Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 018, 31 pages      arXiv:2502.03351      https://doi.org/10.3842/SIGMA.2026.018

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

Alexey Yu. Kokotov and Dmitrii V. Korikov
Department of Mathematics and Statistics, Concordia University, 1400 De Maisonneuve Blvd. W., Montreal, QC H3G 1M8, Canada

Received May 13, 2025, in final form February 03, 2026; Published online February 25, 2026

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 $\pi$.

Key words: polyhedral surfaces; operator $\zeta$-function; Robin mass.

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