On the analytic and geometric aspects of obstruction flatness

TL;DR

This paper explores the properties and existence of obstruction flatness in strongly pseudoconvex CR hypersurfaces, revealing local osculation possibilities, non-spherical examples with symmetry, and the presence of flat points on certain compact hypersurfaces.

Contribution

It provides new results on local osculation by obstruction flat hypersurfaces, constructs non-spherical examples with symmetry, and proves the existence of flat points on specific compact CR hypersurfaces.

Findings

Any strongly pseudoconvex CR hypersurface can be osculated by an obstruction flat one up to a certain order.

Existence of non-spherical, obstruction flat CR hypersurfaces with transverse symmetry in dimension 3.

The unit sphere in a negative line bundle over a Riemann surface always has at least one circle of obstruction flat points.

Abstract

In this paper, we investigate analytic and geometric properties of obstruction flatness of strongly pseudoconvex CR hypersurfaces of dimension $2n-1$. Our first two results concern local aspects. Theorem 3.2 asserts that any strongly pseudoconvex CR hypersurface $M\subset \mathbb{C}^n$ can be osculated at a given point $p\in M$ by an obstruction flat one up to order $2n+4$ generally and $2n+5$ if and only if $p$ is an obstruction flat point. In Theorem 4.1, we show that locally there are non-spherical but obstruction flat CR hypersurfaces with transverse symmetry for $n=2$. The final main result in this paper concerns the existence of obstruction flat points on compact, strongly pseudoconvex, 3-dimensional CR hypersurfaces. Theorem 5.1 asserts that the unit sphere in a negative line bundle over a Riemann surface $X$ always has at least one circle of obstruction flat points.