Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T18:34:53.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Convexity at infinity and bounded harmonic functions

Published online by Cambridge University Press:  17 April 2009

Albert Borbély
Albert Borbély
Affiliation:
Department of Mathematics and Computer ScienceKuwait UniversitySafat 13060 Kuwait e-mail: borbely@mcc.sci.kuniv.edu.kw
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.

It is shown that a complete simply connected negatively curved manifold supports nontrivial bounded harmonic functions if the singular set of the ideal boundary is disconnected.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 1 , August 1997 , pp. 63 - 68
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Ancona, A., ‘Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature’, Rev. Mat. Iberoamericana 10 (1994), 189–220.CrossRefGoogle Scholar
[2]Ancona, A., ‘Negatively curved manifolds, elliptic operators, and the Martin boundary’, Ann. Math. 125 (1987), 495–536.CrossRefGoogle Scholar
[3]Anderson, M.T., ‘The dirichlet problem at infinity for manifolds of negative curvature’, J. Differential Geom. 18 (1983), 701–721.CrossRefGoogle Scholar
[4]Anderson, M.T. and Schoen, R., ‘Positive harmonic functions on complete manifolds of negative curvature’, Ann. Math. 121 (1985), 429–461.CrossRefGoogle Scholar
[5]Borbély, A., ‘A note on the Dirichlet problem at infinity for manifolds of negative curvature’, Proc. Amer. Math. Soc. 114 (1992), 865–872.CrossRefGoogle Scholar
[6]Borbély, A., ‘The nonsolvability of the Dirichlet Problem on negatively curved manifolds’ (to appear).Google Scholar
[7]Choi, H.I., ‘Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds’, Trans. Amer. Math. Soc. 281 (1984), 691–716.CrossRefGoogle Scholar
[8]Eberlain, P. and O'Neill, B., ‘Visibility manifolds’, Pacific J. Math 46 (1973), 45–109.CrossRefGoogle Scholar
[9]Greene, R.E. and Wu, H., ‘C ∞ Approximations of convex, subharmonic and plurisubharmonic functions’, Ann. Sci. École Norm. Sup. 12 (1979), 69–100.CrossRefGoogle Scholar
[10]Kifer, Y., ‘Brownian motion and harmonic functions on manifolds of negative curvature’, Theory Probab. Appl. 21 (1976), 81–95.CrossRefGoogle Scholar
[11]Sullivan, D., ‘The Dirichlet problem at infinity for a negatively curved manifold’, J. Differential Geom. 18 (1983), 723–732.CrossRefGoogle Scholar