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

Sullivan, D.

J. Differ. Geom. 18, 723-732 (1983).

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

Keywords:

Dirichlet problem at infinity; probability on Riemannian manifolds; harmonic functions

Citations:

Zbl 0541.53035; Zbl 0541.53036