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Positive harmonic functions on complete manifolds of negative curvature. (English) Zbl 0587.53045
Let M denote a complete, simply connected Riemannian manifold with sectional curvature K satisfying $$-b^ 2\leq K\leq -a^ 2<0$$ for some positive constants a,b. In this work the authors use a combination of differential geometric and analytic techniques to generalize to M many of the methods and results of harmonic function theory on the unit disc $$(=hyperbolic$$ plane) and other rank 1 symmetric spaces of noncompact type. The curvature bounds allow one to construct a $$C^{\alpha}$$ structure, $$\alpha =a/b$$, on the geometric boundary S($$\infty)$$ that consists of equivalence classes of asymptotic geodesics of M. (A smooth structure on S($$\infty)$$ does not exist in general.) The estimates that occur in the existence proof of the $$C^{\alpha}$$ structure lead to a solution, due to the second author, of the Dirichlet problem for M - constructing a bounded harmonic function on M with prescribed continuous boundary data on S($$\infty)$$. This solution of the Dirichlet problem, which uses the Perron method of super- and subharmonic barrier functions, is simpler than earlier solutions due to the first author [J. Differ. Geom. 18, 701-722 (1983; Zbl 0541.53036)] and to D. Sullivan [ibid. 723-732 (1983; Zbl 0541.53037)].
If Hf denotes the unique harmonic function with boundary values f on S($$\infty)$$, then each point $$x\in M$$ determines a harmonic measure $$\omega^ x$$ on S($$\infty)$$ such that $$(Hf)(x)=\int_{S(\infty)}f(Q) d\omega^ x(Q)$$ for all continuous f on S($$\infty)$$. The measures $$\omega^ x$$ are absolutely continuous and give rise to a Poisson kernel $$K(x,Q)=(d\omega^ x/d\omega^ 0)(Q)$$, where (x,Q)$$\in M\times S(\infty)$$ and 0 is a fixed point of M. The function $$x\to K(x,Q)$$ is actually defined only for Q in a subset of full measure in S($$\infty)$$ with respect to harmonic measure. This function, whenever it exists, is a kernel function for Q, i.e. K(x,Q) is a positive harmonic function on M such that $$K(0,Q)=1$$ and K extends continuously to the zero function on S($$\infty)-Q.$$
In general, a Harnack inequality at infinity (Corollary 5.2) shows that every Q in S($$\infty)$$ determines a unique kernel function at Q. This Harnack inequality then leads to the main result of the paper (Theorem 6.3), which proves the existence of a natural homeomorphism $$\Phi$$ : $${\mathcal M}\to S(\infty)$$, where $${\mathcal M}$$ is the Martin boundary of M. Specifically, given a point Q in S($$\infty)$$ it is shown that lim G(y,x)/G(y,0) exists and equals the kernel function $$\Phi^{-1}(Q)$$ in $${\mathcal M}$$, $$y\to Q$$ where G(y,x) is the Green function on M with pole at y. In particular, $$\Phi^{-1}(Q)$$ is the function $$x\to K(x,Q)$$ for all Q such that K(x,Q) is defined. In addition, $$\Phi^{-1}$$ is $$C^{\alpha}$$, $$\alpha =a/b$$, with respect to a natural metric on $${\mathcal M}$$ and angle measurement on S($$\infty)$$ relative to the fixed origin 0. Theorem 6.3 has since been generalized by A. Ancona [see the next review].
Following the theory of Martin or arguing directly as the authors do one obtains the generalization of a classical representation formula (Theorem 6.5): Given a positive harmonic function u on M there exists a unique finite positive Borel measure $$\mu$$ on S($$\infty)$$ such that $$u(x)=\int_{S(\infty)}K(x,Q) d\mu$$. In section 7 the authors define nontangential convergence in a way that reduces to the usual definition in the unit disk. In Theorem 7.6 they generalize the classical result of Fatou to show that for every positive harmonic function u on M the nontangential limit of u exists almost everywhere on S($$\infty)$$ with respect to harmonic measure. The paper concludes with a geometric description of the Martin boundary of a noncompact quotient manifold X of M such that X admits a compact totally convex set.
Reviewer: P.Eberlein

##### MSC:
 53C20 Global Riemannian geometry, including pinching 31C12 Potential theory on Riemannian manifolds and other spaces 31C35 Martin boundary theory
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