Variational Problems for Fefferman Hypersurface Measure and Volume-Preserving CR Invariants

We derive Euler’s equation for the isoperimetric problem and the extremal hypersurface problem associated with Fefferman hypersurface measure for strongly pseudoconvex real hypersurfaces in ℂ². We use volume-preserving CR invariants to show that the only “torsion-free” solutions of Euler’s equation for the isoperimetric problem are the volume-preserving images of spheres.