Angle structures on pseudo 3-manifolds
Huabin Ge, Longsong Jia, Faze Zhang
TL;DR
The paper addresses when angle structures with area-curvature $(A,\kappa)$ exist on triangulated compact pseudo 3-manifolds, showing that a semi-angle structure with $A\le 0$ and the strict inequality $\chi^*(s)<\chi^{(A,\kappa)}(s)$ for quadrilateral-positive cases suffices (and is equivalent to) the existence of such an angle structure. It leverages Farkas's lemma to convert the problem into a linear-inequality duality, and uses the $\chi^{(A,\kappa)}$ framework to connect combinatorial data to topology. A key corollary is that every compact hyperbolic 3-manifold with totally geodesic boundary admits an angle structure, via Kojima decomposition and a deformation argument to achieve negative area curvature. The results also yield topological consequences, showing absence of essential spheres, tori, Klein bottles, disks, and annuli under the angle-structure regime and enabling 0-efficient triangulations. Overall, the work connects combinatorial angle data to hyperbolic geometry and 3-manifold topology, providing constructive criteria and implications for geometric structures on triangulated spaces.
Abstract
It is still not known whether a hyperbolic 3-manifold admits an angle structure or not. We consider angle structures with area-curvature on triangulated pseudo 3-manifolds M in this article. A suficient and necessary condition for the existence of such angle structures is established. As a consequence, any compact hyperbolic 3-manifold with totally geodesic boundary admits an angle structure. We also derive certain topological information of M from the existence of such angle structures.