完备非紧流形上的几何以及调和函数论从上世纪70年代以来一直是几何分析中的重要研究方向。已经知道对于一般的具有非负Ricci曲率的完备非紧流形，非常数的多项式增长调和函数可能并不存在，并且对于多项式增长调和函数的估计仍然缺乏一致性。本项目以具有极大体积增长和非负Ricci曲率的完备非紧流形上的多项式增长调和函数为研究对象，对此类流形上非常数的多项式调和函数的存在性以及频率估计进行研究。具体内容包括：此类流形无穷远处切锥上的Laplace算子谱的分析和非常数的多项式增长调和函数的存在性的联系，多项式增长调和函数的频率的一致估计，以及积分形式的三圆定理。本项目旨在揭示具有极大体积增长和非负Ricci曲率的完备非紧流形上非常数多项式增长调和函数和该流形的几何性质的关系，从而为此类流形上的几何以及调和函数的研究提供技术工具以及深化认知。

The geometry and harmonic functions on complete noncompact manifolds has been a focus in geometric analysis since 1970's. It is known that on general complete manifolds with nonnegative Ricci curvature, non-constant harmonic functions of polynomial growth possibly do not exist; and the uniform estimate of harmonic functions of polynomial growth is not available yet. This project will study the harmonic functions of polynomial growth on complete manifolds with nonnegative Ricci curvature and maximal volume growth, especially the existence of nonconstant harmonic functions of polynomial growth and their frequency functions. More concretely, we will study the relation between the existence of non-constant harmonic functions of polynomial growth and the analysis of the spectrum of Laplace operator on the tangent cone at infinity of those manifolds, the uniform estimate of frequency functions of harmonic functions with polynomial growth, and the Three Circles Theorem in integral sense. This project will try to discover the deep connection between the geometry and nonconstant harmonic functions of polynomial growth on complete manifolds with nonnegative Ricci curvature and maximal volume growth, which will provide the technical tools and deepen our understanding of the geometry and harmonic functions on those manifolds.