Teichmüller spaces parametrise hyperbolic structures on compact surfaces, punctures and conical singularities complexifying the picture. Some constructions used by R.C. Penner in the late 1980s as convenient tools are avatar of deep links between such Teichmüller spaces and flat singular spacetimes. A point in a Teichmüller space can be seen as space-like slice in a spacetime containing particules. The main goal of this study is to extend the known correspondance obtained from Mess-Bonsante and Mess-Barbot classification theorems to more complex situations.
Alexandrov proved in the early 20th century that a metric on the sphere locally isometric to the boundary of a convex polyedra can be realised globally and in an essentially unique way as the boundary of a convex polyedra. The theorem of Alexandrov dealt with Euclidean geometry on the sphere but has been extend to include more complex compact surfaces and other geometries, namely hyperbolic and spherical geometries. In higher genus, the conical singularities cannot, even locally, be embedded into the euclidean space, but can be embedded into Minkowski space. The last cases of Alexandrov type theorems have been proved in the late 2000s in non constructive ways, then some cases have been solved constructively.
The case of my current interest is genus at least two surfaces, their universal cover naturally embed into Minkowski space as the boundary of a polyedra if the conical singularities are all of negative curvature. Constructive methods can be applied to prove Alexandrov theorem in this case, first proved by François Fillastre, in order to give a new proof and to extends the result. We aim at removing the negative singularity hypothesis by considering singular spacetimes instead of Minkowski space.
Thomas Buchert developped a framework both numerical and theoretical that explains parts of the Dark matter/energy conundrum as coming from inhomogeneities in the universe. The main idea being to consider the exact evolution of averaged instead of pointwise physical quantities, differential geometry integral formulas give some informations on large scale properties of inhomogeneous dark matter/energy. One of the simplest, the Gauss-Bonnet-Chern gives some interesting results, theoretical and numerical.