Building Three-Dimensional Differentiable Manifolds Numerically II: Limitations

TL;DR

The paper extends the method of constructing reference metrics on 3-manifolds by relaxing the dihedral-angle uniformity constraint and expanding the class of triangulations considered. It formulates a set of edge-sum and vertex- continuity constraints on dihedral angles to obtain a globally $C^0$ metric that can be smoothed to $C^1$, and uses a BFGS-based minimization of the constraint norm $||C||$ with a practical tolerance. Results show only a small fraction of the Regina catalog multicubes admit constraint-satisfying dihedral angles, yielding 23 uniformly constrained and 80 non-uniformly constrained cases, of which 17 boldface entries admit non-singular $C^1$ metrics. The findings reveal substantial limitations of the existing approach and suggest that achieving broader applicability will require fundamental changes to how triangulations are transformed into multicube coverings. The work thus clarifies the current boundary of numerically constructing differentiable structures on 3-manifolds and points to directions for overcoming these barriers.

Abstract

Methods were developed in Ref. [1] for constructing reference metrics (and from them differentiable structures) on three-dimensional manifolds with topologies specified by suitable triangulations. This note generalizes those methods by expanding the class of suitable triangulations, significantly increasing the number of manifolds to which these methods apply. These new results show that this expanded class of triangulations is still a small subset of all possible triangulations. This demonstrates that fundamental changes to these methods are needed to further expand the collection of manifolds on which differentiable structures can be constructed numerically.

Figures (1)

• Figure 1: Figure shows the intersection between the corner of a cubic region and a small sphere centered on one of the vertices of that cube. This sphere is depicted as the dashed (blue) curve; the intersection of this cubic region with the sphere is a spherical triangle shown as solid (red) curves; the solid (black) straight lines are the edges of the cube. The dihedral angles $\psi_{\{\alpha\beta\}}$ between the cube faces are also the angles of this spherical triangle. The vertex angles, $\theta_{\{\alpha\}\{\beta\gamma\}}$, are the angles between the edges of the cube, and are also the arc lengths of the sides of this spherical triangle.