A Polyakov formula for sectors
Abstract
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.
Keywords
Cite
@article{arxiv.1411.7894,
title = {A Polyakov formula for sectors},
author = {Clara L. Aldana and Julie Rowlett},
journal= {arXiv preprint arXiv:1411.7894},
year = {2020}
}
Comments
51 pages, 2 figures. Major modification of Lemma 4, it was revised and corrected. Other small misprints were corrected. Accepted for publication in The Journal of Geometric Analysis