相关论文: On conformal metrics of constant positive curvatur…

We show that the prescribed Gaussian curvature equation in $\mathbb{R}^2$ $$-\Delta u= (1-|x|^p) e^{2u},$$ has solutions with prescribed total curvature equal to $\Lambda:=\int_{\mathbb{R}^2}(1-|x|^p)e^{2u}dx\in \mathbb{R}$, if and only if…

In this paper, we prove that if $u$ is a solution to the Liouville equation \begin{align} \label{scalliouville} \Delta u+e^{2u} =0 \quad \mbox{in $\mathbb{R}^2$,} \end{align}then the diameter of $\mathbb{R}^2$ under the conformal metric…

We study the polyharmonic problem $\Delta^m u = \pm e^u$ in ${\mathbb R}^{2m}$, with $m \geq 2$. In particular, we prove that {\sl for any} $V > 0$, there exist radial solutions of $\Delta^m u = -e^u$ such that $$\int_{{\mathbb R}^{2m}} e^u…

We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on the unit disk. This is equivalent to solving the following P.D.E. \begin{equation*}\begin{cases}-\Delta u=2K(z)e^u&\hbox{in}\;\mathbb{D}^2,\\…

In this paper, we investigate the following curvature equation: \begin{equation} \Delta u+e^{u}=8\pi (\delta _{0}+\delta _{\frac{\omega _{k}}{2}})\text{ in } E_{\tau }\text{, }\tau \in \mathbb{H} (0.1) \label{a} \end{equation} Here $E_{\tau…

In this paper, we consider the following curvature equation $$\Delta u+{\rm e}^u=4\pi\biggl((\theta_0-1)\delta_0+(\theta_1-1)\delta_1 +\sum_{j=1}^{n+m}\bigl(\theta_j'-1\bigr)\delta_{t_j}\biggr)\qquad \text{in}\ \mathbb R^2,$$…

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincar\'e metrics (i.e., complete metrics of constant negative curvature)…

In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the…

On the upper hemisphere, we use the Obata-Escobar argument to classify conformal metrics with constant $\sigma_k$ curvature and constant boundary mean curvature in all types of cones including positive and negative cones. This extends a…

Answering a question by M. Struwe (Vietnam J. Math. 2020) related to the blow-up behaviour in the Nirenberg problem, we show that the prescribed $Q$-curvature equation $$\Delta^2 u=(1-|x|^p)e^{4u}\text{ in }\mathbb{R}^4,\quad…

We classify the solutions to the equation (- \Delta)^m u=(2m-1)!e^{2mu} on R^{2m} giving rise to a metric g=e^{2u}g_{R^{2m}} with finite total $Q$-curvature in terms of analytic and geometric properties. The analytic conditions involve the…

We mainly show that for a conformal metric $g=u^{\frac{4}{n-2m}}|dx|^2$ on $\mathbb{R}^n$ with $n\geq 2m+1$, if the higher order Q-curvature $Q^{(2m)}_g$ is positive and has slow decay barrier near infinity, the lower order Q-curvature…

We study finite total curvature solutions of the Liouville equation $\Delta u+e^{2u}=0$ on a complete surface $(M,g)$ with nonnegative Gauss curvature. It turns out that the asymptotic behavior of the solution separates two extremal cases:…

The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a…

We give a complete classification of solutions bounded from above of the Liouville equation $$-\Delta u=e^{2u}\quad\mbox{in}\quad {\mathbf{R}}^2.$$ More generally, solutions in the class $$N:=\{ u:\limsup_{z\to\infty}…

We show that if a planar graph $G$ with minimum degree at least $3$ has positive Lin-Lu-Yau Ricci curvature on every edge, then $\Delta(G)\leq 17$, which then implies that $G$ is finite. This is an analogue of a result of DeVos and Mohar…

We study conformal metrics on $R^3$, i.e., metrics of the form $g_u=e^{2u}|dx|^2$, which have constant $Q$-curvature and finite volume. This is equivalent to studying the non-local equation $$ (-\Delta)^\frac32 u = 2 e^{3u}$$ in $R^3$…

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic…

We study asymptotic behavior of positive smooth solutions of the conformal scalar curvature equation in ${\bf R}^n$. We consider the case when the scalar curvature of the conformal metric is bounded between two positive numbers outside a…

We consider the $Q$-curvature equation \begin{equation}\label{0.1} (-\Delta)^n u = K(x)e^{2nu}\quad\text{in} ~\mathbb{R}^{2n} \ (n \geq 2) \end{equation} where $K$ is a given non constant continuous function. Under mild growth control on…