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The Dirichlet problem at infinity for manifolds of negative curvature. (English) Zbl 0541.53036

Let \(N^ n\) be a simply connected Riemannian manifold with sectional curvature bounded between two negative constants. The author solves the Dirichlet problem at infinity for \(N^ n\), i.e., given a continuous function on the sphere at infinity \(S^{n-1}(\infty)\), there is a continuous harmonic extension to \(\bar N^ n=N^ n\cup S^{n- 1}(\infty)\). As a consequence it follows that \(N^ n\) has a large class of bounded harmonic functions. This result was proved independently by D. Sullivan by a different method [see the next review].
Reviewer: G.Thorbergsson

MSC:

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