A Polyakov formula for angular sectors.

In this talk, I present a variational Polyakov formula for the Dirichlet Laplacian on finite area convex sectors in the Euclidean plane. This formula shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle of the sector. We use conformal transformations to differentiate the determinant with respect to the opening angle. We obtain a closed formula for the derivative of the determinant with respect to the angle of the sector using Carslaw-Sommerfeld heat kernel for the infinite sector. The results presented in the talk are in collaboration with Julie Rowlett.