arXiv research

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.

Proves a Liouville-type theorem for p-Laplacian on manifolds.

The paper proves Liouville-type theorems on Hadamard manifolds.

The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.

We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.

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.

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…

The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.

For $n\geq2,$ we obtain Liouville type theorems for minimal surface equations in half space $\mathbf R^n_+$ with affine Dirichlet boundary value or constant Neumann boundary value.

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.

Optimal Liouville theorem for half-Euclidean space equations.

We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.

Ancient Lagrangian flows get limited convex solutions.

The paper proves constant rank theorems for special Lagrangian equations.

The study confirms Liouville-type theorems for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary.

Paper proves Liouville-type theorems for minimal graphs with capillary boundary.

Paper proves rigidity for Einstein metrics in high dimensions.

Study proves unique compactification of hyperbolic space.

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…

In the present paper we prove Liouville-type theorems: non-existence theorems for conformal mappings of complete Riemannian manifolds. In addition, we give an application of these results to the theory of conharmonic transformations. A part of these results was announced in our reports on the conferences "Differential …

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the t…

Paper proves constant functions for pluriharmonic on certain solitons.

The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.

In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the C…

We establish Liouville type theorems for degenerate conformally invariant equations.

Local gradient estimates for eigenfunctions on conformal solitons improve Liouville theorems.

Researchers prove constant solutions for a specific Finslerian equation.

Let $(M^n,g)$ be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $Δu+au\log u=0,$ where $a$ is a nonzero constant. In particular, for $a<0$, we prove that any bounded positive solution of the…

We generalize the Omori-Yau almost maximum principle of the Laplace-Beltrami operator on a complete Riemannian manifold $M$ to a second-order linear semi-elliptic operator $L$ with bounded coefficients and no zeroth order term. Using this result, we prove some Liouville-type theorems for a real-valued $C^{2}$ function …

On a complete Riemannian manifold M with Ricci curvature satisfying $\textrm{Ric}(\nabla r,\nabla r) \geq -Ar^2(\log r)^2(\log(\log r))^2...(\log^{k}r)^2$ for $r\gg 1$, where A>0 is a constant, and r is the distance from an arbitrarily fixed point in M. we prove some Liouville-type theorems for a C^2 function $f:M\ri…

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \(n\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the $L^…

The paper proves that under certain conditions, solutions to a specific differential inequality are nonnegative.

In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $-Δu=K(x)e^u, in R^2$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite to…

In this paper, we get a Liouville type theorem for the special Lagrangian equation with a certain 'convexity' condition, where Warren-Yuan first studied the condition in [30]. Based on Warren-Yuan's work, our strategy is to show a global Hessian estimate of solutions via the Neumann-Poincar$\mathrm{\acute{e}}$ inequali…

The paper classifies invariant operators and proves a Liouville theorem.

Gradient estimates derived for a specific equation on pseudo-Hermitian manifolds.

We study analytic properties of harmonic maps from Riemannian polyhedra into CAT($κ$) spaces for $κ\in\{0,1\}$. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT($κ$) spaces. We compute a target variation formula that captures the curvature bound in…

The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.

We study minimal graphic functions on complete Riemannian manifolds $\Si$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with …

The paper establishes gradient estimates and Liouville theorems for Φ-Laplacian equations on Riemannian manifolds.

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.

In this paper, we will address to the following parabolic equation $u_t=Δ_fu + F(u)$ on a smooth metric measure space with Bakry-Émery curvature bounded from below. Here $F$ is a differentiable function defined in $\mathbb{R}$. Our motivation is originally inspired by gradient estimates of Allen-Cahn and Fisher equ…

We introduce and study an approximate solution of the p-Laplace equation, and a linearlization $L_ε$ of a perturbed p-Laplace operator. By deriving an $L_ε$-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 …

We prove that there does not exist non-constant positive $f$-harmonic function on the complete gradient shrinking Ricci solitons. We also prove the $L^{p}(p\geq 1 \ {\rm or}\ 0<p\leq 1)$ Liouville theorems on the complete gradient shrinking Ricci solitons.

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $u_t=ΔF(u),$ 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.

In this paper, we prove a mean value formula for bounded subharmonic Hermitian matrix valued function on a complete Riemannian manifold with nonnegative Ricci curvature. As its application, we obtain a Liouville type theorem for the complex Monge-Ampère equation on product manifolds.