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

Abstract

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 $\alpha ={\lim }_{R\to \infty }\frac{\mathrm{Vol}(B_p(R))}{R^n}$. 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 $h_d({\mathbb{R}}^n)$.

Comments are welcome

References in corpus (3)

• Behaviour of the reference measure on $\mathsf{RCD}$ spaces under charts
• Local spectral convergence in $RC{D}^{\ast }(K,N)$ spaces
• Short-time behavior of the heat kernel and Weyl's law on $RC{D}^{\ast }(K,N)$-spaces