Harmonic Functions with Polynomial Growth

Twenty years ago Yau generalized the classical Liouville theo rem of complex analysis to open manifolds with nonnegative Ricci curva ture Speci cally 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 by means of a gradient estimate which implies the Harnack inequality As a consequence of this gradient estimate see 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 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 and see also the excellent survey article by Peter Li