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.
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). 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.
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.
We study the Liouville type theorems for transversally harmonic and biharmonic maps on foliated Riemannian manifolds
We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.
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.
We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
The paper proves Liouville-type theorems on Hadamard manifolds.
problem Non-existence of Killing-Yano tensors, Killing tensors, and harmonic symmetric tensors on Hadamard manifolds.
method Proofs use Liouville-type theorems on non-existence of subharmonic and harmonic functions on complete Riemannian manifolds, modified for Hadamard manifolds.
result Proves several Liouville-type theorems on Hadamard manifolds.
We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.
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.
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. 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 study proves Liouville theorems on curved manifolds with convex boundaries.
problem Proving Liouville theorems on manifolds with nonnegative curvature and strictly convex boundary.
method Analyzing smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary.
result Derives Liouville theorems and verifies a conjecture about eigenvalues and inequalities.
L. Capogna and M. Cowling showed that if φ is 1-quasiconformal on an open subset of a Carnot group G, then composition with φ preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that φ is in fact $C^\inft…
In the present paper we prove Liouville-type theorems: non-existence theorems for complete twisted and warped products of Riemannian manifolds which generalize and complement similar results for compact manifolds.
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. On the slit tangent manifold of a Finsler manifold M are given the vertical and the Liouville foliations. In this paper we define some new types of vertical forms with respect to the Liouville foliation on TM^0. We define a cohomology group of TM^0 using these new forms. We prove a de Rham type theorem.
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.
Study proves unique compactification of hyperbolic space.
problem Proving uniqueness of compactification of hyperbolic space.
method Analyzing one-parameter family of elliptic PDEs on hyperbolic space.
result Euclidean half-plane is the only compactification of hyperbolic space.
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.
Local gradient estimates for eigenfunctions on conformal solitons improve Liouville theorems.
problem Estimating eigenfunctions on conformal solitons.
method Proving local gradient estimates for positive eigenfunctions of L-operator. result Improved Liouville theorems for Lu=0 on conformal solitons. 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. 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.
Paper proves Liouville-type theorems for minimal graphs with capillary boundary.
problem Proves conditions for minimal graphs to be flat over half-spaces with capillary boundaries.
method Uses gradient estimates for mean curvature equation over R+n with capillary boundary condition, adapting maximum principle. result Minimal graphs are flat under specific conditions on growth or boundedness.
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.
In the present paper we prove Liouville-type theorems: non-existence theorems for some complete Riemannian almost product manifolds and special mappings of complete Riemannian manifolds which generalize similar results for compact manifolds.
Paper proves constant functions for pluriharmonic on certain solitons.
problem Proving Liouville type theorems for harmonic functions on gradient Ricci solitons.
method Analyzing pluriharmonic functions on gradient shrinking or steady Kähler-Ricci solitons.
result Any pluriharmonic function with gradient in Lp is constant. The study confirms Liouville-type theorems for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary.
problem Proving Liouville-type theorems for positive harmonic functions on specific types of manifolds.
method Employing the P-function method and a closed conformal vector field inherent to such manifolds.
result Confirms some cases of Wang's conjecture and provides a partial verification of Wang's conjecture on warped product manifolds.
Liouville theorem for minimal graphs on manifolds with specific properties.
problem Characterizing positive minimal graphic functions on specific Riemannian manifolds.
method Using volume doubling property and uniform Neumann-Poincaré inequality.
result Positive minimal graphic functions on the manifold are constants.
The paper proves a hierarchy of Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.
problem Establishing Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.
method A new L2 estimate for the Laplacian of a polyharmonic function, obtained by induction through a cutoff construction combined with a hole-filling argument. result All polyharmonic functions of sublinear growth on manifolds of nonnegative Ricci curvature are constant.
Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cyli…
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.
In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: ut=ΔF(u), with F′(u)>0, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equ…
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.
Paper proves rigidity for Einstein metrics in high dimensions.
problem Einstein metrics on high-dimensional manifolds.
method Liouville type rigidity result for asymptotically hyperbolic metrics.
result Established a rigidity theorem for d≥5. 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…
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.
The Liouville theorem is proven for harmonic maps from a specific type of manifold.
problem Proving Liouville theorem for harmonic maps from a special class of manifolds.
method Gradient estimate and Liouville theorem for harmonic maps from Kasue manifolds.
result Liouville theorem is proven for harmonic maps from Kasue manifolds.
Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of two-dimensional minimal surface graphs in R^n. This leads us to a theorem of Bernstein-Liouvil…
The paper extends Liouville theorems to sub-Riemannian manifolds.
problem Generalizing Liouville theorems to sub-Riemannian manifolds.
method Constructing 'good' cut-off functions and applying a nonnegative generalized curvature-dimension inequality.
result The Liouville theorems are extended to sub-Riemannian manifolds.
Sharp Liouville theorem for minimal graphs on manifolds with nonnegative Ricci curvature.
problem Characterizing smooth solutions to minimal hypersurface equations on manifolds with nonnegative Ricci curvature.
method Gradient estimate for minimal graphs over Σ with small linear growth of the negative parts of graphic functions via iteration. result Every smooth solution u to minimal hypersurface equation on Σ is a constant provided u has sublinear growth for its negative part. 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.
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 (F,F′)p-harmonic maps. result Established a Liouville type theorem for (F,F′)p-harmonic maps. Solves a conjecture using a new formula on conformally Einstein manifolds.
problem Solving a conjecture in conformal geometry.
method Using an Obata type formula established by previous works.
result Solves Hang-Yang conjecture via an Obata-type argument.
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.
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.