Canonical Metrics On Nonsmooth Quasiconvex Domain

In the field of several complex variables and complex geometry,the Kobayashi metric,Caratheodory metric,Bergman metric,and K(?)hler-Einstein metric on complex manifolds are important objects of study.Among them,the Bergman metric and K(?)hler-Einstein metric have been extensively studied and widely applied in complex geometry.For example,on a bounded symmetric domain,the Bergman metric is the K(?)hler-Einstein metric,thus establishing a deep connection between the Bergman metric and K(?)hler-Einstein metric.However,the relationship between the canonical metric,the Bergman metric,and the K(?)hler-Einstein metric is not yet clear,and requires further exploration.This paper aims to study the canonical metric on certain non-homogeneous domains,establish the equivalence between the Bergman metric,K(?)hler-Einstein metric,and canonical metric,and explore their applications in differential geometry and mathematical physics.Specifically,we will delve deeper into the following aspects:Firstly,we will explore the canonical metric on a special class of non-smooth bounded pseudoconvex domains,namely the generalized Hartogs triangles,which are a generalization of the classical Hartogs triangles.We will investigate their geometric properties and introduce new K(?)hler potential functions to consider a new K(?)hler metric.We calculate the explicit expression of Rawnsley’s ε-function and use it to give sufficient and necessary conditions for the generalized Hartogs triangles to have K(?)hler-Einstein and balanced metrics.In particular,we prove that the Bergman metric on the generalized Hartogs triangle is both K(?)hler-Einstein and balanced.Secondly,we will discuss the applications of K(?)hler-Einstein and balanced metrics in geometric quantization and isometric embeddings.On one hand,by using the Calabi distance function associated with the K(?)hler metric mentioned above,we establish the Cauchy-Schwarz inequality and prove that the generalized Hartogs triangle satisfies the Berezin quantization.On the other hand,we analyze the relationship between the holomorphic isometric embeddings and the balanced metrics.By this analysis,we prove that the generalized Hartogs triangle is a K(?)hler-Einstein submanifold on an infinite-dimensional complex projective space.