Parametric CR-umbilical locus of ellipsoids in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>

Foo, Wei-Guo; Merker, Joël; Ta, The-Anh

doi:10.1016/j.crma.2017.11.019

Cited by 5 publications

(8 citation statements)

References 10 publications

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“…In [7], the authors proved the non-existence of CR-umbilical points on the boundaries of Grauert tubes around flat tori by showing the nonvanishing of a certain invariant determinant introduced in [6], which vanishes exactly when the Cartan curvatures vanish. In this paper, we shall use an explicit expression of Cartan curvatures obtained before by the second named author and Sabzevari in [20,21] for locally graphed hypersufaces, and alternatively a formula in [10] for hypersurfaces given as zero locus of implicit functions.…”

Section: Calculation Of the Complex Cartan Curvature Of ∂ω ε ⊂ H Cmentioning

confidence: 99%

“…There are instances where the hypersurfaces actually have much simpler representation by mean of implicit functions. An example is the case of general ellipsoidal hypersurfaces in C 2 considered in [10], where a direct calculation from the formula (5.1) for a graphing function of the ellipsoids gives a very complicated expression for I, while an alternative formula (cf. [10,Corollary 12]) applied to simple implicit defining functions of the ellipsoids allows one to see a whole curve of CR-umbilical points.…”

Section: Calculation Of the Complex Cartan Curvature Of ∂ω ε ⊂ H Cmentioning

confidence: 99%

“…An example is the case of general ellipsoidal hypersurfaces in C 2 considered in [10], where a direct calculation from the formula (5.1) for a graphing function of the ellipsoids gives a very complicated expression for I, while an alternative formula (cf. [10,Corollary 12]) applied to simple implicit defining functions of the ellipsoids allows one to see a whole curve of CR-umbilical points. As the implicit defining function of M ε is also very simple, we shall use the formulation in [10] to verify the nonvanishing of the Cartan curvature of M ε once again.…”

Section: Calculation Of the Complex Cartan Curvature Of ∂ω ε ⊂ H Cmentioning

confidence: 99%

“…Nonetheless, Webster [25], and later Huang and Ji [16] were able to investigate the case of real ellipsoidal hypersurfaces. In recent years, new variants and explicit formulas (see [7,20,21,10]) made it possible to determine the vanishing locus of the Cartan curvatures for new classes of 3-dimensional CR manifolds. For instance, we were able to find a whole explicit curve of points of vanishing Cartan curvature on general ellipsoids in C 2 in [10].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4, we work out the defining function for the Grauert tube around the Poincaré upper half-plane. The formula then will be used in Section 5 to calculate the Cartan curvatures on the boundaries of Grauert tubes of hyperbolic surfaces by explicit expressions given in [20] and [10], and to show that the Cartan curvatures do not vanish for small enough radii. Section 6 explains in details how nonvanishing of the Cartan curvature on the boundary of Grauert tubes around hyperbolic surfaces can be deduced from the calculations in Section 5.…”

Section: Introductionmentioning

confidence: 99%

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

Foo¹,

Merker²,

Ta³

2019

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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 C 6 -smooth hypersurfaces in 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 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.

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Section: Calculation Of the Complex Cartan Curvature Of ∂ω ε ⊂ H Cmentioning

confidence: 99%

Section: Calculation Of the Complex Cartan Curvature Of ∂ω ε ⊂ H Cmentioning

confidence: 99%