arXiv research

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

New Liouville-type results for CR Yamabe equation in Heisenberg group.

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

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

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.

Ancient Lagrangian flows get limited convex solutions.

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.

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.

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

Paper proves rigidity for Einstein metrics in high dimensions.

The paper proves constant rank theorems for special Lagrangian equations.

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.

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 proves Liouville-type theorems on Hadamard manifolds.

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 …

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

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

Study proves unique compactification of hyperbolic space.

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

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress-energy tensor which is naturally linked to conservation law and yield monotonicity formula via the coarea formula and comparison theorem in Riemannian geometr…

Paper derives estimates for Hessian equations under concavity assumptions.

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 …

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…

Optimal Liouville theorem for half-Euclidean space equations.

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…

Proves stability of convex disks close to round caps.

Let $u: (M, g)\to (N, h)$ be a map between Riemannian manifolds $(M, g)$ and $(N, h)$. The $p$-bienergy of $u$ is defined by $E_p(u)=\int_M|τ(u)|^pdν_g$, where $τ(u)$ is the tension field of $u$ and $p>1$. Critical points of $E_p(\cdot)$ are called $p$-biharmonic maps. In this paper we will prove nonexistence result of…

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

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…

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

Symmetry proven for positive solutions of a weighted p-Laplace operator inequality.

We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ Δ^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index solutions (whether positive or sign changing), in the full exponent range. We also compute an…

We establish Liouville type theorems for degenerate conformally invariant equations.

Paper solves curvature prescription problem on surfaces with boundary.

Researchers prove constant solutions for a specific Finslerian equation.

Paper proves constant functions for pluriharmonic on certain solitons.

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^…

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…

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…

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…

In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in $S^1$,} \end{equation} where $(-Δ)^\frac{1}{2}$ stands for the fractional Laplacian and $κ$ is a bounded function. We interpret the above equation as the prescri…

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

The paper proves a conjecture about positivity preserving in Riemannian manifolds.

Geometrically reduces Hamiltonian systems using particular integrals.