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 corner. As an application of the method, we obtain an analogues Polyakov formula for a surface with one conical singularity. We compute the zeta-regularized determinant of rectangular domains of fixed area and prove that it is uniquely maximized by the square.