Holomorphic Immersions of Bi-Disks into 9D Real Hypersurfaces with Levi Signature (2, 2). (1st October 2020)

• Record Type: Journal Article
• Authors: Foo, Wei Guo; Merker, Joël
• Is Part Of: International mathematics research notices. Volume 2022:Number 7(2022)
• Page Start: 5466
• Page End: 5505
• Publication Date: 2020-10-01
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa266
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartan's method to the question of the existence of bi-disk $\mathbb{D}^{2}$ in a smooth $9$ D real-analytic real hypersurface $M^{9}\subset \mathbb{C}^{5}$ with Levi signature $(2, 2)$ passing through a fixed point. The result is that the lift to $M^{9}\times U(2)$ of the image of the bi-disk in $M^{9}$ must lie in the zero set of two complex-valued functions in $M^{9}\times U(2)$ . We then provide an example where one of the functions does not identically vanish, thus obstructing holomorphic immersions.