发表机构
Steklov Mathematical Institute of Russian Academy of Sciences; Concordia University（俄罗斯科学院列别捷夫数学研究所; 康考迪亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对带边界的亏格$\frak{g}$光滑可定向2维黎曼流形，利用其二重曲面的全纯微分周期，给出了Dirichlet-to-Neumann映射的zeta正则行列式相关共形不变量的初等表达式，建立了长度谱与全纯微分周期的联系。
AI 中文摘要
设$(M,g)$为亏格$\frak{g}$的光滑可定向2维黎曼流形，带有黎曼度量$g$和连通边界$\Gamma$。记$\frak{Lambda}$为$\frak{Gamma}$上的Dirichlet-to-Neumann映射，$\frak{det}_\frak{zeta}(\frak{Lambda})$为其（排除零模的修正）$\frak{zeta}$正则行列式。已知量$\frak{det}_\frak{zeta}(\frak{Lambda})/|\frak{Gamma}|$（$|\frak{Gamma}|$为$\frak{Gamma}$的长度）是共形不变量。Edward和Wu（文献[EV]）证明，当$\frak{g}=0$时该不变量等于1；当$\frak{g}>0$时，Guillarmou和Guillopé（文献[Guillarmou]）通过来自$(M,g)$共形类的两个负常曲率曲面的Ruellezeta函数和Selberg zeta函数，给出了该不变量的两个显式表达式：一个是无限体积且完备的，另一个带有测地边界。本文仅利用$M$的二重曲面$2M$上全纯微分的周期，给出Guillarmou和Guillopé公式的初等对应版本。本文方法基于$M$的Hilbert变换性质（文献[B,HilbKor]）及伪微分算子行列式的Kontsevich-Vishik-Friedlander-Guillemin正则化（文献[KV,Ww,F]），特别建立了（单值化的）$M$、$2M$的长度谱与$2M$上全纯微分周期之间的联系。
英文摘要
Let $(M,g)$ be a smooth orientable $2d$ Riemannian manifold of genus $\mathfrak{g}$ with Riemannian metric $g$ and connected boundary $Γ$. Let $Λ$ be the Dirichlet-to-Neumann map on $Γ$ and let ${\rm det}_ζ(Λ)$ be its (modified, i. e. with zero mode excluded) $ζ$-regularized determinant. It is well-known that the quantity ${\rm det}_ζ(Λ)/|Γ|$ (where $|Γ|$ is the length of $Γ$) is a conformal invariant. It was shown by Edward and Wu (\cite{EV}) that this invariant equals one for $\mathfrak{g}=0$; in the case $\mathfrak{g}>0$ Guillarmou and Guillopé \cite{Guillarmou} found two explicit expressions for this invariant through the Rouelle and (respectively) the Selberg zeta-functions of the two surfaces of negative constant curvature from the conformal class of $(M,g)$: one is of infinite volume and complete whereas another has geodesic boundary. We present an elementary counterpart of the formulae of Guillarmou and Guillopé using the periods of holomorphic differentials on the double $2M$ of $M$ only. Our approach is based on the properties of the Hilbert transform of $M$ \cite{B,HilbKor} and the Kontsevich-Vishik-Friedlander-Guillemin regularization of the determinants of pseudodifferenial operators \cite{KV,Ww,F}. In particular, a connection between the length spectra of (uniformized) $M$, $2M$ and the periods of holomorphic differentials on $2M$ is established.
