0000000001156291
AUTHOR
Päivi Lammi
Compactness of a conformal boundary of the Euclidean unit ball
We study conformal metrics d‰ on the Euclidean unit ball B n : We assume that either the density ‰ associated with the metric d‰ satisfies a logarithmic volume growth condition for small balls or that ‰ satisfies a Harnack inequality and a suitable sub-Euclidean volume growth condition. We prove that the ‰-boundary @‰ B n is homeomorphic to S ni1 if and only if @‰ B n is compact. In the planar case, the compactness of @‰ B 2 is further equivalent to local connectivity of the ‰-boundary together with the boundedness of (B 2 ;d‰):
Quasihyperbolic boundary condition: Compactness of the inner boundary
We prove that if a metric space satisfies a suitable growth condition in the quasihyperbolic metric and the Gehring–Hayman theorem in the original metric, then the inner boundary of the space is homeomorphic to the Gromov boundary. Thus, the inner boundary is compact. peerReviewed
Gromov hyperbolicity and quasihyperbolic geodesics
We characterize Gromov hyperbolicity of the quasihyperbolic metric space (\Omega,k) by geometric properties of the Ahlfors regular length metric measure space (\Omega,d,\mu). The characterizing properties are called the Gehring--Hayman condition and the ball--separation condition. peerReviewed