Minimal surfaces and particles in 3-manifolds
Kirill Krasnov, Jean-Marc Schlenker
TL;DR
This work develops a unifying framework in which 3-manifolds of constant curvature (hyperbolic, AdS, dS, Minkowski) — including singular cone-manifolds representing particles — are encoded by data on a canonical surface via minimal or constant-mean-curvature (CMC) foliations. Central to the framework is the identification of moduli spaces with (open subsets of) the cotangent bundle T^*T_g (or T^*T_{g,n}) of Teichmüller space, with holomorphic quadratic differentials (HQDs) serving as the natural complex-analytic data associated to second fundamental forms. The paper establishes precise parameterizations (including two distinct AdS descriptions due to Mess and via maximal surfaces) and extends them to singular settings, showing that, in AdS (and in many other cases), the gravity symplectic form matches the canonical symplectic form on T^*T_g. It also analyzes analytic continuations between settings, geometric dualities (e.g., via equidistant foliations and Fock-type metrics), and obtains explicit metric descriptions (e.g., the Fock metric) that enable explicit reconstruction of the 3-manifolds from Teichmüller data. Overall, the results articulate a coherent picture in which Teichmüller-theoretic data govern 3d gravity moduli across several geometries, with extensions to cone-manifolds and implications for quantum gravity through the tractability of T^*T_g data.
Abstract
We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic, anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these manifolds admit ``nice'' foliations and explicit metrics, and whether the space of these metrics has a simple description in terms of Teichmüller theory. In the hyperbolic settings both questions have positive answers for a certain subset of the quasi-Fuchsian manifolds: those containing a closed surface with principal curvatures at most 1. We show that this subset is parameterized by an open domain of the cotangent bundle of Teichmüller space. These results are extended to ``quasi-Fuchsian'' manifolds with conical singularities along infinite lines, known in the physics literature as ``massive, spin-less particles''. Things work better for globally hyperbolic anti-de Sitter manifolds: the parameterization by the cotangent of Teichmüller space works for all manifolds. There is another description of this moduli space as the product two copies of Teichmüller space due to Mess. Using the maximal surface description, we propose a new parameterization by two copies of Teichmüller space, alternative to that of Mess, and extend all the results to manifolds with conical singularities along time-like lines. Similar results are obtained for de Sitter or Minkowski manifolds. Finally, for all four settings, we show that the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichmüller space is the same as the 3-dimensional gravity one.