Simple Numerical Solutions to the Einstein Constraints on Various Three-Manifolds

TL;DR

This paper develops and computes numerical solutions to Einstein constraint equations on various three-manifolds with complex topologies, including constant and non-constant mean curvature cases, linking to the Yamabe problem.

Contribution

It introduces a straightforward numerical approach to solve Einstein constraints on diverse three-manifolds with non-trivial topology, expanding the set of known solutions.

Findings

Constructed solutions on manifolds from three Thurston classes

Found constant mean curvature solutions related to the Yamabe problem

Demonstrated numerical methods for complex topologies

Abstract

Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated non-constant mean curvature solution are computed on example manifolds from three of the eight Thursten geometrization classes. The constant mean curvature solutions found here are also solutions to the Yamabe problem that transforms a geometry into one with constant scalar curvature.