发表机构
Concordia University; Steklov Mathematical Institute of Russian Academy of Sciences; Leonhard Euler International Mathematical Institute in Saint Petersburg（康科迪亚大学; 俄罗斯科学院列别捷夫数学研究所; 圣彼得堡列昂哈德·欧拉国际数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过引入LC-图推导了Dirichlet-to-Neumann共形不变量的显式初等公式，其结果与Wentworth的渐近结果一致，确定了积分常数。
AI 中文摘要
利用Mandelstam-Giddings-Wolpert图的类比，我们将复平面$\boldsymbol{\text{C}}$中任意连通性的有界区域的共形类的新典范代表引入为带有测地边界的平坦锥面（即“截断光锥图”，下文简称LC-图）。这些图的空间可配备自然坐标。我们推导了关于这些坐标的LC-图上Dirichlet边值问题行列式的变分公式。接着，通过LC-图的Schottky双，利用Burghelea-Friedlander-Kappeler公式以及全纯微分模空间上拉普拉斯算子行列式的已知变分公式，我们得到了Dirichlet-to-Neumann（DN）共形不变量$\frac{\text{det}\boldsymbol{\text{Λ}}}{|\boldsymbol{\text{Γ}}|}$的显式计算（其中$\boldsymbol{\text{Λ}}$是多连通区域边界$\boldsymbol{\text{Γ}}$上的DN算子，$|\boldsymbol{\text{Γ}}|$为边界长度）。所得公式仅使用区域Schottky双的周期，与Guillarmou和Guillopé通过Ruelle和Selberg ζ函数表达DN不变量的公式形成了显著初等的对应。当除一个边界分量外的所有边界分量收缩时，我们的公式与Wentworth关于DN不变量渐近性的最新结果一致，我们利用该结果确定了公式中未确定的积分常数。
英文摘要
Using an analogue of the Mandelstam-Giddings-Wolpert diagrams, we introduce a new canonical representative of the conformal class of a bounded domain of arbitrary connectivity in $\mathbb{C}$ as a flat conical surface with geodesic boundary (a "truncated light-cone diagram", simply LC-diagram in the sequel). The space of diagrams carries natural coordinates. We derive variational formulas for the determinant of the Dirichlet Laplacian on a LC-diagram with respect to these coordinates. Then, passing to the Schottky double of the LC-diagram, making use of the Burghelea-Friedlander-Kappeler formula and the known variational formulas for determinants of Laplacians on the moduli space of holomorphic differentials, we compute the Dirichlet-to-Neumann (DN) conformal invariant $\frac{{\rm det}'Λ}{|Γ|}$ (here $Λ$ is the DN operator on the boundary, $Γ$, of a multiply connected domain and $|Γ|$ is the length of the boundary). The resulting formula (which uses the periods of the Schottky double of the domain only) provides an elementary counterpart to the formulas of Guillarmou and Guillopé who had expressed the DN invariant through the Ruelle and Selberg zeta-functions. Our formula agrees with the recent result of Wentworth on the asymptotics of the DN invariant as all but one boundary components shrink; we have used this result to fix the undetermined constant of integration in our formula. As a corollary, we derive an explicit formula for the determinant of the Dirichlet Laplacian in a multiply connected domain.
CommentsRemarks 1.1, 1.2, 4.2 and 5.1 are added; Proof of relation (55) is improved
