2019/04/23 by Wei Guo Foo, Foo, Wei Guo, Joël Merker +3
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1904.10203
openalex publication_date 2019/04/23 · openalex created_date 2019/05/03 · openalex updated_date 2026/07/28
We show that the boundaries of thin strongly pseudoconvex Grauert tubes, with respect to the Guillemin-Stenzel Kähler metric canonically associated with the Poincaré 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 C6-smooth hypersurfaces in ℂ2. 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 ℂ2 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.