关于多复变量泊松核及其相关的复蒙日-安培方程
On the pluricomplex Poisson kernel and the associated complex Monge--Ampère equation
浏览论文内容
中文总结 AI 辅助
本文刻画了有界强线性凸域的多复变量泊松核，证实了Bracci等人的猜想，并构造一族连续解表明相关复蒙日-安培方程解不唯一，与格林函数情形形成对比。
中文摘要 AI 辅助
我们给出了有界强线性凸域在$\mathbb C^{n+1}$中的多复变量泊松核的几何刻画，该刻画基于由所有复测地圆盘构成的叶状结构，这些圆盘的闭包包含一个固定的边界点，从而证实了Bracci等人在2009年提出的猜想，甚至在更一般的情况下也成立。证明关键依赖于我们之前在一系列两篇论文中获得的结果。我们还重新审视了与多复变量泊松核相关的齐次复蒙日-安培方程，并表明其解在一般情况下远非唯一，通过构造$\mathbb C^{n+1}$中单位球上的一族成对非成比例连续解。这与多复变量格林函数对应方程解的唯一性形成鲜明对比。
英文摘要
We give a geometric characterization of the pluricomplex Poisson kernel of bounded strongly linearly convex domains in $\mathbb C^{n+1}$ in terms of their foliation by all complex geodesic discs whose closure contains a fixed boundary point, thus confirming a conjecture posed by Bracci et al. in 2009, even in greater generality. The proof relies crucially on our previous results obtained in a series of two papers. We also revisit the homogeneous complex Monge--Ampère equation associated with the pluricomplex Poisson kernel and show that its solutions are far from unique in general by constructing a family of pairwise nonproportional continuous solutions on the unit ball in $\mathbb C^{n+1}$. This sharply contrasts with the uniqueness of solutions to the corresponding equation for the pluricomplex Green function.
发表机构
- CAS Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院和吴文俊数学重点实验室）
机构由 AI 辅助整理，请以论文原文为准。
