arXiv 2109.075342021-09-15math.DG

非负里奇曲率流形上多项式增长的调和函数

Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature

Xian-Tao Huang

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摘要：

设$(M,g)$是一个具有维数$n$、非负里奇曲率、最大体积增长和无穷远处唯一切锥的黎曼流形.在这种情况下，无穷远处的切锥$C(X)$是横截面$X$上的欧几里得锥。用$α=\lim_{r\rightarrow\infty}\frac{\mathrm{Vol}(B_{r}(p))}{r^{n}}$表示渐近体积比。设$h_{k}=h_{k}(M)$是增长阶至多为$k$的多项式增长的调和函数空间的维数。本文利用$X$的特征值的计数函数证明了$h_{k}$的一个上界.作为推论，我们得到了$\lim_{k\rightarrow\infty}k^{1-n}h_{k}=\frac{2α}{(n-1)!ω_{n}}$。这些结果是清晰的，因为它们恢复了$h_{k}(\mathbb{R}^{n})$的相应的公知性质。特别是，这些结果举行的非负截面曲率和最大体积增长的流形。

英文摘要：

Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the cross-section $X$. Denote by $α=\lim_{r\rightarrow\infty}\frac{\mathrm{Vol}(B_{r}(p))}{r^{n}}$ the asymptotic volume ratio. Let $h_{k}=h_{k}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $k$. In this paper, we prove a upper bound of $h_{k}$ in terms of the counting function of eigenvalues of $X$. As a corollary, we obtain $\lim_{k\rightarrow\infty}k^{1-n}h_{k}=\frac{2α}{(n-1)!ω_{n}}$. These results are sharp, as they recover the corresponding well-known properties of $h_{k}(\mathbb{R}^{n})$. In particular, these results hold on manifolds with nonnegative sectional curvature and maximal volume growth.

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