To appear in J.Diff.Geom. COUNTING MASSIVE SETS AND DIMENSIONS OF HARMONIC FUNCTIONS

In a previous article of the authors [L-W1], they introduced the notion of dmassive sets. Using this, they proved a structural theorem for polynomial growth harmonic maps into a Cartan-Hadamard manifold with strongly negative sectional curvature. When d = 0, the maximum number of disjoint 0-massive sets m0(M) admissible on a complete manifold M is known [G] to be the same as the dimension h0(M) of bounded harmonic functions on the manifold. This relationship is not established and it is unclear if it is at all true for d > 0. However, the theory of estimating md(M) has an analogous counterpart in the theory of estimating hd(M). This was first indicated in [L-W1]. The purpose of this article is to give sharp estimates on both of these quantities in various situations. Before we outline the main results in this article, let us recall some definitions and set up the appropriate notations. Throughout this article, M is assumed to be an n-dimensional, complete, noncompact manifold without boundary. The operator ∆ denotes the Laplacian with respect to the given Riemannian metric. Let p ∈ M be a fixed point in M and rp(x) the geodesic distance function from x ∈ M to the point p.