The deformation space of Delaunay triangulations of the sphere

Yanwen Luo, Tianqi Wu and Xiaoping Zhu

Vol. 323 (2023), No. 1, 115–127

DOI: 10.2140/pjm.2023.323.115

Abstract

We determine the topology of the spaces of convex polyhedra inscribed in the unit $2$-sphere and the spaces of strictly Delaunay geodesic triangulations of the unit $2$-sphere. These spaces can be regarded as discretized groups of diffeomorphisms of the unit $2$-sphere. Hence, it is natural to conjecture that these spaces have the same homotopy types as those of their smooth counterparts. The main result of this paper confirms this conjecture for the unit $2$-sphere. It follows from an observation on the variational principles on triangulated surfaces developed by I. Rivin.

On the contrary, the similar conjecture does not hold in the cases of flat tori and convex polygons. We will construct simple examples of flat tori and convex polygons such that the corresponding spaces of Delaunay geodesic triangulations are not connected.

Keywords

geodesic triangulations, angle structures, Delaunay triangulations, spaces of embeddings

Mathematical Subject Classification

Primary: 05C62, 58D10

Milestones

Received: 22 April 2022

Revised: 21 November 2022

Accepted: 7 March 2023

Published: 29 May 2023