Holomorphic immersions of bi-disks into $9$ dimensional real hypersurfaces with Levi signature $(2, 2)$

TL;DR

This paper investigates the existence of holomorphic bi-disk immersions into 9-dimensional real hypersurfaces with Levi signature (2,2), using Cartan's method to identify obstructions and providing explicit examples.

Contribution

It applies Cartan's method to analyze holomorphic immersions of bi-disks into high-dimensional hypersurfaces with specific Levi signatures, revealing obstructions to such immersions.

Findings

Lift of bi-disk image lies in zero set of two functions

Explicit example shows one function does not vanish identically

Obstructions prevent certain holomorphic immersions

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 $9$-dimensional 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.