AI 中文总结
本文针对具非负Ricci曲率的局部共形平坦流形，证实丘成桐提出的关于多项式增长调和函数空间维数的第二个问题，即其维数不超过对应欧几里得空间的该空间维数的精确上界成立。
AI 中文摘要
设$\boldsymbol{\textit{H}}_d(M)$表示完备黎曼流形$M$上次数至多为$d$的多项式增长调和函数构成的空间。丘成桐对具非负Ricci曲率的完备流形上的$\boldsymbol{\textit{H}}_d(M)$提出两个基本问题：其一为$\boldsymbol{\textit{H}}_d(M)$的有限维性，已由Colding与Minicozzi证实；其二为是否存在其欧几里得对应$\boldsymbol{\textit{H}}_d(\boldsymbol{\textit{R}}^n)$的维数给出的精确上界。本文验证，若$(M,g)$是具非负Ricci曲率的局部共形平坦流形，则该精确欧几里得界成立，即$\text{dim}(\boldsymbol{\textit{H}}_d(M))\boldsymbol{\boldsymbol{\text{dim}}}(\boldsymbol{\textit{H}}_d(\boldsymbol{\textit{R}}^n))$。
英文摘要
Let $\mathcal{H}_d(M)$ denote the space of harmonic functions with polynomial growth of degree at most $d$ on a complete Riemannian manifold $(M,g)$. Yau raised two fundamental questions regarding $\mathcal{H}_d(M)$ on complete manifolds with nonnegative Ricci curvature. The first question is the finite dimensionality of $\mathcal{H}_d(M)$, which was confirmed by Colding and Minicozzi. The second question asks whether a sharp upper bound given by its Euclidean analog $\operatorname{dim}\mathcal{H}_{d}(\mathbb{R}^n)$ holds. We verify that the second question is true on locally conformally flat manifolds. Indeed, one can precisely determine the value of $\dim \mathcal{H}_d(M)$ case by case.
