Regularized $\hat{I}{\P }_{\hat{I}}”(1)$ for Polyhedra

Abstract

Let $X$ be a compact polyhedral surface (a compact Riemann surface with flat conformal metric $\mathfrak{T}$ having conical singularities). The $\hat{I}\P$-function $\hat{I}{\P }_{\hat{I}}”(s)$ of the Friedrichs Laplacian on $X$ is meromorphic in $\mathbb{C}$ with a single simple pole at $s=1$. We define $\mathrm{reg}\hat{I}{\P }_{\hat{I}}”(1)$ as $\underset{s\to 1}{\lim }(\hat{I}{\P }_{\hat{I}}”(s)-\frac{\mathrm{Area}(X,\mathfrak{T})}{4\ddot{I}€(s-1)})$. We derive an explicit expression for this spectral invariant through the holomorphic invariants of the Riemann surface $X$ and the (generalized) divisor of the conical points of the metric $\mathfrak{T}$. We study the asymptotics of $\mathrm{reg}\hat{I}{\P }_{\hat{I}}”(1)$ for the polyhedron obtained by sewing two other polyhedra along segments of small length. In addition, we calculate $\mathrm{reg}\hat{I}\P (1)$ for a family of (non-Friedrichs) self-adjoint extensions of the Laplacian on the tetrahedron with all the conical angles equal to $\ddot{I}€$.