多边形的模空间和有边界多面体的变形

Pub Date : 2023-12-13 DOI:10.1007/s10711-023-00834-7

Sasha Anan’in, Dmitrii Korshunov

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引用次数: 0

摘要

我们证明了伊恩-阿戈尔（Ian Agol）的一个猜想：有边界的多面体表面的所有等距变现都会在与边界同构的多边形的卡波维奇-米尔森模空间中扫出一个各向同性子集。对于一般的多面体圆盘，我们证明其等距实现的边界构成了一个拉格朗日子集。作为这一结果的应用，我们得出结论：一般等边多边形不可能是穹顶的（在肯扬、格拉兹林和帕克问题的意义上）。

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Moduli spaces of polygons and deformations of polyhedra with boundary

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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