# Hyperbolic geometry in hyperbolically k-convex regions

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

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

## Teaser

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...