Affiliation:
1. School of Mathematics, Sun Yat-sen University, Guangzhou510275, P. R. China
Abstract
AbstractSuppose {(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)}.
In this paper, we study the behavior of {h_{d}(M)} when d is large.
More precisely, suppose {(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 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})}.
Funder
National Natural Science Foundation of China
Subject
Applied Mathematics,General Mathematics
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