Building Three-Dimensional Differentiable Manifolds Numerically

TL;DR

This paper introduces an algorithmic method for constructing differentiable three-dimensional manifolds using multicube structures, solving biharmonic equations to ensure smoothness across interfaces, and demonstrates its application to various complex manifolds.

Contribution

The paper presents a novel numerical approach for building differentiable 3D manifolds with multicube structures, including an automated code implementation and examples from multiple geometrization classes.

Findings

Successfully constructed manifolds from five Thurston classes.

Automated code generates reference metrics with increasing differentiability.

Examples include well-known spaces like Poincare dodecahedral and Seifert-Weber.

Abstract

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 Poincare dodecahedral space, and the Seifert-Weber space.