Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-07-07T03:20:48.273Z Has data issue: false hasContentIssue false

On the area growth of a hyperbolic surface

Published online by Cambridge University Press:  17 April 2009

Pui-Fai Leung
Affiliation:
Department of MathematicsNational University of Singapore10 Kent Ridge CrescentSingapore 0511
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We conjecture that if the rate of area growth of a geodesic disc of radius r on a smooth simply-connected complete surface with non-positive Gaussian curvature is faster than r2(logr)1+e for some ε ≥ 0, then the surface is hyperbolic. We prove this under an additional assumption that the surface is rotationally symmetric.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1989

References

[1] Greene, R.E. and Wu, H., Function Theory on Manifolds which Possess a Pole: Lecture Notes in Mathematics 699, (Springer-Verlag, Berlin, Heidelberg, New York, 1979).CrossRefGoogle Scholar
[2] Hicks, N.J., Notes on Differential Geometry (Van Nostrand, Princeton, N.J., 1965).Google Scholar
[3] Karp, L., ‘Subharmonic functions on real and complex manifolds’, Math Z 179 (1982), 535554.CrossRefGoogle Scholar
[4] Milnor, J., ‘On deciding whether a surface is parabolic or hyperbolic’, Amer. Math. Monthly 84 (1977), 4345.CrossRefGoogle Scholar
[5] Struik, D.J., Lectures on classical differential geometry (Addison-Wesley, Cambridge, Mass., 1950).Google Scholar