4 vertices (and 6 edges). This class gives a remarkable vector field on the space of bi-vector fields on Rd. The evolution with respect to the t…
Study vector fields on hyperbolic spaces to create Ricci-Bourguignon solitons.
problem Characterize vector fields on hyperbolic spaces Hn that transform them into Ricci-Bourguignon solitons. method Detailed geometric study of vector fields in dimensions n=2,3 and n≥3, focusing on dual forms in odd dimensions. result Dual forms of these vectors are contact forms in odd dimensions.
For a vector field X on a smooth manifold M there exists a smooth but not necessarily Hausdorff manifold MR and a complete vector field XR on it which is the universal completion of (M,X).
In this paper, we define a certain "proportional volume property" for an unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum and this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the…
Characterizes Lie-Backlund vector fields in infinite jet bundles that can be exponentiated.
problem Characterizing Lie-Backlund vector fields in infinite dimensional jet bundles that can be exponentiated.
method Characterization through conditions for exponentiation and providing non-trivial examples.
result Conditions for exponentiation of Lie-Backlund vector fields in J∞(Rn,Rm), with different results for m=1 and m>1. Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.
problem Existence of non-vanishing steady Euler flows and Beltrami fields in high dimensions.
method Using open books, proved existence of non-vanishing steady solutions to the Euler equations for vector fields in odd dimensions.
result Existence of non-vanishing steady Euler flows and Beltrami fields in high dimensions.
Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.
problem Diagonalizing the Toda flow on matrices with simple spectrum.
method Lie theoretic methods applied to complex semisimple Lie algebras and their real forms.
result Decouples the Toda vector field into simpler components.
Surveying recent progress on flows of G2-structures on 7-manifolds.
problem Preserving metrics while modifying G2-structures on 7-manifolds. method Heat flows and other approaches in terms of 3-forms, octonions, vector fields, and geometric structures. result Comparison of different perspectives on G2-structure flows. The paper generalizes relations between dynamical series and resolvents of vector fields.
problem Analyzing dynamical series using resolvents of vector fields.
method Derives the general form of relations involving intersection of kernel with integration currents for any smooth flow.
result Computes values of dynamical series and their relation with topological invariants.
We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic is hyperbolic), then we can…
We study the twisted Ruelle zeta function ζX(s) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove Fried conjecture, relating Reidemeister torsion and ζX(0). In higher dimensions, we show more generally that ζX(0) is locally constant with…