A family of compact strictly pseudoconvex hypersurfaces in C^2 without umbilical points

Abstract

We prove the following: For ϵ>0, let Dϵ be the bounded strictly pseudoconvex domain in ℂ2 given by (log|z|)2+(log|w|)2<ϵ2. The boundary Mϵ:=∂Dϵ⊂ℂ2 is a compact strictly pseudoconvex CR manifold without umbilical points. This resolves a long-standing question in complex analysis that goes back to the work of S.-S. Chern and J. K. Moser in 1974.