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

Abstract

We show that the boundaries of thin strongly pseudoconvex Grauert tubes, with respect to the Guillemin-Stenzel K\"{a}hler metric canonically associated with the Poincar\'e metric on closed hyperbolic real-analytic surfaces, has nowhere vanishing Cartan CR-curvature. This result provides a wealth of examples of compact $3$-dimensional Levi nondegenerate CR manifolds having no CR-umbilical point. We provide two proofs utilizing two recent formulas for determining the Cartan 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 with Sabzevari, and it is an expanded explicit formula, valid for locally graphed hypersurfaces, containing millions of terms. The other formula, which we published in 2018 when studying Webster's ellipsoidal hypersurfaces, is not expanded, but more suitable for calculations with a hypersurface in $\mathbb{C}^2$ that is represented as the zero locus of some implicit (but simple in some sense, e.g. quadratic) defining function. We also discuss Grauert tubes constructed with respect to extrinsic metrics depending on embeddings in complex surfaces, together with a certain combinatorics of product metrics.