球凸性与双曲度量
Spherical convexity and hyperbolic metric
摘要：
设$Ω$是$\mathbb{C}$中具有高斯曲率的双曲度量$λ_Ω(z)|dz|$的域，$-4.$ Mejia和Minda在1990年的论文中证明了$Ω$是（欧几里德）凸当且仅当对于$z\inΩ,$为$d(z,\partialΩ)λ_Ω(z)\ge1/2$其中$d(z,\partialΩ)$表示从$z$到边界$\partialΩ.$的欧几里德距离在本说明中，我们将根据双曲度量的球面密度给出球面凸域的类似刻画。
英文摘要：
Let $Ω$ be a domain in $\mathbb{C}$ with hyperbolic metric $λ_Ω(z)|dz|$ of Gaussian curvature $-4.$ Mejia and Minda proved in their 1990 paper that $Ω$ is (Euclidean) convex if and only if $d(z,\partialΩ)λ_Ω(z)\ge1/2$ for $z\inΩ,$ where $d(z,\partialΩ)$ denotes the Euclidean distance from $z$ to the boundary $\partialΩ.$ In the present note, we will provide similar characterizations of spherically convex domains in terms of the spherical density of the hyperbolic metric.
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