Regularized $\zeta_{\Delta}(1)$ for Polyhedra

TL;DR

This paper introduces a regularized spectral invariant for the Laplacian on polyhedral surfaces, providing explicit formulas and asymptotic analysis, with applications to polyhedra and conical singularities.

Contribution

It derives an explicit expression for the regularized zeta function at one for polyhedral surfaces and studies its asymptotics under geometric degenerations.

Findings

Explicit formula for regularized $oldsymbol{ ext{zeta}_ riangle(1)}$ in terms of Riemann surface invariants.

Asymptotic behavior of the spectral invariant during polyhedral sewing.

Calculation of $oldsymbol{ ext{reg} ext{zeta}(1)}$ for specific Laplacian extensions on the tetrahedron.

Abstract

Let $X$ be a compact polyhedral surface (a compact Riemann surface with flat conformal metric $\mathfrak{T}$ having conical singularities). The $\zeta$-function $\zeta_\Delta(s)$ of the Friedrichs Laplacian on $X$ is meromorphic in ${\mathbb C}$ with a single simple pole at $s=1$. We define $\operatorname{reg}\zeta_\Delta(1)$ as $\lim\limits_{s\to 1} \bigl( \zeta_\Delta(s)-\frac{ {\rm Area}(X,\mathfrak{T}) }{4\pi(s-1)}\bigr)$. 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 $\operatorname{reg}\zeta_\Delta(1)$ for the polyhedron obtained by sewing two other polyhedra along segments of small length. In addition, we calculate $\operatorname{reg}\zeta(1)$ for a family of (non-Friedrichs) self-adjoint extensions of the Laplacian on the tetrahedron with all the conical angles equal to $\pi$.