Three circles theorems for harmonic functions

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• 作者：Guoyi Xu
• 关键词：Mathematics Subject Classification35B40 ; 58J05 ; 53C23 ; 35A01
• 刊名：Mathematische Annalen
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：366
• 期：3-4
• 页码：1281-1317
• 全文大小：719 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1807
• 卷排序：366

文摘

We proved two three circles theorems for harmonic functions on manifolds in integral sense. As one application, on manifold with nonnegative Ricci curvature, whose tangent cone at infinity is the unique metric cone with unique conic measure, we showed the existence of nonconstant harmonic functions with polynomial growth. This existence result recovered and generalized the former result of Ding, and led to a complete answer of Ni’s conjecture. Furthermore in similar context, combining the techniques of estimating the frequency of harmonic functions with polynomial growth, which were developed by Colding and Minicozzi, we confirmed their conjecture about the uniform bound of frequency.