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.
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 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. 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.
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.
Study on nonlocal Liouville equation blow-up and quantization.
problem Analyzing a nonlocal Liouville-type equation and its blow-up behavior.
method Interpreted as a prescribed curvature equation for a curve in conformal parametrization, established relation to analogous equation in R.
result Established blow-up and quantization analysis for the nonlocal Liouville-type equation.
Study proves Liouville theorem for specific curvature equations with boundary conditions.
problem Proving Liouville theorem for σk-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 σk-curvature equations in R+n and boundary conditions. The paper proves Liouville theorems and nonexistence results for semilinear equations on pseudo-Hermitian manifolds.
problem Analyzing semilinear elliptic equations and inequalities on pseudo-Hermitian manifolds.
method Using a generalized Jerison-Lee's formula and volume estimates.
result Established Liouville theorems and nonexistence results for specific equations and inequalities.
Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.
problem Proving a Liouville theorem for a generalized elliptic equation on H-type groups.
method Proof based on an a priori integral estimate and a generalized differential identity.
result Obtained a Liouville type theorem for the semilinear subcritical elliptic equation on H-type groups.
Liouville theorems extended to graphs with bounded geometry.
problem Ancient solutions of subexponential growth on graphs.
method Extended Mosconi's results to graphs with bounded geometry.
result Nonnegative ancient solutions are stationary and harmonic.
Study on nonlinear Dirac equations on manifolds with formulas and theorems.
problem Qualitative behavior of nonlinear Dirac equations on Riemannian manifolds.
method Derivation of monotonicity formulas and Liouville theorems.
result Extension to Dirac-harmonic maps with curvature term.
Unified treatment of spacelike and timelike minimal surfaces via Liouville equation.
problem Investigating minimal surfaces in Lorentz-Minkowski space.
method Complex and paracomplex analysis, Möbius-type transformations, pseudo-isometries.
result Unified approach to both spacelike and timelike minimal surfaces.
We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.
Gradient estimates and Liouville theorems proven for a specific nonlinear elliptic equation.
problem Gradient estimates and Liouville theorems for positive solutions to a nonlinear elliptic equation.
method Analyzes the gradient estimates and Liouville type theorems for positive solutions to the equation Δu + au log u = 0.
result For a specific condition on the constant a and the lower bound of Ricci curvature, any bounded positive solution must be constant.
Proves uniqueness of solutions for a nonlocal Liouville equation with finite Q-curvature.
problem Proving uniqueness of solutions for a specific nonlocal Liouville equation.
method Connection to Calogero--Moser derivative NLS and ground state solitons.
result Uniqueness of solutions in the Gaussian case and general positive, symmetric-decreasing K. Study Liouville equation on Riemannian surfaces, linking volume growth to classification results.
problem Classifying solutions and manifolds of the Liouville equation on Riemannian surfaces.
method Analyzing the Liouville equation −Δu=eu on Riemannian surfaces with non-negative Ricci curvature, considering asymptotic volume growth. result Established classification results for solutions and manifolds, revealing a connection between volume growth and classification.
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.
Study on solutions to complex equations, proving strong comparison and Liouville theorems.
problem Analyzing continuous viscosity solutions to fully nonlinear elliptic equations.
method Proving strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations.
result Liouville theorem for entire solutions, showing they are either constants or standard bubbles.
The paper provides new gradient estimates and Liouville theorems for specific Poisson equations.
problem Addressing gradient estimates and Liouville theorems for specific Poisson equations on smooth metric measure spaces.
method Analyzing the parabolic equation \(u_t = \Delta_f u + F(u)\) on smooth metric measure spaces with Bakry-Émery curvature bounded from below.
result New gradient estimates and Liouville theorems for positive or bounded solutions to the equation when \(F\) is specific functions.
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…
We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.
Gradient bounds and Liouville theorems for quasi-linear equations on manifolds with nonnegative Ricci curvature.
problem Establishing bounds and theorems for solutions to quasi-linear elliptic equations on compact manifolds with nonnegative Ricci curvature.
method Gradient bounds, Liouville-type theorems, local splitting theorem, Harnack-type inequality, ABP estimate.
result Gradient bounds and Liouville-type theorems for solutions to quasi-linear equations on compact manifolds with nonnegative Ricci curvature.
The paper proves a formula for complex Monge-Ampère equations on manifolds.
problem Proving a mean value formula for complex Monge-Ampère equations.
method Using subharmonic Hermitian matrix valued functions and Liouville type theorems.
result Obtained a Liouville theorem for complex Monge-Ampère equations.
Paper proves nonpositive boundary integral for Liouville's equation, zero only for discs.
problem Properties of Liouville's equation and boundary integrals.
method Polyhomogeneous expansions and rigidity/gap theorems.
result Boundary integral is nonpositive and zero only for discs.
New Liouville-type results for CR Yamabe equation in Heisenberg group.
problem Characterizing solutions to CR Yamabe equation in Heisenberg group.
method Integral estimates combined with divergence formula.
result Liouville-type results for bounded solutions in n=2 and solutions with pointwise decay assumption in n≥3. We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.
We prove that the space of smooth initial data and the set of smooth solutions of the Liouville equation are homeomorphic.
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.
Paper finds solutions to a complex equation on surfaces with boundary conditions.
problem Existence of solutions to a super-Liouville equation on compact Riemannian surfaces with boundary.
method Introduced a weighted Dirac operator and constructed a Nehari manifold to show existence of non-trivial solutions.
result Existence of non-trivial solutions to the super-Liouville equation.
Study proves no nontrivial minimal surfaces in half-space with specific boundary conditions.
problem Proving the nonexistence of minimal surfaces in half-space with certain boundary conditions.
method Analyzes minimal surface equations in half-space with specific boundary conditions.
result Establishes Liouville type theorems for minimal surfaces in half-space.
The paper studies gradient estimates and Liouville theorems for a nonlinear elliptic equation on metric measure spaces.
problem Gradient estimates and Liouville theorems for positive solutions to a specific nonlinear elliptic equation.
method Analyzes the nonlinear elliptic equation \( \Delta_{V}u^{m} + \mu(x)u + p(x)u^{\alpha} = 0 \) on smooth metric measure spaces with bounded Bakry-Émery curvature.
result Establishes gradient estimates and related Liouville theorems and Harnack inequalities.
The paper proves a Liouville theorem for special Lagrangian equations with convexity conditions.
problem Proving Liouville theorems for special Lagrangian equations with specific conditions.
method Using Neumann-Poincaré inequality, mean value inequality for superharmonic functions, and geometric measure theory.
result Derives global and interior Hessian estimates for solutions of special Lagrangian equations.
Sharp inequalities and existence results for Liouville equations on spheres.
problem Sharp inequalities and existence results for Liouville equations on spheres.
method Sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on the sphere.
result Existence of critical points and sufficient conditions under symmetry or nondegeneracy assumptions.
This paper proves Liouville theorems for conformally invariant fully nonlinear equations.
problem Positive entire solutions of certain fully nonlinear equations are unique.
method Derives necessary and sufficient conditions for Liouville-type theorems.
result Enhanced understanding of solutions near isolated singularities.
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
Prove rigidity and classification results for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.
problem Quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.
method Prove rigidity and classification results for the quasilinear Liouville equation associated with the n-Laplacian on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. result Under a sharp logarithmic lower bound, the ambient manifold must be isometric to the Euclidean space and the solution must be one of the standard bubbles.
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…
Paper proves Liouville theorem for curvature equation with boundary conditions.
problem Proving Liouville theorem for curvature equation with boundary conditions.
method Using Chern--Gauss--Bonnet formula and boundary conditions.
result Extends Wei's earlier result by removing the infinity condition.
In this note, we derive a Liouville theorem for the complex Monge-Ampère equation. Our result states that if the global solution u of the complex Monge-Ampère equation with constant right-hand side differs from a quadratic polynomial solution by $o(\abs{x}^2)$ at infinity, then u is a quadratic polynomial.
In this paper we study the Kato' inequality on locally finite graph. We also study the application of Kato inequality to Ginzburg-Landau equations on such graphs. Interesting properties of Schrodinger equation and a Liouville type theorem are also derived.
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.
Study on solutions of conformal equations, proving bounds and profiles.
problem Analyzing solutions of conformally invariant equations on Euclidean domains.
method Established blow-up profiles and heights of solutions around blow-up points.
result Proved bounds on distances and heights of solutions around blow-up points.
Suggestions concerning the generalization of the geometric quantization to the case of nonlinear field theories are given. Results for the Liouville field theory are presented.
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we …
Gradient estimates for solutions on Riemannian manifolds.
problem Gradient estimates for solutions to the Allen-Cahn equation on Riemannian manifolds.
method Derive gradient estimates for bounded positive solutions to the Allen-Cahn equation on complete noncompact Riemannian manifolds.
result Derive a Liouville type theorem on manifolds with nonnegative Ricci curvature.
The paper derives estimates and proves theorems for a specific type of nonlinear parabolic equation.
problem Analyzing solutions to a weighted nonlinear parabolic equation on metric measure spaces.
method Derives elliptic gradient estimates and proves Liouville-type theorems.
result Establishes conditions for the existence of positive ancient solutions.
Study finite curvature solutions on surfaces with nonnegative Gauss curvature.
problem Finite total curvature solutions of Liouville equation on surfaces with nonnegative Gauss curvature.
method Analyzes asymptotic behavior of solutions on complete surfaces.
result Two extremal cases identified: Euclidean plane or flat cylinder, with specific decay conditions.
Buser's inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold M in terms of the Cheeger constant h(M). Agol later gave a quantitative improvement of Buser's inequality. Agol's result is less transparent since it is given implicitly by a set of equations, one of which is …