Three circles theorems for harmonic functions
Three circles theorems for harmonic functions
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DOI：
10.1007/s00208-016-1366-5
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发表时间：
2016-01
影响因子：
1.4
通讯作者：
Guoyi Xu
中科院分区：
文献类型：
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作者：
Guoyi Xu
在积分意义下证明了流形上调和函数的两个三圆定理。作为一个应用，在具有非负Ricci曲率的流形上，证明了具有多项式增长的非常数调和函数的存在性，其中无穷远处的切锥是具有唯一锥测度的唯一度量锥.这一结果恢复和推广了Ding的结果，从而完整地回答了Ni猜想。在类似的背景下，结合Colding和Minicozzi提出的利用多项式增长估计调和函数频率的方法，我们进一步证实了他们关于频率一致界的猜想.
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.