有界对称域上某些Hartogs域的Kaehler-Einstein度量
The Kaehler-Einstein metric for some Hartogs domains over bounded symmetric domains
摘要：
本文研究了Hartogs域$\widetilde Ω$的完备Kähler-Einstein度量，它是由不可约有界对称域$Ω$的膨胀得到的，并利用$Ω$的泛范数的幂$N^μ$. Kähler-Einstein度规的生成函数满足一个带边界条件的复蒙日-安培方程。域$\widetilde Ω$一般不是齐次的，但它有一个自同构子群，其轨道由$X\in\lbrack0,1[$参数化。这允许将蒙日-安培方程简化为具有极限条件的常微分方程。该方程可以显式求解为$μ$的特殊值$μ_{0}$，称为临界指数。我们工作的两个特殊的对称域的细节。临界指数似乎也是相关的其他不变度量的属性，如伯格曼度量;一个猜想，这是证明例外域。
英文摘要：
We study the complete Kähler-Einstein metric of a Hartogs domain $\widetilde Ω$, which is obtained by inflation of an irreducible bounded symmetric domain $Ω$, using a power $N^μ$ of the generic norm of $Ω$. The generating function of the Kähler-Einstein metric satisfies a complex Monge-Ampère equation with boundary condition. The domain $\widetilde Ω$ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by $X\in\lbrack0,1[$. This allows to reduce the Monge-Ampère equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value $μ_{0}$ of $μ$, called the critical exponent. We work out the details for the two exceptional symmetric domains. The critical exponent seems also to be relevant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
