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.
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48 results for “Liouville currents”
The study defines Finsler metrics on special surfaces and constructs geodesic currents.
problem Defining and studying Finsler metrics on specific geometric structures.
method Defined compatible Finsler distances, studied geodesics, and constructed Liouville currents.
result Constructs a Liouville current for each metric, encoding curve lengths.
The paper connects currents and entropy in hyperbolic 3-manifolds.
problem Understanding the entropy of negatively curved 3-manifolds.
method Intersection of geodesic and conformal currents, proving sharp bounds.
result New proofs of Liouville entropy, minimal surface entropy, and Mostow Rigidity Theorem.
Study confirms conjecture on extremal length of hyperbolic metrics.
problem Determining the extremal length of hyperbolic metrics on Riemann surfaces.
method Analyzes the topology of closed hyperbolic Riemann surfaces to find extremal lengths.
result Extremal length is topology-dependent and has a specific upper bound.
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.
New inequality shows energy growth and decay in geometric problems.
problem Understanding energy behavior in geometric problems.
method Introduced a symmetric (log-)epiperimetric inequality.
result Energy growth and decay observed in geometric problems.
In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields …
Let be a complete borderless infinite area hyperbolic surface. We introduce Thurston's boundary to the Teichmüller space of the surface using Liouville (geodesic) currents. Thurston's boundary to is identified with the space of projective bounded measured laminations on $X…
The paper explores the -Liouville property on graphs and its connections to stochastic completeness.
problem Investigating the -Liouville property on graphs and its implications.
method Characterization of -Liouville property in terms of Green function, equivalence with stochastic completeness, and comparison theorems based on inner-outer curvatures.
result Equivalence of -Liouville property and stochastic completeness on model graphs, and introduction of Dirichlet -Liouville property.
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.
Liouville structuresmath-ph
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.
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.
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.
Flexible metrics found on a genus 2 surface.
problem Identifying non-rigid hyperbolic cone metrics on a genus 2 surface.
method Using a theorem by Erlandsson, Leininger, and Sadanand.
result Nine mapping class group orbits of non-rigid metrics found.
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.
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 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 -harmonic function, -harmonic 1 form, and harmonic form (with ).
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.
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 - harmonic maps, - symphonic maps, and ---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.
Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometrymath.DG
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 -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.
Liouville theorem for --symphonic map , --harmonic map and and - harmonic map with free boundarymath.DG
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 --symphonic, --harmonic, and - harmonic maps.
result Established Liouville theorem for the specified harmonic maps with free boundary.
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 -Laplacian Liouville equation on the half-space with positive nonlinear Neumann boundary condition.
result The classification of solutions extends previous results for and .
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 -th eigenvalue grows linearly with , with a constant determined by the Liouville area and a specific .
Special Liouville metrics with Ricci-like conditions are determined by elliptic functions.
problem Characterizing Liouville metrics with Ricci-like conditions in complex space forms.
method Analyzing necessary conditions for induced metrics of parallel mean curvature surfaces and proving the existence of specific Liouville metrics.
result Explicit determination of special Liouville metrics with Ricci-like conditions by elliptic functions.
We study the Liouville action for quasi-Fuchsian groups with parabolic and elliptic elements. In particular, when the group is Fuchsian, the contribution of elliptic elements to the classical Liouville action is derived in terms of the Bloch-Wigner functions. We prove the first and second variation formulas for the cla…
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.
Ancient Lagrangian flows get limited convex solutions.
problem Controlling convex solutions of Lagrangian flows at antiquity.
method Proving a Liouville type theorem with quadratic growth restrictions.
result Ancient convex solutions are unique.
Liouville Theorem for curvature equation in half space with fully nonlinear boundary conditionmath.DG
Study proves Liouville theorem for specific curvature equations with boundary conditions.
problem Proving Liouville theorem for -curvature equations in half spaces with nonlinear boundary conditions.
method Established using positive constant curvature equations and variational functional approach.
result Proved Liouville theorem for positive constant -curvature equations in and boundary conditions.
Gradient estimates and Liouville theorems for the \(Φ\)-Laplacian equations on Riemannian manifoldsmath.DG
The paper establishes gradient estimates and Liouville theorems for Φ-Laplacian equations on Riemannian manifolds.
problem Gradient estimates and Liouville theorems for Φ-Laplacian equations on Riemannian manifolds.
method Nonlinear Φ-Bochner formula and Nash-Moser iteration technique for gradient bounds; maximum principle for parabolic case.
result Unified framework for gradient estimates and Liouville theorems for Φ-Laplacian equations.
Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler-Lagrange equations, culminating in a blo…
Paper proves a Liouville theorem for solitons with constant curvature.
problem Understanding harmonic functions on specific geometric structures.
method Proved a Liouville theorem without gradient estimates.
result Finite dimensionality of harmonic functions with polynomial growth.
New GLPs split Lévy bridges into non-overlapping subprocesses.
problem Creating multivariate stochastic processes with specific properties.
method Defining GLPs by splitting Lévy bridges and using time changes.
result GLPs have terminal values and increments with generalised multivariate Liouville distributions.