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# Mathematical Research Letters

## Volume 25 (2018)

### Number 1

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

Pages: 75 – 84

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n1.a4

#### Authors

#### Abstract

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

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