Dimension of polynomial growth harmonic functions on locally conformally flat manifolds with nonnegative Ricci curvature

Abstract

Let ${\mathcal{H}}_d(M)$ denote the space of harmonic functions with polynomial growth of degree at most $d$ on a complete Riemannian manifold $(M,g)$. Yau raised two fundamental questions regarding ${\mathcal{H}}_d(M)$ on complete manifolds with nonnegative Ricci curvature. The first question is the finite dimensionality of ${\mathcal{H}}_d(M)$, which was confirmed by Colding and Minicozzi. The second question asks whether a sharp upper bound given by its Euclidean analog $\dim {\mathcal{H}}_d({\mathbb{R}}^n)$ holds. We verify that the second question is true on locally conformally flat manifolds. Indeed, one can precisely determine the value of $\dim {\mathcal{H}}_d(M)$ case by case.