Submanifolds in Lorentzian conformal geometry and CR-geometry
Conformal Lorentz geometry, in both its intrinsic and extrinsic aspects, has played an important role since the work of H. Weyl in general relativity and of W. Blaschke
and G. Thomsen in the classical geometries of Laguerre, Möbius and Lie. In the 1980s, the subject has been considered in the twistor approach to gravity by Penrose and Rindler and more recently in the study of cyclic cosmological models and in the regularization of the Kepler problem. In addition, the Fefferman fibration establishes a bridge between the world of Lorentzian conformal geometry and that of the Cauchy-Riemann geometry
We are interested in studying integrable geometric flows, variational problems, deformation and rigidity of low-dimensional sub-manifolds in the context of Lorentzian and Cauchy-Riemann geometries.
Using the method of moving frames and the theory of Lie transformation groups, one can investigate the local differential invariants and build the appropriate frame reductions. By means of adapted frames and the differential invariants one can rephrase the geometrical problems within the theory of exterior differential systems in involution and, ultimately, in the framework of integrable Hamiltonian systems.
The reformulation within the Hamiltonian context allows us to find families of explicit solutions. Their determination brings into play the theory of special functions. The search for possible closed solutions within a given family is often due to the rationality of the periods of abelian differentials on real cycles of elliptic or hyperelliptic curves. This last problem requires both analytical and algebraic geometry techniques.