Polyakov conjecture and 2+1 dimensional gravity coupled to particles

TL;DR

This paper proves Polyakov's conjecture concerning the auxiliary parameters in an SU(1,1) Riemann-Hilbert problem, linking it to the uniformization of punctured spheres and the Hamiltonian structure of 2+1 dimensional gravity.

Contribution

It provides a proof of Polyakov's conjecture for elliptic singularities, connecting complex analysis with the Hamiltonian formulation of 2+1 gravity.

Findings

Proof of Polyakov's conjecture for elliptic singularities

Connection between Riemann-Hilbert problem and uniformization

Implications for the Hamiltonian structure of 2+1 gravity

Abstract

A proof is given of Polyakov conjecture about the auxiliary parameters of the SU(1,1) Riemann-Hilbert problem for general elliptic singularities. Such a result is related to the uniformization of the the sphere punctured by n conical defects. Its relevance to the hamiltonian structure of 2+1 dimensional gravity in the maximally slicing gauge is stressed.