Symmetries and Mapping in Real and Complex Geometry

Project title: Symmetries and Mappings in Real and Complex Geometry

Principal investigator: A/Prof. Gerd Schmalz (UNE)

Other participants: A/Prof. Vladimir Ejov (Flinders), Prof. Dmitri Alekseevsky (Moscow), A/Prof. Martin Kolar (Brno), Dr Alessandro Ottazzi (UNSW)

Funding body: Australian Research Council (2013-2016)

Project description:
This project is focused on a wide range of interrelated areas of Differential and Cauchy-Riemann geometry and Geometric Analysis.  Cauchy-Riemann-geometry has its roots in the study of the geometric properties of boundaries of domains in complex space. In recent time it has received much attention in Differential and Riemannian geometry. On the one hand, Cauchy-Riemann manifolds serve as test objects for the general machinery of parabolic geometry, on the other hand, there is a deep relation between CR geometry and Riemannian, pseudo-Riemannian and conformal geometry, which was established by Fields prize winner (and 2016 Wolf prize winner) Charles Fefferman.  Particular aims of the project are: studying mappings and embeddings of CR-manifolds into other CR-manifolds and complex space, exploring the relation between CR-manifolds and its generalisations with shear-free Lorentzian geometry, finding new classes of homogeneous CR-manifolds and classifying flat Sasakian manifolds.

Related publications:

  1. OTTAZZI, A., SCHMALZ, G., Singular multicontact structures. J. Math. Anal. Appl. 443:2, 1220-1231, 2016.
  2. EJOV, V., SCHMALZ, G., The zero curvature equation for rigid CR-manifolds, Complex Variables and Elliptic equations, 61:4, 443-447, 2016.
  3. JOO, Ch.-J., KIM, K.-T., SCHMALZ, G.,  On the generalization of Forelli's theorem, Mathematische Annalen, 365:3-4, 1187-1200, 2016.
  4. EJOV, V., SCHMALZ, G., Explicit description of spherical rigid hypersurfaces in C^2,  Complex Analysis and its Synergies,  1.1,  Paper no. 2, 10 pp, 2015.
  5. DE HOOG, F., SCHMALZ, G., GUREYEV, T.E., An uncertainty inequality, Appl. Math. Lett., 38, 84-86, 2014.
  6. EJOV, V., SCHMALZ, G., Spherical rigid hypersurfaces C^2, Differential Geom. Appl., 33, suppl., 267-271, 2014.
  7. JOO, Ch.-J., KIM, K.-T., SCHMALZ, G., A generalization of Forelli's theorem, Mathematische Annalen, 355, no. 3, 1171-1176, 2013.
  8. EJOV, V., Kolar, M., SCHMALZ, G., Normal forms and symmetries of real hypersurfaces of finite type in C^2, Indiana Univ. Math. J., 62, no. 1, 1-32, 2013.
  9. SCHMALZ, G., SLOVAK, J., Free CR distributions, Cent. Eur. J. Math., 10,  1896-1913, 2012.