$\mathbb{C}^n$ -- II中一类蛋的Wu度量分析
Analyzing the Wu metric on a class of eggs in $\mathbb{C}^n$ -- II
摘要：
研究了\[ E_{2m} = \big\{ z \in \mathbb{C}^n : \vert z_1 \vert^{2m} + \vert z_2 \vert^2 + \ldots + \vert z_{n-1} \vert^2 + \vert z_n \vert^{2} <1 \big \}, \]型非凸区域的吴度量，其中$ 0 < m < 1/2$ .得到了这种伪蛋$E_{2m}$上Kobayashi度量和Wu度量的显式表达式.证明了Wu度量是$ E_{2m} $上的连续厄米度量，并且除了沿复超曲面$ Z = \{ (0, z_2, \ldots, z_n ) \in E_{2m} \} $外，它处处是实解析的.我们还表明，这个非紧家庭的伪凸域的吴度量的全纯截面曲率有界以上的电流由一个独立的负常数$m$。这验证了S. Kobayashi和H.吴为这样的$E_{2m}$。
英文摘要：
We study the Wu metric for the non-convex domains of the form \[ E_{2m} = \big\{ z \in \mathbb{C}^n : \vert z_1 \vert^{2m} + \vert z_2 \vert^2 + \ldots + \vert z_{n-1} \vert^2 + \vert z_n \vert^{2} <1 \big \}, \] where $ 0 < m < 1/2$. Explicit expressions for the Kobayashi metric and the Wu metric on such pseudo-eggs $E_{2m}$ are obtained. The Wu metric is then verified to be a continuous Hermitian metric on $ E_{2m} $ which is real analytic everywhere except along the complex hypersurface $ Z = \{ (0, z_2, \ldots, z_n ) \in E_{2m} \} $. We also show that the holomorphic sectional curvature of the Wu metric for this non-compact family of pseudoconvex domains is bounded above in the sense of currents by a negative constant independent of $m$. This verifies a conjecture of S. Kobayashi and H. Wu for such $E_{2m}$.
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