3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry

TL;DR

This paper investigates the 3D combinatorial Yamabe flow in hyperbolic geometry, establishing conditions under which it converges to a ball packing with zero curvature and showing non-existence results for certain configurations.

Contribution

It proves existence and convergence of the combinatorial Yamabe flow under specific tetrahedral incident conditions in hyperbolic 3-manifolds.

Findings

Flow converges exponentially fast to the ball packing with zero curvature when conditions are met.

No such ball packing exists if the incident tetrahedra count is at most 22.

Existence depends on the number of tetrahedra incident to each vertex.

Abstract

We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist real or virtual ball packings with vanishing (extended) combinatorial scalar curvature, i.e. the (extended) solid angle at each vertex is equal to 4{\pi}. In this case, if such a ball packing is real, then the (extended) combinatorial Yamabe flow converges exponentially fast to that ball packing. Moreover, we prove that there is no real or virtual ball packing with vanishing (extended) combinatorial scaler curvature if the number of tetrahedra incident to each vertex is at most 22.