Bergman Logarithmically Flat and Obstruction Flat Hypersurfaces and Their CR Structures

The purpose of this chapter is twofold. We first survey recent developments on strictly pseudoconvex Bergman logarithmically flat and obstruction flat hypersurfaces in complex manifolds and provide some new insight into the CR geometry of such hypersurfaces. Then we establish some new results for the two notions of flatness. Among other things, we prove there exists a family \(\mathcal {F}\) , parameterized by the real numbers (and, hence, is uncountably infinite), of (noncompact) real analytic homogeneous CR hypersurfaces in \(\mathbb {C}^{n+1}, n \geq 2,\) with transverse symmetry that are all obstruction flat and Bergman logarithmically flat, and such that, moreover, the CR hypersurfaces in \(\mathcal {F}\) have mutually distinct local CR structures.