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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].

##### 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