Polyhedral surfaces in flat (2+1)-spacetimes and balanced cellulations on hyperbolic surfaces

21.11.2023 09:45 - 11:15

Roman Prosanov (University of Vienna)

Alexandrov showed that every flat metric on the 2-sphere with cone-singularities of positive curvature admits a unique isometric realization at the boundary of a convex polyhedron in the Euclidean 3-space. The Gauss map allows to translate this statement to a purely 2-dimensional problem of finding a balanced geodesic cellulation on the round 2-sphere, from which the flat metric can be easily recovered. I will discuss a generalization of this result to surfaces of higher genus. We consider flat metrics with cone-singularities of negative curvature and we look for celluations on hyperbolic surfaces. The respective isometric realization problem takes place in so-called flat GHMC (2+1)-spacetimes, which have interesting relation to Teichmüller theory. Further, I will discuss the problem of simultaneous realization of a pair of such metrics. This is a joint work with François Fillastre.

A. Keating, B. Szendroi, V. Vertesi

SR 7, 2. OG., OMP 1