标题： 关于求积区域的拓扑和奇点 显示英文标题

标题： On topology and singularities of quadrature domains

摘要： 我们证明了在求积域边界上的奇点数量有一个线性上界，改进了Gustafsson \cite{Gus88}所知的二次上界。 这个边界二重点数量的线性上界也加强了Lee和Makarov \cite{LM16}给出的求积域连通性（即补成分的数量）的界限。 我们的证明使用了共形动力学和双曲几何的论证。 最后，我们引入了一种新的动态方法来构造多连通的求积域。 显示英文摘要

摘要： We prove a linear upper bound for the number of singular points on the boundary of a quadrature domain, improving a previously known quadratic bound due to Gustafsson \cite{Gus88}. This linear upper bound on the number of boundary double points also strengthens the bound on the connectivity (i.e., the number of complementary components) of a quadrature domain given by Lee and Makarov \cite{LM16}. Our proofs use conformal dynamics and hyperbolic geometry arguments. Finally, we introduce a new dynamical method to construct multiply connected quadrature domains.