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Moduli spaces of polygons and deformations of polyhedra with boundary

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Abstract

We prove a conjecture of Ian Agol: all isometric realizations of a polyhedral surface with boundary sweep out an isotropic subset in the Kapovich-Millson moduli space of polygons isomorphic to the boundary. For a generic polyhedral disk we show that boundaries of its isometric realizations make up a Lagrangian subset. As an application of this result, we conclude that a generic equilateral polygon cannot be domed (in the sense of a problem of Kenyon, Glazyrin and Pak).

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  1. S. Anan’in

Authors and Affiliations

  1. Universidade de São Paulo, São Carlos, Brazil

    Sasha Anan’in

  2. Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

    Dmitrii Korshunov

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  1. Sasha Anan’in
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  2. Dmitrii Korshunov
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Correspondence to Dmitrii Korshunov.

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To Fedor Bogomolov with all our admiration.

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Sasha Anan’in is deceased.

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Anan’in, S., Korshunov, D. Moduli spaces of polygons and deformations of polyhedra with boundary. Geom Dedicata 218, 26 (2024). https://doi.org/10.1007/s10711-023-00834-7

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  • DOI: https://doi.org/10.1007/s10711-023-00834-7

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