Curating the Mercosul Biennial in Brazil: An Interview with the Blanton's Gabriel Pérez-Barreiro

Latin American Studie

Drive Thru—An Interview with Artist Matheus Rocha-Pitta

Latin American Studie

A Polyakov formula for sectors

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.Comment: 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 Analysi