Partial Rigidity of CR Embeddings of Real Hypersurfaces into Hyperquadrics with Small Signature Difference

TL;DR

This paper investigates the rigidity of holomorphic CR embeddings of real hypersurfaces into hyperquadrics with small signature difference, showing that under certain conditions, the images are contained in low-dimensional complex planes.

Contribution

It establishes new partial rigidity results for CR embeddings into hyperquadrics with small signature difference, linking degeneracy levels to geometric constraints.

Findings

Mappings with limited degeneracy are contained in a complex plane.

Degeneracy of mappings relates to the signature difference of the target hyperquadric.

Results hold for CR hypersurfaces with bounded CR complexity.

Abstract

We study the rigidity of holomorphic mappings from a neighborhood of a Levi-nondegenerate CR hypersurface $M$ with signature $l$ into a hyperquadric $Q_{l'}^{N} \subseteq \mathbb{CP}^{N+1}$ of larger dimension and signature. We show that if the CR complexity of $M$ is not too large then the image of $M$ under any such mapping is contained in a complex plane with dimension independent of $N$. This result follows from two theorems, the first demonstrating that for sufficiently degenerate mappings, the image of $M$ is contained in a plane, and the second relating the degeneracy of mappings into different quadrics.