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

TL;DR

This paper constructs a family of compact, strictly pseudoconvex hypersurfaces in complex two-space that lack umbilical points, answering a longstanding question in complex analysis about their existence.

Contribution

It provides explicit examples of compact strictly pseudoconvex CR manifolds without umbilical points, resolving a question posed since 1974.

Findings

Constructed a family of hypersurfaces $M_\epsilon$ without umbilical points

Proved these hypersurfaces are strictly pseudoconvex and compact

Resolved a long-standing open problem in complex analysis

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<\epsilon^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.