The Gauss Map and 2+1 Gravity

TL;DR

This paper demonstrates that the Gauss map of constant mean curvature surfaces in Minkowski space is harmonic and uses this to connect geometric and physical variables in 2+1 gravity, enabling a new approach to solving the theory.

Contribution

It establishes the harmonicity of the Gauss map for such surfaces and introduces a canonical transform linking ADM and holonomy variables in 2+1 gravity.

Findings

Gauss map of constant mean curvature surfaces is harmonic

A canonical transform between ADM and holonomy variables is constructed

Potential to solve evolution equations without explicit constraint solutions

Abstract

We prove that the Gauss map of a surface of constant mean curvature embedded in Minkowski space is harmonic. This fact will then be used to study 2+1 gravity for surfaces of genus higher than one. By considering the energy of the Gauss map, a canonical transform between the ADM reduced variables and holonomy variables can be constructed. This allows one to solve (in principle) for the evolution in the ADM variables without having to explicitly solve the constraints first.