论文标题
几乎双曲线3个manifolds的Ruelle Zeta功能为零
The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds
论文作者
论文摘要
我们表明，对于带有betti数字$ b_1 $的紧凑型双曲线3个manifold $σ$的通用共形度度扰动，零Zeta功能的消失令等于$ 4-B_1 $，而在毛额情况下，它等于4-2b_1 $ $ 4-2B_1 $。这与二维情况相反，即消失的顺序是拓扑不变的。该证明使用微局部方法来动态ZETA函数，从而在双曲线情况下在0处对pollicott-ruelle谐振差异形式进行了几何描述，并使用第一个变化来进行扰动。为了表明第一个变化通常是非零的，我们引入了一种新的身份，将谐振和核心2形的产品的推送在球体束$sς$上，并用$σ$上的谐波1形。
We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $Σ$ with Betti number $b_1$, the order of vanishing of the Ruelle zeta function at zero equals $4-b_1$, while in the hyperbolic case it is equal to $4-2b_1$. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott-Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle $SΣ$ with harmonic 1-forms on $Σ$.
