## 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

### Citations:

Zbl 0541.53035; Zbl 0541.53037
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