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

TL;DR

This paper investigates how the dimension of harmonic functions with polynomial growth behaves asymptotically on certain Riemannian manifolds with nonnegative Ricci curvature, especially when the growth order is large.

Contribution

It provides new estimates for the dimension of harmonic functions with polynomial growth on manifolds with maximal volume growth and a unique tangent cone at infinity.

Findings

Derived estimates of $h_{d}(M)$ in terms of $d$, $n$, and volume ratio $eta$.

Revealed the asymptotic behavior of $h_{d}(M)$ matching Euclidean space when volume ratio is maximal.

Extended understanding of harmonic functions' growth on manifolds with specific geometric conditions.

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 $\alpha=\lim_{R\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(R))}{R^{n}}$. 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 $h_{d}(\mathbb{R}^{n})$.