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Three circles theorems for harmonic functions

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Abstract

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.

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Acknowledgments

The author was partially supported by NSFC 11401336. We thank Jiaping Wang for his interest and continuous encouragement, Xian-Tao Huang, William P. Minicozzi II, Christina Sormani, Shing-Tung Yau for their comments, and Liqun Zhang for sending the offprint [49] to us. We are indebted to Bo Yang for his comments and pointing out the relation between Conjecture 1.4 and the result in [22] to us in 2012. We are grateful to Shouhei Honda for his detailed comments and enthusiastic suggestions on the paper. Last but not least, we particularly thank Gang Liu for carefully reading the earlier version of the paper and pointing out some gaps, and we benefit from several long conversations with him.

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  1. Yau Mathematical Sciences Center, Jin Chun Yuan West Building, Tsinghua University, Beijing, 100084, People’s Republic of China

    Guoyi Xu

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Xu, G. Three circles theorems for harmonic functions. Math. Ann. 366, 1281–1317 (2016). https://doi.org/10.1007/s00208-016-1366-5

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