We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete Riemannian manifolds with nonnegative Ricci curvature and maximal volume growth, further assume the dimension of the manifold is not less than 3, we prove that quantitative strong unique continuation yields the existence of nonconstant polynomial growth harmonic functions. Also the uniform bound of frequency for linear growth harmonic functions on such manifolds is obtained, and this confirms a special case of Colding–Minicozzi's conjecture on frequency.

中文翻译：

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谐波函数的增长率

我们从两个方面研究谐波函数的增长率：梯度估计和频率。我们获得了具有非负曲率的完整表面测地球中正谐波函数的尖锐梯度估计。在具有非负Ricci曲率和最大体积增长的完整黎曼流形上，进一步假设流形的尺寸不小于3，我们证明了定量强唯一连续会产生非常数多项式增长调和函数的存在。还获得了此类流形上线性增长谐波函数的频率统一边界，这证实了Colding–Minicozzi频率猜想的特殊情况。