Rough isometry and the asymptotic Dirichlet problem

Abstract

We propose a new asymptotic Dirichlet problem for harmonic functions via the rough isometry on a certain class of Riemannian manifolds. We prove that this problem is solvable for naturally defined class of functions. This result generalizes those of Schoen and Yau and of Cheng. 1. Introduction. The asymptotic Dirichlet problem for harmonic functions on a noncompact complete Riemannian manifold is to find the harmonic function satisfying the given Dirichlet boundary condition at infinity. It has a long history, and by now, it is well understood by the works of the first author, M. Anderson, D. Sullivan, R. Schoen and others, when M is a Cartan-Hadamar d manifold with sectional curvature — b2<KM< —a2<0. (By a Cartan-Hadamar d manifold, we mean a complete simply connected manifold of nonpositive sectional curvature.) In (Ch), the first author posed the asymptotic Dirichlet problem and proved that it is solvable when a Cartan-Hadamar d manifold M with sectional curvature KM<—a2<0 satisfies the convex conic neighborhood condition. In (A), Anderson constructed a convex neighborhood, thereby solving the problem when the sectional curvature satisfies — b2<KM< — a2<0. At the same time, Sullivan (S) also solved this problem using the probabilistic approach. In (A-S), Anderson and Schoen showed that the Martin boundary of M can be identified with M(oo) which is naturally defined to be the set of the asymptotic classes of geodesic rays. By constructing the Poisson kernel, they obtained the representation formula for harmonic functions, and proved the Fatou-type theorem. The essence of all of the above works is that the curvature assumption enables one to control the angle via the Toponogov comparison theorem and the convexity property near the boundary at infinity M(oo). There have been many attempts to generalize the above results. A typical approach is to relax the curvature assumption to allow curvature decay at a certain rate. But the basic method of proof still remains the same and the improvements are mostly techni- cal. One interesting generalization that does not directly involve the curvature bound