A family of compact strictly pseudoconvex hypersurfaces in ${\mathbb{C}}^2$ without umbilical points

Abstract

We prove the following: For $\epsilon >0$, let $D_{\epsilon }$ be the bounded strictly pseudoconvex domain in ${\mathbb{C}}^2$ given by \begin{equation*} (\log|z|)^2+(\log|w|)^2<ε^2. \end{equation*} The boundary $M_{\epsilon }:=\partial D_{\epsilon }\subset {\mathbb{C}}^2$ 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.

Cited by in corpus (2)

• On the Chern-Moser-Weyl tensor of real hypersurfaces
• The CR umbilical locus of a real ellipsoid in ${\mathbb{C}}^2$