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

We prove the following: For $\epsilon \gt 0$, let $D_{\epsilon}$ be the bounded strictly pseudoconvex domain in $\mathbb{C}^2$ given by\[{(\log \lvert z \rvert)}^2 + {(\log \lvert w \rvert)}^2 \lt \epsilon^2 \; \textrm{.}\]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.

2010 Mathematics Subject Classification

30F45, 32V05

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The first author was supported in part by the NSF grant DMS-1600701.

The second-named author was partially supported by the Qatar National Research Fund, NPRP project 7-511-1-098. Part of this work was done while the second-named author visited UC San Diego in August 2016, which he would like to thank for partial support and hospitality.

Received 18 September 2016

Accepted 14 March 2017

Published 4 June 2018