$\mathbb{C}^n$ -- I中一类蛋的Wu度量分析
Analyzing the Wu metric on a class of eggs in $\mathbb{C}^n$ -- I
摘要：
研究了\[ E_{2m} = \big\{ z \in \mathbb{C}^n : \vert z_1 \vert^{2m} + \vert z_2 \vert^2 + \ldots + \vert z_{n-1} \vert^2 + \vert z_n \vert^{2} <1 \big\} \]型凸蛋域上的吴度量，其中$m \geq 1/2, m \neq 1$ .吴度量被证明是真正的解析无处不在，除了一个低维的子品种，它不能是$C^2$ -光滑。然而，总体而言，吴度规被证明是连续的时，$m=1/2$，甚至$C^1$ -光滑的每个$m>1/2$，并在所有情况下，一个非凯勒厄米特度规的全纯曲率强负的意义上的电流。这给了S的一个猜想一个自然的答案。Kobayashi和H. Wu对于这样的$E_{2m}$。
英文摘要：
We study the Wu metric on convex egg 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 $m \geq 1/2, m \neq 1$. The Wu metric is shown to be real analytic everywhere except on a lower dimensional subvariety where it fails to be $C^2$-smooth. Overall however, the Wu metric is shown to be continuous when $m=1/2$ and even $C^1$-smooth for each $m>1/2$, and in all cases, a non-Kähler Hermitian metric with its holomorphic curvature strongly negative in the sense of currents. This gives a natural answer to a conjecture of S. Kobayashi and H. Wu for such $E_{2m}$.
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。
