标题： K$\ddot{\operatorname{a}}$几何不变度量在二维伪凸域上的性质 显示英文标题

标题： K$\ddot{\operatorname{a}}$hlerity of invariant metrics on pseudoconvex domain of dimension two

摘要： 对于二维有界拟凸域，其类型有限，我们通过K$\ddot{\operatorname{a}}$凯勒- Kobayashi 度量或K$\ddot{\operatorname{a}}$凯勒- Carath$\acute{\operatorname{e}}$odory 度量，以及三阶准有限几何，证明了统一化定理。特别地，有限类型的拟凸 Reinhardt 域是单位球当且仅当 Bergman 度量是 Kobayashi 度量或 Carath$\acute{\operatorname{e}}$odory 度量的标量倍数。此外，我们建立了一个关于 Bergman 度量的全纯截面曲率和 Lu 常数的刚性定理。 显示英文摘要

摘要： For a two dimensional bounded pseudoconvex domain of finite type, we prove uniformization theorems via K$\ddot{\operatorname{a}}$hler-Kobayashi metric or K$\ddot{\operatorname{a}}$hler-Carath$\acute{\operatorname{e}}$odory metric with quasi-finite geometry of order three. In particular, a pseudoconvex Reinhardt domain of finite type is the unit ball if and only if the Bergman metric is a scalar multiple of the Kobayashi metric or Carath$\acute{\operatorname{e}}$odory metric. Moreover, we establish a rigidity theorem concerning holomorphic sectional curvature of Bergman metric and Lu constant.