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).