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.
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.
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…
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.
We provide a variational description of any Liouville (i.e. volume preserving) autonomous vector fields on a smooth manifold. This is obtained via a ``maximal degree'' variational principle; critical sections for this are integral manifolds for the Liouville vector field. We work in coordinates and provide explicit for…
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…
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.
Unified geometric framework for quantum states using dual number algebras.
problem Representing quantum states in a geometrically unified way.
method Smooth embeddings into higher-order dual number algebras and algebraic flows.
result Established nilpotent dual algebras as a geometric landscape for quantum kinematics.
Paper derives estimates for Hessian equations under concavity assumptions.
problem Estimating solutions to Hessian equations with concavity constraints.
method Exploiting concavity of sums of Hessian operators, deriving Pogorelov estimates.
result Entire admissible solutions in R^n with quadratic growth are quadratic polynomials.
Characterizes rigid and flexible hyperbolic cone metrics and billiards.
problem Understanding the rigidity and flexibility of hyperbolic cone metrics and their billiard dynamics.
method Characterization through Liouville currents and deformation spaces.
result Generic rigidity and parameterization of deformation spaces for flexible metrics.
On the basis of Liouville theorem the generalization of the Nambu mechanics is considered. Is shown, that Poisson manifolds of n-dimensional multi-symplectic phase space have inducting by (n-1) Hamiltonian k-vector fields, each of which requires of (k)-hamiltonians.
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. 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.
On a symplectic manifold a family of generalized Poisson brackets associated with powers of the symplectic form is studied. The extreme cases are related to the Hamiltonian and Liouville dynamics. It is shown that the Dirac brackets can be obtained in a similar way.
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.
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).
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.
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…
The theory of derivative of noninteger order goes back to Leibniz, Liouville and Riemann. Derivatives of fractional order have found many applications in recent studies in mechanics, physics, economics. In this paper we define the fractional tangent bundle on a manifold, using a method of Radu Miron. The fractional Lei…
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.
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.
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.
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 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. 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.
We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled tables that differ by an affine map. The main tool is a new theorem that establi…
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.
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. This note provides a new proof of the real analyticity of the Liouville map.
problem Real analyticity of the Liouville map on Riemann surfaces.
method Complex analysis approach.
result Real analyticity of the Liouville map proved using complex analysis.
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.
Liouville domains have become central objects in symplectic and contact geometry. However, the auxiliary data they involve --- namely, Liouville forms --- and the non-compactness of their completions generate some inconvenience. The notion of ideal Liouville domains is designed to suppress these awkward aspects and to …
The paper proves a Liouville theorem for heat flows on manifolds with specific curvature conditions.
problem Investigating heat flows on manifolds with specific curvature conditions.
method Gradient estimate and Liouville type theorem for ancient solutions.
result Established a Liouville theorem for V-harmonic heat flows. The paper extends Liouville's theorem to calibrated geometries in various dimensions.
problem Extending Liouville's theorem to calibrated geometries in different dimensions.
method Analyzing Sobolev mappings and calibrations in calibrated geometries.
result Calibrations in certain dimensions have the Liouville property.
The paper proves a Liouville theorem for specific harmonic maps with free boundary.
problem Analyzing harmonic maps with free boundary conditions.
method Developed Liouville theorem for φ-F-symphonic, φ-F-harmonic, and φ-ΦS,p,ε harmonic maps. result Established Liouville theorem for the specified harmonic maps with free boundary.
Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.
problem Investigate semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces.
method Analyze eigenfunctions in low, critical, and high energy regimes using quantum ergodicity and equidistribution.
result Eigenfunctions in different regimes converge to distinct measures: invariant, Liouville, or equidistributed.
Liouville entropy increases strictly along Ricci flow on surfaces.
problem Understanding the behavior of Liouville entropy under Ricci flow.
method New expression for Liouville entropy derivative, proof of positivity in specific directions.
result Liouville entropy is strictly increasing along normalized Ricci flow for 1/6-pinched metrics.
Improved Liouville theorems for ancient solutions to V-harmonic map heat flows.
problem Establishing Liouville theorems for ancient solutions to V-harmonic map heat flows.
method Refined gradient estimates and exponential growth conditions.
result Better Liouville theorems for ancient solutions to V-harmonic map heat flows.
New method shows some 3D shapes can't be filled in certain ways.
problem Obstructing Liouville and weak fillability of contact structures.
method Introducing a new method to obstruct fillability.
result Various rational homology 3-spheres admit strongly fillable contact structures without Liouville fillings.
The paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.
problem Classifying solutions to a Liouville equation with a nonlinear Neumann boundary condition.
method Analyzing the n-Laplacian Liouville equation on the half-space R+n with positive nonlinear Neumann boundary condition. result The classification of solutions extends previous results for n=2 and p=n. Researchers found a Weyl law for Liouville quantum gravity eigenvalues.
problem Understanding the spectral geometry of Liouville quantum gravity.
method Obtained a Weyl law for eigenvalues of Liouville Brownian motion.
result The n-th eigenvalue grows linearly with n, with a constant determined by the Liouville area and a specific cγ.