On the Riemann Mapping Theorem

Every real ellipsoid in ℂ² admits CR umbilical points

“…It may not be surprising that it is more likely to find umbilical points on a hypersurface in C 2 than to find umbilical points for a hypersurface in C n (n ≥ 3). The proof of Theorem 1.2 uses Chern's inhomogeneous coordinates for the projective G-structure bundle of the Segre family of a real analytic strongly pseudoconvex hypersurface [C], [CJ2], and a formula derived in Huang-Ji-Yau [HJY,Theorem 3.1] for the complexified Cartan fundamental curvature tension represented under these coordinates. The formula of [HJY] seems to fit particularly well with the computation here.…”

An explicit formula for the Webster pseudo-Ricci curvature on real hypersurfaces and its application for characterizing balls in {$\Bbb C\sp n$}

“…The characterization for balls in C n+1 is always an interesting subject [8,15,19,23,29]. Formula (1.7) in Theorem 1.1 and the main theorem in [23] on characterizing D to be a ball in C n+1 lead us to the second main purpose of this paper by using the pseudo-scalar curvature to characterize a strictly pseudo-convex domain to be a ball.…”

“…It suffices to consider M = ∂D, where D is a smoothly bounded strictly pseudo-convex domain in C n+1 . In addition, it was proved by Chern and Ji [8] that if D is simply connected and local spherical then D must global spherical, or D is biholomorphical to the unit ball in C n+1 . In this case, one can easily construct a contact form θ with constant pseudo-scalar curvature (see formula in Theorem 1.1 below).…”

Higher Order Asymptotic Behavior of Certain Kähler Metrics and Uniformization for Strongly Pseudoconvex Domains

“…As a generalization of Chern-Ji's theorem ( [2]), Nemirovskii-Shaffikov proved in [11,12] that a strongly pseudoconvex domain Ω with C ∞ -smooth boundary is covered by the unit ball if every boundary point is spherical in the sense that all the CR invariants vanish identically on ∂Ω (cf. [3]).…”