论文标题
紧凑型谎言组的剩余不变指标的直径和拉普拉斯特征值估计值
Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups
论文作者
论文摘要
让$ g $成为一个紧凑的连接谎言组$ m $。一旦固定了$ g $的双重变量度量，$ g $上的剩余不变的度量标准与$ m \ times m $阳性确定的对称矩阵相应。我们估计了与$ g $上的剩余的左右公制相关的拉普拉斯 - 贝特拉米操作员的直径和最小的正征值，就相应的正定义对称矩阵的特征值而言。结果，我们对Eldredge，Gordina和Saloff-Coste的猜想给出了部分答案。也就是说，我们给出了剩余不变的$ \ Mathcal m $ on $ g $的$ \数学s $ s $，以使得存在一个积极的实际数字$ c $，具体取决于$ g $和$ \ nathcal s $ S $。 $ \ Mathcal S = \ Mathcal M $的常数$ C $的存在是原始猜想。
Let $G$ be a compact connected Lie group of dimension $m$. Once a bi-invariant metric on $G$ is fixed, left-invariant metrics on $G$ are in correspondence with $m\times m$ positive definite symmetric matrices. We estimate the diameter and the smallest positive eigenvalue of the Laplace-Beltrami operator associated to a left-invariant metric on $G$ in terms of the eigenvalues of the corresponding positive definite symmetric matrix. As a consequence, we give partial answers to a conjecture by Eldredge, Gordina and Saloff-Coste; namely, we give large subsets $\mathcal S$ of the space of left-invariant metrics $\mathcal M$ on $G$ such that there exists a positive real number $C$ depending on $G$ and $\mathcal S$ such that $λ_1(G,g)\operatorname{diam}(G,g)^2\leq C$ for all $g\in\mathcal S$. The existence of the constant $C$ for $\mathcal S=\mathcal M$ is the original conjecture.
