In this paper, we discuss the invariant complete metric on the Cartan-Hartogs domain of the fourth type. Firstly, we find a new invariant complete metric, and prove the equivalence between Bergman metric and the new m...In this paper, we discuss the invariant complete metric on the Cartan-Hartogs domain of the fourth type. Firstly, we find a new invariant complete metric, and prove the equivalence between Bergman metric and the new metric; Secondly, the Ricci curvature of the new metric has the super bound and lower bound; Thirdly,we prove that the holomorphic sectional curvature of the new metric has the negative supper bound; Finally, we obtain the equivalence between Bergman metric and Einstein-Kahler metric on the Cartan-Hartogs domain of the fourth type.展开更多
The main point is the calculation of the Bergman kernel for the so-called Cartan-Hartogs domains. The Bergman kernels on four types of Cartan-Hartogs domains are given in explicit formulas. First by introducing the id...The main point is the calculation of the Bergman kernel for the so-called Cartan-Hartogs domains. The Bergman kernels on four types of Cartan-Hartogs domains are given in explicit formulas. First by introducing the idea of semi-Reinhardt domain is given, of which展开更多
Let YI be the Cartan-Hartogs domain of the first type. We give the generating function of the Einstein-Kahler metrics on YI, the holomorphic sectional curvature of the invariant Einstein-Kahler metrics on YI. The comp...Let YI be the Cartan-Hartogs domain of the first type. We give the generating function of the Einstein-Kahler metrics on YI, the holomorphic sectional curvature of the invariant Einstein-Kahler metrics on YI. The comparison theorem of complete Einstein-Kahler metric and Kobayashi metric on YI is provided for some cases. For the non-homogeneous domain YI, when K = mn+1/m+n, m ＞ 1, the explicit forms of the complete Einstein-Kahler metrics are obtained.展开更多
The authors give the condition that the Bergman kernel function on the first type of Cartan-Hartogs domain exists zeros.If the Bergman kernel function of this type of domain has zeros,the zero set is composed of sever...The authors give the condition that the Bergman kernel function on the first type of Cartan-Hartogs domain exists zeros.If the Bergman kernel function of this type of domain has zeros,the zero set is composed of several path-connected branches,and there exists a continuous curve to connect any two points in the non-zero set.展开更多
Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equatio...Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain of the second type is discussed by using the analytic method in this paper. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear second-order ordinary differential equation (ODE) by using quite different method. Secondly, the solution of the Dirichlet problem is given in semi-explicit formula, and under a special case the exact solution is obtained. These results may be helpful for the numerical method of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain.展开更多
In this paper,we compute the complete Einstein-Kahler metric with explicit formula for the Cartan-Hartogs domain of the fourth type in some cases.Under this metric the holomorphic sectional curvature is given,which in...In this paper,we compute the complete Einstein-Kahler metric with explicit formula for the Cartan-Hartogs domain of the fourth type in some cases.Under this metric the holomorphic sectional curvature is given,which intervenes between-2k and-1.This is the sharp estimate.展开更多
文摘In this paper, we discuss the invariant complete metric on the Cartan-Hartogs domain of the fourth type. Firstly, we find a new invariant complete metric, and prove the equivalence between Bergman metric and the new metric; Secondly, the Ricci curvature of the new metric has the super bound and lower bound; Thirdly,we prove that the holomorphic sectional curvature of the new metric has the negative supper bound; Finally, we obtain the equivalence between Bergman metric and Einstein-Kahler metric on the Cartan-Hartogs domain of the fourth type.
文摘The main point is the calculation of the Bergman kernel for the so-called Cartan-Hartogs domains. The Bergman kernels on four types of Cartan-Hartogs domains are given in explicit formulas. First by introducing the idea of semi-Reinhardt domain is given, of which
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.10071051 and 10171068)Natural Science Foundation of Beijing(Grant Nos.1002004 and 1012004).
文摘Let YI be the Cartan-Hartogs domain of the first type. We give the generating function of the Einstein-Kahler metrics on YI, the holomorphic sectional curvature of the invariant Einstein-Kahler metrics on YI. The comparison theorem of complete Einstein-Kahler metric and Kobayashi metric on YI is provided for some cases. For the non-homogeneous domain YI, when K = mn+1/m+n, m ＞ 1, the explicit forms of the complete Einstein-Kahler metrics are obtained.
基金supported by the National Natural Science Foundation of China(No.11871044)the Natural Science Foundation of Hebei Province(No.A2019106037)
文摘The authors give the condition that the Bergman kernel function on the first type of Cartan-Hartogs domain exists zeros.If the Bergman kernel function of this type of domain has zeros,the zero set is composed of several path-connected branches,and there exists a continuous curve to connect any two points in the non-zero set.
基金supported by the Research Foundation of Beijing Government(Grant No.YB20081002802)National Natural Science Foundation of China(Grant No.10771144)
文摘Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain of the second type is discussed by using the analytic method in this paper. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear second-order ordinary differential equation (ODE) by using quite different method. Secondly, the solution of the Dirichlet problem is given in semi-explicit formula, and under a special case the exact solution is obtained. These results may be helpful for the numerical method of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain.
文摘In this paper,we compute the complete Einstein-Kahler metric with explicit formula for the Cartan-Hartogs domain of the fourth type in some cases.Under this metric the holomorphic sectional curvature is given,which intervenes between-2k and-1.This is the sharp estimate.
