# Hyperbolic geometry in hyperbolically k-convex regions

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This paper is the third and final part of a trilogy dealing with the concept of k-convexity in various geometries.

Tipo de documento: Artículo - Article

Palabras clave: Trilogy, concept of k-convexity, various geometries

Fuente: http://www.bdigital.unal.edu.co

## Introducción

Revista Colombiana de Matematicas (1991) pgs.
123 - 142 Vol.
XXV HYPERBOLIC GEOMETRY K-CONVEX IN HYPERBOLICALLY REGIONS by Diego MEJIA and David MINDA 1 §I.
Introduction. This paper is the third and final part of a trilogy dealing with the concept of k-convexity in various geometries. clidean Our first geometry spherical geometry. convexity relative paper and [MM1] the In second this to hyperbolic paper dealt with k-convexity [MM2] we geometry treat in eu- with k-convexity the concept on the unit disk in of k[) = {z : Izi l}. We assume that the reader is familiar with both [MM1] and [M M 2]; we frequently omit details of proofs when they are similar to proofs of analogous results in either one of these papers. 1.
Research partially supported by a National Science Foundation grant (DMS-9008051) MEJIA and MINDA A Jordan .n in the unit disk [) is region called hyperbolically k-convex if the hyperbolic curvature of the boundary is at least k at each point of a.n.
This assumes that the boundary of .n is smooth. A definition of hyperbolic k-convexity that applies to arbitrary regions is given in section III. Our goal is to obtain sharp hyperbolic geometric estimates for various quantities in hyperbolically distortion and constant for k-convex covering disk [) onto §II Hyperbolic = 0, f about hyperbolic metric is AID(z) Idzl The group of conformal = Au t([» hyperbolic unit metric disk - IzI2); it has automorphisms = {T(z) e i8(z - a) I- the az :e is invariant some [). unit basic The hyper- Gaussian curvature of [) is E IR, a under the that is, T* (AID(Z) Idzl) = AID(Z) Idzl, or equivalently, 1-(1 - Iz 12) for all TEA u t( [) ). The hyperbolic a.be [) is defined of by recalling on the Idzlj(1 mappings a) of K h(k, regions. We begin geometry = conformal convex Geometry. lead to sharp Bloch-Landau set) for the family a) k- hyperbolically facts T...