AI 中文总结
研究强凸域中 Kobayashi 测地线的指数收敛性，通过证明特定条件下两条测地线的指数收敛不等式，利用此性质借助广义挤压函数刻画强拟凸域，给出了强拟凸域的一种新判定方法。
AI 中文摘要
在本文中，我们证明了某些凸域的逼近测地线性质的一个定性版本。具体而言，我们证明了如果\(\Omega \subset \mathbb{C}^{d}\)是一个具有\(\mathcal{C}^3\)边界的有界强凸域，并且\(\gamma_{1}, \gamma_{2}:[0, \infty) \to \Omega\)是两条测地线，使得\(\gamma_{1}(\infty)=\gamma_{2}(\infty)=\xi \in \partial \Omega\)。那么存在\(A\big(\gamma_{1}(0), \gamma_{2}(0) \big)>0\)和\(T \in \mathbb{R}\)，使得对于所有\(t \geq 0\)，有\(K_{\Omega}(\gamma_{1}(t), \gamma_{2}(t+T))\leq Ae^{-\frac{t}{2}}\)。此外，利用这个性质，我们通过双全纯不变函数即广义挤压函数给出了强拟凸域的一个刻画。我们证明了：对于每个\(\alpha>2\)，存在\(\epsilon(d,\alpha)>0\)使得以下成立：如果\(\Omega \subset \mathbb{C}^d\)是一个具有\(\mathcal{C}^{2,\alpha}\)边界的有界凸域，并且在\(\Omega\)的一个紧子集之外\(T_{\Omega}^{D}(z)\geq 1-\epsilon\)，其中\(D \Subset \mathbb{C}^{d}\)是一个具有\(\mathcal{C}^{3}\)边界的平衡强凸域，并且\(T_{\Omega}^{D}\)是\(\Omega\)关于域\(D\)的挤压函数，那么\(\Omega\)是强拟凸的。
英文摘要
In this paper, we have proved a quantitative version of the approaching geodesic property for certain convex domains. We have proved that that if $Ω\subset \mathbb{C}^{d}$ is a bounded strongly convex domain with $\mathcal{C}^3$ boundary and $γ_{1}, γ_{2}:[0, \infty) \to Ω$ are two geodesic rays such that $γ_{1}(\infty)=γ_{2}(\infty)=ξ\in \partial Ω$. Then if the images of $γ_{1}$ and $γ_{2}$ are contained in the same complex geodesic, then there exists $T\in \mathbb{R}$ \[ \lim_{t \to \infty} \frac{1}{t} \log K_Ω\big(γ_{1}(t), γ_{2}(t+T)\big) = -2, \] otherwise \[ \lim_{t \to \infty} \frac{1}{t} \log K_Ω\big(γ_{1}(t), γ_{2}(t+T)\big) = -1. \] Furthermore, using this property we provided a characterization of strongly pseudoconvex domain via a biholomorphic invariant function namely generalized squeezing function. We have proved that: For every $α>0$ there exists $ε(d,α)>0$ such that the following holds: if $Ω\subset \mathbb{C}^d$ is a bounded convex domain with $\mathcal{C}^{2,α}$-boundary and \[ T_Ω^{D}(z)\geq 1-ε\] outside a compact subset of $Ω$, where $D \Subset \mathbb{C}^{d}$ is a balanced strongly convex domain with $\mathcal{C}^{3}$ boundary and $T_Ω^{D}$ is the squeezing function of $Ω$ with respect to the domain $D$ then $Ω$ is strongly pseudoconvex. We also establish exponential convergence of a certain family of quasi-geodesics in the unit ball of $\mathbb{C}^{d}$. We further show that the study of this family of quasi-geodesics provides a useful tool that allows the exponential convergence property of geodesics to be transferred from local subdomains to the ambient domain, as well as in the reverse direction.
CommentsTheorem 1.10 and Theorem 1.11 are added
