Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature

Abstract

Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the cross-section $X$. Denote by $\alpha ={\lim }_{r\to \infty }\frac{\mathrm{Vol}(B_r(p))}{r^n}$ the asymptotic volume ratio. Let $h_k=h_k(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $k$. In this paper, we prove a upper bound of $h_k$ in terms of the counting function of eigenvalues of $X$. As a corollary, we obtain ${\lim }_{k\to \infty }{k}^{1-n}{h}_k=\frac{2\alpha }{(n-1)!{\omega }_n}$. These results are sharp, as they recover the corresponding well-known properties of $h_k({\mathbb{R}}^n)$. In particular, these results hold on manifolds with nonnegative sectional curvature and maximal volume growth.

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