Harmonic functions with polynomial growth

Twenty years ago Yau, [56], generalized the classical Liouville theorem of complex analysis to open manifolds with nonnegative Ricci curvature. Specifically, he proved that a positive harmonic function on such a manifold must be constant. This theorem of Yau was considerably generalized by Cheng-Yau (see [15]) by means of a gradient estimate which implies the Harnack inequality. As a consequence of this gradient estimate (see [13]), one has that on such a manifold even a harmonic function of sublinear growth must be constant. In order to study further the analytic properties of these manifolds one would like to restrict the class of functions to be considered as much as possible while minimizing loss of information (cf. [22], [26]). From the results of Cheng and Yau, it follows that a natural candidate is the class of harmonic functions of polynomial growth (note that they must be of at least linear growth). In fact, in his study of these functions, Yau was motivated to make the following conjecture (see [58], [59], and [60]; see also the excellent survey article by Peter Li, [37]): Conjecture 0.1. (Yau). For an open manifold with nonnegative Ricci curvature the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional.