Orateur
Guillaume Tahar
Description
On a topological sphere endowed with a flat metric with conical singularities, the curvature gap quantifies the obstruction to realize a partition of the set of conical singularities into two sets of equal total angle defect. Unless its curvature gap is equal to zero, such a flat sphere cannot contain a simple closed geodesic. Drawing on the Delaunay decompositions of flat surfaces, we give a quantitative generalization of the latter statement. We prove that when the curvature gap is nonzero, there is an explicit upper bound on the lengths of simple geodesic trajectories in the flat sphere. If time allows, we will give extensions of this result to affine structures on punctured spheres. This is a work in collaboration with Kai Fu.
