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.
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48 results for “Liouville energy”
Derives stress-energy identities in Liouville theory on compact surfaces.
problem Stress-energy tensor correlation functions on compact Riemann surfaces.
method Varying correlation functions with respect to background metric, treating different types of variations separately.
result Stress-energy correlation functions expressed as differential operators acting on primary field correlation functions.
The paper proves unique constant solutions for maps with p-Ginzburg-Landau energy.
problem Finding unique constant solutions for maps with p-Ginzburg-Landau energy.
method Assuming growth conditions or asymptotic conditions for the p-Ginzburg-Landau energy, the paper establishes Liouville type theorems.
result Establishes unique constant solutions for constant Dirichlet boundary value problems on starlike domains.
The study proves that under certain geometric conditions, biharmonic maps are harmonic if their energy is small.
problem Characterizing biharmonic maps with small energy.
method Proving a Liouville-type theorem for biharmonic maps with specific geometric constraints.
result Biharmonic maps are harmonic if their -energy is sufficiently small and other conditions are met.
The paper introduces a new energy density function and proves Liouville type theorems for various maps.
problem Proving Liouville type theorems for holomorphic, harmonic, and pluri-harmonic maps.
method Introducing a new energy density function and deriving Hessian estimates.
result No non-constant holomorphic map exists between certain Hermitian manifolds with specific curvature conditions.
Holographic principle matches deformed Liouville theory action.
problem Matching deformed Liouville theory action in holography.
method Developed a holographic scheme involving bending energy.
result Perfect match between deformed theory actions on field and gravity sides.
Paper analyzes solutions to equations on surfaces with boundary singularities.
problem Analyzing solutions to super-Liouville equations on surfaces with boundary singularities.
method Developed a new method to deduce the removability of boundary singularities due to the vanishing of the Pohozaev constant.
result Established energy quantization for solutions to super-Liouville type equations.
The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.
problem Improving energy decay estimates for Dir-stationary Q-valued functions.
method Establishing improved decay estimates and applying them to derive Liouville-type theorems and continuity.
result Dir-stationary Q-valued functions exhibit the Lebesgue property and reside in a generalized Campanato-Morrey space.
Liouville-type results for stationary maps of a class of functional related to pullback metricsmath.DG
Study Liouville-type results for stationary maps related to pullback metrics.
problem Analyzing stationary maps and their properties related to pullback metrics.
method Derive first variation formula, stress-energy tensor, and monotonicity formula.
result Derive Liouville-type results and investigate boundary value problems.
In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below. Our proof does not rely on the uniformization theorem and the Onofri inequality, thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow…
In this paper, we study harmonic functions on weighted manifolds and harmonic maps from weighted manifolds into Hadamard spaces introduced by Korevaar and Schoen. We prove Liouville theorems for these harmonic maps with finite energy.
Constructs a path integral for quantum Mabuchi K-energy.
problem Quantum corrections to classical Mabuchi K-energy.
method Coupling Liouville action and Mabuchi K-energy via probabilistic tools.
result Obtains a path integral with Weyl anomaly including K-energy term.
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.
The paper proves that certain maps from complex space to Kahler manifolds are holomorphic.
problem Characterizing harmonic maps from complex space to Kahler manifolds.
method Proves that harmonic maps from to any Kahler manifold must be holomorphic under a specific condition.
result Harmonic maps from to Kahler manifolds are holomorphic under an energy density assumption.
We introduce and study an approximate solution of the p-Laplace equation, and a linearlization of a perturbed p-Laplace operator. By deriving an -type Bochner's formula and a Kato type inequality, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact …
Researchers prove constant solutions for a specific Finslerian equation.
problem Investigating exponentially harmonic functions on Finslerian spaces.
method Analyzing the exponential energy functional and using nonnegative Ricci curvature conditions.
result Any bounded solution to the Finslerian equation is constant.
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.
Unique solutions found for a specific flow equation.
problem Finding unique solutions for a specific flow equation.
method Used a uniform bound for the Liouville energy and a natural space-time -bound for the time derivative of the solution.
result Uniqueness of classical solutions for the normalised two-dimensional Hamilton-Ricci flow.
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.
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.
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…
Gradient Ricci solitons can be extended to non-gradient Ricci solitons using energy function.
problem Extending the geometry of gradient Ricci solitons to non-gradient Ricci solitons.
method Using energy function to study the geometry.
result A non-steady Ricci soliton with symmetric covariant derivative is gradient.
Abstract notes on energy inequalities for harmonic maps with potential.
problem Characterizing the qualitative behavior of solutions to nonlinear Poisson equations.
method Generalization of results for harmonic maps with potential between Riemannian manifolds.
result Gradient estimates, monotonicity formulas, and Liouville theorems under curvature and energy assumptions.
Researchers prove spectral rigidity of Liouville tori under specific conditions.
problem Spectral rigidity of Liouville tori under generic conformal classes.
method Noncancellation of wave trace and analysis of second order variational formula for energy.
result Laplace isospectral deformations of Liouville metrics on torus are trivial.
Modified geometric quantization simplifies quantum mechanics on manifolds.
problem Quantum mechanics on Riemannian manifolds with scalar curvature.
method Introducing an arbitrary connection on the polarization bundle for real polarizations.
result Obtained an energy operator without the scalar curvature term.
Research proves the semi-classical limit of Liouville conformal field theory, describing deterministic geometry from random fluctuations.
problem Proving the semi-classical limit of Liouville conformal field theory.
method Probabilistic definition of Liouville theory, proving existence of semi-classical limit, defining classical stress-energy tensor.
result Existence and description of the semi-classical limit in terms of a massive Gaussian free field with Robin boundary conditions.
Obstructs complete metrics with positive scalar curvature on non-compact manifolds.
problem Obstructing complete metrics with positive scalar curvature on non-compact manifolds.
method Using minimal hypersurfaces and MOTS, the study provides topological obstructions and proves the Liouville theorem.
result The Liouville theorem for locally conformally flat n-manifolds of non-negative scalar curvature follows from the impossibility of positive scalar curvature metrics.
In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a f…
In this paper, we introduce the stress-energy tensors of the partial energies E'(f) and E"(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps …
Estimates for Poisson equation on manifolds with weighted Poincare inequality.
problem Existence and estimates of Poisson equation solutions on manifolds.
method Develops Green's function estimate using weighted Poincare inequality and Ricci curvature.
result Proves Liouville property for finite energy holomorphic functions on Kähler manifolds.
The paper proves a Liouville theorem for a specific type of harmonic maps on foliated manifolds.
problem Investigating harmonic maps on foliated Riemannian manifolds.
method First variational formulas, generalized Weitzenböck type formula, and Liouville type theorem for -harmonic maps.
result Established a Liouville type theorem for -harmonic maps.
Researchers prove zero solutions for certain p-Laplacian equations in convex cones.
problem Proving zero solutions for anisotropic Finsler p-Laplacian equations in convex cones.
method Doubling argument, blowing-up method, Liouville theorems.
result All nonnegative solutions must be zero without boundedness assumption.
We define a formal Riemannian metric on a given conformal class of metrics on a closed Riemann surface. We show interesting formal properties for this metric, in particular the curvature is nonpositive and the Liouville energy is geodesically convex. The geodesic equation for this metric corresponds to a degenerate ell…
Study on super-Toda system on Riemann surfaces, analyzing solutions with four types of bubbling.
problem Analyzing solutions to the super-Toda system on Riemann surfaces.
method Blow-up analysis, energy gap results for spinor parts of solutions.
result Showed energy identities for spinor parts of a blow-up sequence of solutions.
The paper studies geometric properties of -harmonic maps and proves Liouville type results.
problem Exploring geometric properties of -harmonic maps.
method Unified geometric analytic methods, first and second variation formulas, stress-energy tensor, conservation law, monotonicity formula, asymptotic assumption, extrinsic average variational method.
result Proves Liouville type results for -harmonic maps.
Global existence and convergence of heat flow for p-harmonic maps.
problem Global existence and convergence of heat flow for p-harmonic maps between manifolds.
method Analysis of heat flow equations for p-harmonic maps.
result Global existence and convergence of heat flow for p-harmonic maps under certain conditions.
Study on Sturm-Liouville problems with zero potential and Neumann boundary conditions.
problem Understanding properties of Sturm-Liouville problems with zero potential.
method Developed simple criteria for assessing properties of regular Sturm-Liouville problems in terms of coefficient functions.
result Proved various properties of Sturm-Liouville problems with zero potential under Neumann boundary conditions.
Study on 4D Riemannian manifolds solves curvature problem.
problem Resonant prescribed T-curvature problem on compact manifolds.
method Variational theory, energy and gradient estimates, Morse lemma, Liouville technique.
result New existence results for critical points at infinity.
We investigate the analogy between the large N expansion in normal matrix models and the asymptotic expansion of the determinant of the Hilb map, appearing in the study of critical metrics on complex manifolds via projective embeddings. This analogy helps to understand the geometric meaning of the expansion of matrix m…
Study Liouville currents for Hitchin representations using root flows.
problem Understanding Liouville currents in Hitchin representations.
method Investigate simple root flows and Liouville currents.
result Derive a Liouville volume rigidity result and construct a Liouville pressure metric.
New proof removes decay assumptions for spacetime positive mass theorem.
problem Proving the rigidity of the spacetime positive mass theorem without additional decay assumptions.
method Uses spacetime harmonic functions and Liouville's theorem, and an alternative proof based on Killing development.
result Removes additional decay assumptions for the spacetime positive mass theorem.
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.
Study the topology of energy surfaces in a specific mathematical case.
problem Determine the topology of isoenergy surfaces in the Kovalevskaya integrable case on Lie algebra so(4).
method Use Fomenko-Zieschang invariants of Liouville foliations.
result Classify diffeomorphisms of three-dimensional regular level surfaces.
Symplectic Tate homologymath.SG
For a Liouville domain satisfying , we propose in this note two versions of symplectic Tate homology and which are related by a canonical map $κ\colon \underrightarrow{H}\underleftarrow{T}(W) \to \underleftarrow{H}\under…
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.
Let be a strictly increasing function with . We unify the concepts of -harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduce -Yang-Mills fields, -degree, -lower degree, and generalized Yang-Mills-Born-Infeld…
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…