Kobayashi 度量局部 Gromov 双曲性对边界类型的最优线性依赖

中文摘要 AI 辅助

度量空间 \\((X,d)\\) 的 Gromov 双曲常数是所有使得 \\((X,d)\\) 为 \\(\delta\\)-双曲的 \\(\delta\ge0\\) 的下确界。对于 Kobayashi 双曲域 \\(\Omega\subset\C^n\\) 及边界点 \\(p\in\partial\Omega\\)，令 \\(\delta_{\mathrm{loc}}(\Omega,p)\\) 表示通过将点限制在 \\(p\\) 的任意小欧几里得邻域内而得到的局部 Gromov 双曲常数，其中距离仍由环境 Kobayashi 距离 \\(K_\Omega\\) 度量。对每个偶数整数 \\(M\ge2\\)，令 \\(\mathfrak H(M)\\) 为所有复维数 \\(n\ge2\\) 且边界在指定点附近光滑、凸且在该点 D'Angelo 型至多为 \\(M\\) 的域上这些常数的上确界。我们证明 \\[ \frac{\log 2}{2} M \le \mathfrak H(M) < 36M. \\] 因此，局部 Gromov 双曲常数对边界类型的依赖的最优普适关系是线性的。我们还表明，即使对于光滑有界强凸域，全局 Gromov 双曲常数也不存在类似的界。

英文摘要

The Gromov hyperbolicity constant of a metric space \((X,d)\) is the infimum of all \(δ\ge0\) such that \((X,d)\) is \(δ\)-hyperbolic. For a Kobayashi hyperbolic domain \(Ω\subset\C^n\) and a boundary point \(p\in\partialΩ\), let \(δ_{\mathrm{loc}}(Ω,p)\) denote the local Gromov hyperbolicity constant obtained by restricting the points to arbitrarily small Euclidean neighborhoods of \(p\), while distances are still measured by the ambient Kobayashi distance \(K_Ω\). For each even integer \(M\ge2\), let \(\mathfrak H(M)\) be the supremum of these constants over all complex dimensions \(n\ge2\) and all domains whose boundary is smooth and convex near the distinguished point and has D'Angelo type at most \(M\) at that point. We prove \[ \frac{\log 2}{2} M \le \mathfrak H(M) < 36M. \] Thus the optimal universal dependence of the local Gromov hyperbolicity constant on boundary type is linear. We also show that no analogous bound holds for the global Gromov hyperbolicity constant, even among smooth bounded strongly convex domains.

发表机构

• School of Science, Beijing University of Posts and Telecommunications（北京邮电大学理学院）
• Key Laboratory of Mathematics and Information Networks (Beijing University of Posts and Telecommunications), Ministry of Education, China（教育部数学与信息网络重点实验室（北京邮电大学））

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