Real Hypersurfaces in Unimodular Complex Surfaces

A unimodular complex surface is a complex 2-manifoldX endowed with a holomorphic volume form Υ. A strictly pseudoconvex real hypersurface in X inherits not only a CR-structure but a canonical coframing as well. In this article, this canonical coframing is defined, its invariants are discussed and interpreted geometrically, and its basic properties are studied. A natural evolution equation for strictly pseudoconvex real hypersurfaces in unimodular complex surfaces is defined, some of its properties are discussed, and several examples are computed. It is shown that a real-analytic 3-manifold endowed with a real-analytic coframing satisfying the structure equations can be real-analytically embedded as a pseudoconvex hypersurface in a unimodular complex surface in such a way that the induced canonical coframing is the given one. Moreover, this embedding is essentially unique up to unimodular biholomorphism. The locally homogeneous examples are determined and used to illustrate various features of the geometry of the induced structure on the hypersurface. The invariants of the underlying CR-structure are expressed in terms of the invariants of the coframing.