Arnold-Liouville systems cannot be bi-Hamiltonian generically.
problem The bi-Hamiltonian structure of Arnold-Liouville systems.
method Proving that a specific class of smooth functions is a meagre subset for the Fréchet topology, which implies Arnold-Liouville systems cannot be bi-Hamiltonian.
result Generically, Arnold-Liouville systems cannot be bi-Hamiltonian.
Investigates integrable systems with linear periodic integral for e(3) Lie algebra.
problem Analyzes singularities and topological properties of integrable systems.
method Examines singularities of Liouville foliation, bifurcation diagram, transformations of Liouville tori, and isoenergy surfaces.
result Discovers topological properties of integrable systems with linear periodic integral.
Classifies geodesic flows on projective plane with potential field.
problem Classifying geodesic flows on a projective plane with a potential field.
method Liouville classification and calculation of Fomenko--Zieschang invariants.
result All Fomenko--Zieschang invariants of the system are calculated.
For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact" 2-parameter family of flat polygons equipped with certain pairing of sides. For the…
Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.
problem Finding invariant measures for contact Hamiltonian systems.
method Splitting the system into Reeb and Liouville dynamics; using invariant measures and symplectic sandwiches.
result Invariant measure found for Reeb dynamics; characterization of Liouville dynamics invariant measure.
Study non-Weinstein Liouville geometry via hyperbolic dynamics, proving rigidity results.
problem Characterize non-Weinstein Liouville geometry with persistent transverse skeleton.
method Anosov 3-flows, Liouville Interpolation Systems, non-singular partially hyperbolic flows, hyperbolic dynamics.
result Mitsumatsu's examples characterize 4D non-Weinstein Liouville geometry with 3D persistent transverse skeleton.
Adaptive neural network approximates stochastic system densities.
problem Approximating high-dimensional stochastic dynamical systems.
method Temporal KRnet (tKRnet) trained with adaptive collocation points and temporal decomposition.
result Improves density approximation for stochastic systems without curse of dimensionality.
Contact Lie systems analyze integral curves of Hamiltonian vector fields.
problem Analyzing integral curves of Hamiltonian vector fields.
method Define and study contact Lie systems, including conservative systems.
result Develop Liouville theorems, contact reductions, and Gromov non-squeezing theorems.
Let A=(aij)n×n be an invertible matrix and A−1=(aij)n×n be the inverse of A. In this paper, we consider the generalized Liouville system: \label{abeq1} Δ_g u_i+\sum_{j=1}^n a_{ij}ρ_j(\frac{h_j e^{u_j}}{\int h_j e^{u_j}}-1)=0\quad\text{in \,}M, where 0<hj∈C1(M) and $ρ_j\in \mathb…
We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. W…
Construct Lax pairs for BKM equations and related integrable hierarchies.
problem Integrability of BKM equations and related systems.
method Construction of Lax pairs for BKM equations and related integrable hierarchies using Sturm-Liouville operators.
result Lax pairs for many integrable hierarchies including previously studied ones.
We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) two-dimensional sphere. The second integral is cubic in the momenta.
We study the wave analog of the Liouville equation and the constant mean curvature equations in 2 space dimensions, which are energy critical. We exhibit a blow-up criteria for the former using tools from conformal geometry, and we exhibit finite time blow-up for the latter under suitable assumptions on the initial dat…
The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if th…
The abstract shows how constant mean curvature surfaces in hyperbolic space are linked to the Liouville equation.
problem The constant mean curvature one equation for surfaces in hyperbolic 3 space.
method Group theoretic constructions and equivalence of quotient representations.
result The Darboux integrability of the equations shows that they admit equivalent quotient representations.
We study here the action of subgroups of PSL(2,R) on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem: Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on…
Study magnetic geodesic flows on spheres, describing their bifurcations.
problem Analyzing magnetic geodesic flows on 2-spheres.
method Generic pair of functions (f,Λ), Liouville fibration, Fomenko-Zieschang invariant, bifurcation diagrams. result Bifurcation diagrams consist of two curves in the (h,k)-plane. Study Toda systems blowup masses linked to Weyl groups.
problem Understanding blowup phenomena in Toda systems.
method Concrete examples of Toda systems solutions and blowup masses.
result Blowup masses correspond to Weyl groups.
We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg-de Vries type), geometry of Liouville-ty…
The paper explores the L1-Liouville property on graphs and its connections to stochastic completeness.
problem Investigating the L1-Liouville property on graphs and its implications. method Characterization of L1-Liouville property in terms of Green function, equivalence with stochastic completeness, and comparison theorems based on inner-outer curvatures. result Equivalence of L1-Liouville property and stochastic completeness on model graphs, and introduction of Dirichlet L1-Liouville property. Flow approach solves Toda system equations.
problem Solving the Toda system equations.
method Introducing Toda flow to study the system.
result Global existence and convergence conditions established.
We give exposition of a Liouville theorem established in \cite{Li3} which is a novel extension of the classical Liouville theorem for harmonic functions. To illustrate some ideas of the proof of the Liouville theorem, we present a new proof of the classical Liouville theorem for harmonic functions. Applications of the …
Study explores Liouville's theorem and SLP for harmonic functions on cones and surfaces.
problem Investigating Liouville's theorem and SLP for harmonic functions on Riemannian cones and surfaces.
method Reinterprets classical Liouville property in terms of radial eigenfunctions, providing explicit estimates and constructing examples.
result Explicit estimates for slowest-growing nonconstant harmonic functions and a unified geometric perspective on Liouville phenomena.
Equal diagonal energies proven on Liouville surfaces.
problem Diagonal energies on Liouville surfaces.
method Analyzing parameter curves and rectangles on Liouville surfaces.
result Diagonal energies are equal in n-dimensional Liouville manifolds.
Extends integrability to cosymplectic manifolds.
problem Integrability of Hamiltonian systems on cosymplectic manifolds.
method Extended Arnold-Liouville and noncommutative integrability to cosymplectic manifolds, proved a variant of non-commutative integrability for specific fields, constructed action-angle variables.
result Variant of non-commutative integrability for evaluation and Reeb vector fields on cosymplectic manifolds.
The paper explores rigidity and splitting theorems for sub-static spaces with minimal hypersurfaces.
problem Rigidity and splitting problems for sub-static systems with boundary.
method Local and global splitting theorems, boundary integral inequalities, Liouville theorem.
result Improvements in rigidity and splitting results for sub-static spaces, including vacuum and non-vacuum cases.
A 'Liouville structure' is a structure isomorphic to a cotangent vector fibration. A Liouville structure is an essential ingredient of every variational formulation of a physical theory. For reasons of interpretation the Liouville structure can not be replaced by the corresponding cotangent fibration. We give a precise…
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.
A large class of variational equations for geometric objects is studied. The results imply conformal monotonicity and Liouville theorems for steady, polytropic, ideal flow, and the regularity of weak solutions to generalized Yang-Mills and Born-Infeld systems.
The Liouville theorem is proven for V T-harmonic map heat flow.
problem Proving Liouville theorems for V T-harmonic maps.
method Analyzing heat flow on manifolds with specific properties.
result Liouville theorems established for V T-harmonic maps.
The paper studies how to transform a sequence of cmc planes into a minimal surface.
problem Transforming a sequence of constant mean curvature planes into a minimal surface.
method Using algebraic-geometric correspondence and solving the Gauss-Codazzi equations.
result A sequence of solutions to the sinh-Gordon system converges to a solution of Liouville's equation, which is related to the Korteweg-de Vries system.
Study Liouville action for harmonic maps between Riemann surfaces.
problem Optimizing harmonic maps between Riemann surfaces.
method Derive variational formula for Liouville action.
result Found variational formula for harmonic diffeomorphisms.
We study simple root flows and Liouville currents for Hitchin representations. We show that the Liouville current is associated to the measure of maximal entropy for a simple root flow, derive a Liouville volume rigidity result, and construct a Liouville pressure metric on the Hitchin component.
Geometrically reduces Hamiltonian systems using particular integrals.
problem Hamiltonian systems with invariant zero-level submanifolds.
method Develops a reduction mechanism using particular integrals in the Hamiltonian context.
result Direct bridge between particular integrals and presymplectic reduction.
The paper classifies invariant operators and proves a Liouville theorem.
problem Classifying invariant operators and proving Liouville theorem.
method Classified Möbius invariant differential operators and established a Liouville type theorem.
result Established a Liouville type theorem for Möbius invariant equations.
Study topological properties of integrable case on Lie algebra so(4).
problem Topological analysis of integrable case for Euler's equations on so(4).
method Construction of bifurcation diagrams, determination of critical points, description of Liouville tori bifurcations, computation of loop molecules.
result Some topological properties of Kovalevskaya case can be derived from the case on so(4).
Study relates symplectic homology capacity to periodic orbits in Liouville domains.
problem Relating symplectic homology capacity to periodic orbits in Liouville domains.
method Uses positive symplectic homology and Hofer-Zehnder capacity to establish bounds and existence of periodic points.
result Non-zero positive symplectic homology implies finite upper bound for Hofer-Zehnder capacity relative to skeleton and Hamiltonian diffeomorphisms.
We suggest a construction that, given a trajectorial diffeomorphism between two Hamiltonian systems, produces integrals of them. As the main example we treat geodesic equivalence of metrics. We show that the existence of a non-trivially geodesically equivalent metric leads to Liouville integrability, and present explic…
The universal Liouville action equals the renormalized volume of a hyperbolic 3-manifold.
problem Understanding the geometric significance of the universal Liouville action.
method Analyzing the Weil-Petersson universal Teichmüller space and its relation to hyperbolic 3-manifolds.
result The gradient flow of the universal Liouville action converges to the origin, providing a bound on Weil-Petersson distance.
Study shows that deformed Liouville metrics on tori remain Liouville.
problem Tackles the conjecture that only Liouville metrics are integrable on tori.
method Examines deformations of non-flat Liouville metrics and proves they remain Liouville.
result For a broad class of deformations, the deformed metric remains Liouville.
Proves a Liouville-type theorem for p-Laplacian on manifolds.
problem Proving Liouville-type theorems for p-Laplacian on manifolds.
method Proved a Liouville-type result for the p-Laplacian on complete Riemannian manifolds.
result Proved a Liouville-type theorem for the p-Laplacian on complete non-compact Riemannian manifolds.
Study Liouville theorem for specific harmonic maps.
problem Proving Liouville theorem for harmonic maps.
method Establish gradient estimates under specific conditions.
result Proved Liouville theorem for harmonic maps.
The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.
problem Investigating Liouville type properties of harmonic forms on Riemannian manifolds.
method Normalized integral Ricci curvature and BiRic curvature.
result Established Liouville theorems for p-harmonic function, p-harmonic 1 form, and harmonic q form (with q≥2). Optimal Liouville theorem for half-Euclidean space equations.
problem Optimal Liouville-type theorems for conformally invariant equations.
method Established optimal Liouville-type theorems for conformally invariant second-order elliptic equations.
result Proved an optimal Liouville-type theorem for equations in the half-Euclidean space.
The paper proves Liouville rigidity for Hessian equations, characterizing geometric conditions for constant solutions.
problem Characterizing geometric conditions for constant solutions in Hessian equations.
method Recursive geometric condition (Liouville admissibility) and anisotropic constructions.
result The Liouville-type property is characterized as a geometric property of the admissible set.
This work generalizes Hamiltonian mechanics using closed differential forms.
problem Hidden invariants in classical Hamiltonian mechanics.
method Establishes a novel correspondence between generalized Hamiltonian mechanics and multisymplectic geometry.
result Key theorems linking classical and generalized Hamiltonian systems.
Generative model improved using Liouville PDE-based sliced-Wasserstein flow.
problem Improving generative models for fair regression.
method Transformed sliced-Wasserstein flow into Liouville PDE-based formalism, handling density estimation with normalizing flows of neural ODE.
result Outperforms in convergence and fairness with reduced variance.
The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.
problem Deriving Liouville theorems for generalized maps on Riemannian manifolds.
method Using conservation laws and monotonicity formulas, the paper derives Liouville theorems for different types of maps under various conditions.
result The paper establishes Liouville theorems for several types of generalized maps, including φ-F harmonic maps, φ-F symphonic maps, and φ-F-V-harmonic maps.