Moduli spaces of polygons and deformations of polyhedra with boundary

TL;DR

This paper proves a conjecture relating isometric realizations of polyhedral surfaces with boundary to isotropic subsets in moduli spaces, revealing geometric properties and solving a problem about spanning domes of curves in 3D.

Contribution

It establishes that all isometric realizations of a polyhedral surface with boundary form an isotropic subset in the moduli space, confirming a conjecture of Ian Agol.

Findings

Realizations form isotropic subsets in the moduli space

Boundaries of realizations form a Lagrangian subset for generic polyhedral disks

Provides a new solution to Kenyon's spanning dome problem

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 obtain a new solution to the problem of Richard Kenyon about spanning domes of piecewise linear curves comprised of unit intervals in R^3.