Spaces of Harmonic Functions
Spaces of Harmonic Functions
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调和函数空间
DOI：
10.1112/s0024610700008759
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
Jiaping Wang
中科院分区：
文献类型：
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作者：
C. Sung;Luen;Jiaping Wang
研究黎曼流形上的调和函数是一个重要而有趣的问题。在Li和的早期工作[21]中，证明了各种有界调和函数和正调和函数空间的维度与流形的端点数密切相关。对于n维完备黎曼流形Mn上所有n次多项式增长的调和函数组成的线性空间，记为Hd(Mn)，Li和[20]证明了当M具有非负Ricci曲率时，空间H1(M)的维度总是满足dimH1(M)⩽dimH1(Rn).他们继续提出，作为Yau[32]的一个猜想的改进，是否对所有d.Colding和Minicozzi在一系列论文[5-11]中证明了当M具有非负Ricci曲率时，dimHd(M)是有限的，从而对这个问题做出了重要贡献。另一方面，在一篇非常引人注目的论文[16]中，李提出了一个优雅而有力的论点来证明以下几点。回想一下，对于某些常数A和ν，如果1.1VX(R)⩽A(RR)νVX(R)，则M满足弱体积增长条件。
It is important and interesting to study harmonic functions on a Riemannian manifold. In an earlier work of Li and Tam [21] it was demonstrated that the dimensions of various spaces of bounded and positive harmonic functions are closely related to the number of ends of a manifold. For the linear space consisting of all harmonic functions of polynomial growth of degree at most d on a complete Riemannian manifold Mn of dimension n, denoted by Hd(Mn), it was proved by Li and Tam [20] that the dimension of the space H1(M) always satisfies dimH1(M) ⩽ dimH1(Rn) when M has non‐negative Ricci curvature. They went on to ask as a refinement of a conjecture of Yau [32] whether in general dim Hd(Mn) ⩽ dimHd(Rn)for all d. Colding and Minicozzi made an important contribution to this question in a sequence of papers [5–11] by showing among other things that dimHd(M) is finite when M has non‐negative Ricci curvature. On the other hand, in a very remarkable paper [16], Li produced an elegant and powerful argument to prove the following. Recall that M satisfies a weak volume growth condition if, for some constant A and ν, 1.1 Vx(R)⩽A(Rr)νVx(r)