Invariants of Transformation Groups Acting on Real Hypersurfaces of Complex Spaces.

Abstract

We find invariants of real hypersurfaces of $mathbb{C}^{2}$ with respect to the group of volume-preserving biholomorphic transformations and with respect to the group of projective transformations. We use these invariants to solve problems associated with each group. In particular, we investigate the isoperimetric problem associated with Fefferman hypersurface measure. We use the volume-preserving invariants to show that the only torsion-free solutions of the associated Euler's equation are the volume-preserving images of spheres. We later use projective invariants to classify all hypersurfaces in $mathbb{C}^{2}$ that are projectively homogeneous with symmetry algebras of dimension larger than 3.