论文标题
与liouville方程相关的一些几何不平等现象
Some geometric inequalities related to Liouville equation
论文作者
论文摘要
在本文中，我们证明，如果$ u $是解决liouville方程的解决方案\ begin {align} \ label {scalliouville}ΔU+e+e^{2u} = 0 \ quad \ quad \ mbox {in $ \ mathbb {r Mathbb {r}共形度量$ g = e^{2U}δ$在下面以$π$为界。这里$δ$是$ \ mathbb {r}^2 $中的欧几里得公制。此外，我们明确构建了一个解决方案家族，以便$ \ mathbb {r}^2 $的相应直径超过$ [π，2π）$。 我们还讨论了超级措施。我们表明，如果$ u $是超级解决方案，而$ \ int _ {\ mathbb {r}^2} e^{2U} dx <\ infty $，则在$ \ e^{2U}δ$ best and best and best and best and best bed或celuse cubs y y belim corplore或celuce $ cum cum cum cum cum cum cum cum cum cum cum cum cum cum cum cum cum cum cum cum cum cum cum y hym y mathbb {r}^2 $的直径。 对于径向超滤波，我们使用分析和几何方法来证明涉及$ \ Mathbb {r}^2 $的共形长度和磁盘区域的某些不平等现象。我们还讨论了上述结果与球体涵盖不平等的连接，而高斯曲率下面的限制为$ 1 $。 还讨论了更高的概括。
In this paper, we prove that if $u$ is a solution to the Liouville equation \begin{align} \label{scalliouville} Δu+e^{2u} =0 \quad \mbox{in $\mathbb{R}^2$,} \end{align}then the diameter of $\mathbb{R}^2$ under the conformal metric $g=e^{2u}δ$ is bounded below by $π$. Here $δ$ is the Euclidean metric in $\mathbb{R}^2$. Moreover, we explicitly construct a family of solutions such that the corresponding diameters of $\mathbb{R}^2$ range over $[π,2π)$. We also discuss supersolutions. We show that if $u$ is a supersolution and $\int_{\mathbb{R}^2} e^{2u} dx<\infty$, then the diameter of $\mathbb{R}^2$ under the metric $e^{2u}δ$ is less than or equal to $2π$. For radial supersolutions, we use both analytical and geometric approaches to prove some inequalities involving conformal lengths and areas of disks in $\mathbb{R}^2$. We also discuss the connection of the above results with the sphere covering inequality in the case of Gaussian curvature bounded below by $1$. Higher dimensional generalizations are also discussed.
