CR Embedding Research Articles

The topology of Stein CR manifolds and the Lefschetz theorem

Necessary topological conditions are given for the closed CR embedding of a CR manifold into a Stein manifold or into a complex projective space.

On the local solution of the tangential Cauchy-Riemann equations

We study the solution operators P and homotopy formula introduced by G. M. Henkin for the tangential Cauchy-Riemann complex of a suitable small domain D on a strictly pseudoconvex real hypersurface in complex n-space. The main difficulties stem from the fact that P is an integral operator with a rather complicated kernel. For U ⊂⊂ D, we derive a Ck-norm estimate of the form ∥Pφ∥U, k ≦ K∥φ∥D, k, where the constant K blows up as U increases to D. We obtain careful control of the rate of this blow-up and of the dependence of K on the derivatives of the function defining the real hypersurface. Our estimates are sufficient for application to the local CR embedding problem.

Partial rigidity of CR embeddings of real hypersurfaces into hyperquadrics with small signature difference

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.