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The Dirichlet problem at infinity for a negatively curved manifold. (English) Zbl 0541.53037

Let M be a simply connected Riemannian manifold with sectional curvature bounded between two negative constants. The author uses probability theory to prove that each continuous function on the sphere at infinity \(\partial \bar M\) has a continuous harmonic extension to \(M\cup \partial \bar M\). As a consequence it follows that there are many non-constant bounded harmonic functions on M. This result was proved independently by M. T. Anderson by a different method [see the preceding review].
Reviewer: G.Thorbergsson

MSC:

53C20 Global Riemannian geometry, including pinching
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
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