Building three-dimensional differentiable manifolds numerically

•Numerical methods for building three-dimensional differentiable manifolds.•Constructs C1 reference metrics for manifolds built from triangulations.•Includes examples: Seifert-Weber, Hantzsche-Wendt, and Poincare dodecahedral spaces.•Pseudo-spectral methods for solving 2D and 3D biharmonic equations. A method is developed here for building differentiable three-dimensional manifolds on multicube structures. This method constructs a sequence of reference metrics that determine differentiable structures on the cubic regions that serve as non-overlapping coordinate charts on these manifolds. It uses solutions to the two- and three-dimensional biharmonic equations in a sequence of steps that increase the differentiability of the reference metrics across the interfaces between cubic regions. This method is algorithmic and has been implemented in a computer code that automatically generates these reference metrics. Examples of three-manifolds constructed in this way are presented here, including representatives from five of the eight Thurston geometrization classes, plus the well-known Hantzsche-Wendt, the Poincaré dodecahedral space, and the Seifert-Weber space.