Regularized ζ△(1) for Polyhedra

Alexey Kokotov

doi:10.3842/SIGMA.2026.018

Regularized ζ△(1) for Polyhedra

2026-01-01

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DOI：10.3842/SIGMA.2026.018

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Let X be a compact polyhedral surface (a compact Riemann surface with flat conformal metric T having conical singularities). The zeta-function zeta triangle(s) of the Friedrichs Laplacian on X is meromorphic in C with a single simple pole at s = 1. We define reg zeta triangle(1) as lim (zeta triangle(s)-Area(X,T)). We derive an explicit expression for this spectral invariant s -> 1 4 pi(s-1) through the holomorphic invariants of the Riemann surface X and the (generalized) divisor of the conical points of the metric T. We study the asymptotics of reg zeta triangle(1) for the polyhedron obtained by sewing two other polyhedra along segments of small length. In addition, we calculate 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.

Keywords:

polyhedral surfaces

operator zeta-function

Robin mass

Regularized ζ△(1) for Polyhedra

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