Positive harmonic functions on complete manifolds of negative curvature.

*(English)*Zbl 0587.53045Let 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.

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 |