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arXiv · 2504.17652

On an infinitesimal Polyakov formula for genus zero polyhedra

Alexey Kokotov·Dmitrii Korikov
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Abstract

Let $X$ be a genus zero compact polyhedral surface (the Riemann sphere equipped with a flat conical metric $m$). We derive the variational formulas for the determinant of the Laplacian, ${\rm det}\,Δ^m$, on $X$ under infinitesimal variations of the positions of the conical points and the conical angles (i. e. infinitesimal variations of $X$ in the class of polyhedra with the same number of vertices). Besides having an independent interest, this derivation may serve as a somewhat belated mathematical counterpart of the well-known heuristic calculation of ${\rm det}\,Δ^m$ performed by Aurell and Salomonson in the 90-s.

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Alexey Kokotov, Dmitrii Korikov. 2025-12-16. On an infinitesimal Polyakov formula for genus zero polyhedra. https://arxiv.org/abs/2504.17652

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