English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We show that the boundaries of thin strongly pseudoconvex Grauert tubes, withrespect to the Guillemin-Stenzel K\"{a}hler metric canonically associated withthe Poincar\'e metric on closed hyperbolic real-analytic surfaces, has nowherevanishing Cartan CR-curvature. This result provides a wealth of examples ofcompact $3$-dimensional Levi nondegenerate CR manifolds having no CR-umbilicalpoint.  We provide two proofs utilizing two recent formulas for determining theCartan CR-curvature of any local $\mathcal{C}^6$-smooth hypersurfaces in$\mathbb{C}^2$. One was obtained in 2012 by the second named author joint withSabzevari, and it is an expanded explicit formula, valid for locally graphedhypersurfaces, containing millions of terms. The other formula, which wepublished in 2018 when studying Webster's ellipsoidal hypersurfaces, is notexpanded, but more suitable for calculations with a hypersurface in$\mathbb{C}^2$ that is represented as the zero locus of some implicit (butsimple in some sense, e.g. quadratic) defining function.  We also discuss Grauert tubes constructed with respect to extrinsic metricsdepending on embeddings in complex surfaces, together with a certaincombinatorics of product metrics.

Nonvanishing of Cartan CR curvature on boundaries of Grauert tubes around hyperbolic surfaces