On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth

Suppose $(M^{n},g)$ is a Riemannian manifold with nonnegative Ricci curvature, and let $h_{d}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $d$. Colding and Minicozzi proved that $h_{d}(M)$ is finite. Later on, there are many researches which give better estimates of $h_{d}(M)$. We study the behavior of $h_{d}(M)$ when $d$ is large in this paper. More precisely, suppose that $(M^{n},g)$ has maximal volume growth and has a unique tangent cone at infinity, then when $d$ is sufficiently large, we obtain some estimates of $h_{d}(M)$ in terms of the growth order $d$, the dimension $n$ and the the asymptotic volume ratio $\alpha=\lim_{R\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(R))}{R^{n}}$. When $\alpha=\omega_{n}$, i.e., $(M^{n},g)$ is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of $h_{d}(\mathbb{R}^{n})$.