论文标题
Carnot组的紧凑型商的收缩不平等与POPP的体积
Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume
论文作者
论文摘要
在本文中，我们为Carnot组$γ\ Backslash G $的商空间提供了收缩不平等。也就是说，我们表明存在正常数$ c $，因此$γ\ backslash g $的收缩率小于$ {\ rm cvol}（γ\ backslash g）^{\ frac {1} {1} {q}}} $，其中$ q $是$ q $ hausdorff dimension。此外，常数仅取决于Lie代数$ \ Mathfrak {g} = \ bigoplus v_i $的等级的维度。为了证明这一事实，在POPP的定义中引入的$ G $上的标量产品起着关键作用。
In this paper, we give a systolic inequality for a quotient space of a Carnot group $Γ\backslash G$ with Popp's volume. Namely we show the existence of a positive constant $C$ such that the systole of $Γ\backslash G$ is less than ${\rm Cvol}(Γ\backslash G)^{\frac{1}{Q}}$, where $Q$ is the Hausdorff dimension. Moreover, the constant depends only on the dimension of the grading of the Lie algebra $\mathfrak{g}=\bigoplus V_i$. To prove this fact, the scalar product on $G$ introduced in the definition of Popp's volume plays a key role.
