We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a closed hyperbolic surface is connected. We give an upper bound on the number of edge flips that are necessary to transform any geometric triangulation on such a surface into a Delaunay triangulation. We conduct experiments in genus 2 and conjecture that the above bound is largely overestimated.
This is joint work with Vincent Despré, Loïc Dubois, Benedit Kolbe, and Jean-Marc Schlenker.