Spherical convexity and hyperbolic metric

2017/04/26 by Toshiyuki Sugawa, Sugawa, Toshiyuki
Mathematics · #51M10 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Primary 30F45 #Secondary 30C80

paper · pdf · doi:10.48550/arxiv.1704.07944

openalex publication_date 2017/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω be a domain in 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,∂Ω)λ_Ω(z)≥1/2 for z∈Ω, where d(z,∂Ω) denotes the Euclidean distance from z to the boundary ∂Ω. In the present note, we will provide similar characterizations of spherically convex domains in terms of the spherical density of the hyperbolic metric.

Citations

• Spherical density of hyperbolic metric and uniform perfectness

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