Regularized $ζ_Δ(1)$ for polyhedra
Alexey Kokotov, Dmitrii Korikov
TL;DR
This work develops a robust framework to compute the regularized value ${\it reg}\,\zeta_\Delta(1)$ for polyhedral surfaces endowed with a flat Troyanov metric, expressing it explicitly via holomorphic data of the underlying Riemann surface and the conical divisor. It extends Steiner's short-time heat-kernel analysis to cone singularities by constructing a conical heat kernel parametrix and evaluating the Robin mass through a Verlinde-type formula, yielding a concrete integral formula for ${\it reg}\,\zeta_\Delta(1)$. The paper also analyzes pinch degenerations, establishing an additive law for ${\it reg}\,\zeta_\Delta(1)$ after removing the zero mode and the first vanishing eigenvalue, and provides exact computations for self-adjoint extensions on a tetrahedron with conical angles $\pi$ in terms of modular data via Dedekind eta functions. These results connect spectral invariants of polyhedral surfaces to the moduli of Riemann surfaces and offer precise closed-form formulas for notable degenerate geometries. Overall, the approach unifies geometric, analytic, and algebro-geometric techniques to elucidate spectral properties of polyhedral Laplacians.
Abstract
Let $X$ be a compact polyhedral surface (a compact Riemann surface with flat conformal metric $\mathfrak{T}$ having conical singularities). The zeta-function $ζ_Δ(s)$ of the Friedrichs Laplacian on $X$ is meromorphic in ${\mathbb C}$ with only (simple) pole at $s=1$. We define ${\it reg\,}ζ_Δ(1)$ as $$\lim\limits_{s\to 1} \left( ζ_Δ(s)-\frac{ {\rm Area\,}(X,\mathfrak{T}) }{4π(s-1)}\right).$$ 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 ${\it reg\,}ζ_Δ(1)$ for the polyhedron obtained by sewing two other polyhedra along segments of small length. In addition, we calculate ${\it reg\,}ζ(1)$ for a family of (non-Friedrichs) self-adjoint extensions of the Laplacian on the tetrahedron with all the conical angles equal to $π$.