We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw–Sommerfeld’s heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.

中文翻译：

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扇区的波利亚科夫公式。

我们考虑有限面积凸欧几里得圆扇形。我们证明了变分波利亚科夫公式，该公式显示了拉普拉斯算子的 zeta 正则化行列式如何随张角变化。改变角度对应于在原点处具有对数奇点的共形因子方向上的共形变形。我们明确计算来自该部门不同部分对该公式的所有贡献。在此过程中，我们使用 Carslaw-Sommerfeld 热核获得了无限面积扇区上热核的显式表达式。我们还计算了单位面积矩形域的 zeta 正则化行列式，并证明它是由正方形唯一最大化的。