Journal article, 2018

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.

Sector

Spectrum

Heat kernel

Conical singularity

Zeta-regularized determinant

Laplacian

Rectangle

Polyakov formula

Angular variation

University of Luxembourg

Chalmers, Mathematical Sciences, Analysis and Probability Theory

University of Gothenburg

1050-6926 (ISSN)

Vol. 28 2 1773-1839Algebra and Logic

Geometry

Mathematical Analysis

10.1007/s12220-017-9888-y