The Dirichlet problem at infinity for manifolds of nonpositive curvature.

*(English)*Zbl 0786.53021
Gu, C. H. (ed.) et al., Differential geometry. Proceedings of the symposium in honour of Professor Su Buchin on his 90th birthday, Shanghai, China, September 17-23, 1991. Singapore: World Scientific. 48-58 (1993).

M. T. Anderson [J. Differ. Geom. 18, 701-722 (1983; Zbl 0541.53036)] and D. Sullivan [ibid., 723-732 (1983; Zbl 0541.53037)] proved that a complete Riemannian manifold \(M\) admits a wealth of bounded harmonic functions provided that the sectional curvature \(\text{Riem}(M)\) satisfies the inequality \(-b^ 2\leq \text{Riem}(M)\leq - a^ 2\) for some constants \(a,b>0\). However the classical harmonic theory on symmetric domains cannot be included in the Anderson or Sullivan’s theory since the symmetric domain \(\Omega\) endowed with Bergmann metric has \(- b^ 2\leq \text{Riem}(\Omega)\leq 0\) for some constant \(b > 0\).

S.-T. Yau and R. Schoen [Differential geometry (Chinese) (Science Press) (1988)] posed the following problem: For any simply connected Riemannian manifold \(M\) with \(-b^ 2 \leq \text{Riem}(M)\leq 0\) under what weaker conditions does there exist nontrivial bounded harmonic functions or Poisson kernel function? And what about Martin boundary? In the paper under review the author establishes a new Laplacian comparison theorem and proves an existence theorem of bounded harmonic functions on a complete Riemannian manifold of nonpositive curvature which gives an answer to the above problem.

For the entire collection see [Zbl 0800.00026].

S.-T. Yau and R. Schoen [Differential geometry (Chinese) (Science Press) (1988)] posed the following problem: For any simply connected Riemannian manifold \(M\) with \(-b^ 2 \leq \text{Riem}(M)\leq 0\) under what weaker conditions does there exist nontrivial bounded harmonic functions or Poisson kernel function? And what about Martin boundary? In the paper under review the author establishes a new Laplacian comparison theorem and proves an existence theorem of bounded harmonic functions on a complete Riemannian manifold of nonpositive curvature which gives an answer to the above problem.

For the entire collection see [Zbl 0800.00026].

Reviewer: V.V.Goldberg (Newark / New Jersey)

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

31C35 | Martin boundary theory |

60J50 | Boundary theory for Markov processes |