On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth
On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth
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多项式增长调和函数空间维数的渐近行为
DOI：
10.1515/crelle-2018-0029
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Xian-Tao Huang
中科院分区：
文献类型：
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作者：
Xian-Tao Huang
设（Mn,g）为具有非负Ricci曲率的黎曼流形，设hd(M)为生长阶不超过d的多项式增长调和函数空间的维数。Colding和Minicozzi证明了hd(M)是有限的。后来，有许多研究对hd(M)给出了更好的估计。本文研究了d较大时hd(M)的行为。更准确地说，假设（Mn,g）有最大的体积增长，并且在无穷远处有唯一的切锥。然后，当d足够大时，我们得到了hd(M)关于生长阶数d、维数n和渐近体积比α=limR→∞Vol(Bp(R))/Rn的一些估计。当α=ωn，即（Mn,g）与欧几里德空间等距时，本文所得到的渐近性质恢复了hd（Rn）的一个著名的渐近性质。
Suppose (Mn,g) is a Riemannian manifold with nonnegative Ricci curvature, and let hd(M) be the dimension of the space of harmonic functions with polynomial growth of growth order at most d. Colding and Minicozzi proved that hd(M) is finite. Later on, there are many researches which give better estimates of hd(M). In this paper, we study the behavior of hd(M) when d is large. More precisely, suppose (Mn,g) has maximal volume growth and has a unique tangent cone at infinity. Then when d is sufficiently large, we obtain some estimates of hd(M) in terms of the growth order d, the dimension n and the asymptotic volume ratio α=limR→∞Vol(Bp(R))/Rn. When α=ωn, i.e., (Mn,g) is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of hd(Rn).