Rigidity of CR-immersions into spheres

Abstract

We consider local CR-immersions of a strictly pseudoconvex real hypersurface $M\subset\bC^{n+1}$, near a point $p\in M$, into the unit sphere $\mathbb S\subset\bC^{n+d+1}$ with $d>0$. Our main result is that if there is such an immersion $f:(M,p)\to \mathbb{S}$ and $d<n/2$, then $f$ is {\em rigid} in the sense that any other immersion of $(M,p)$ into $\mathbb{S}$ is of the form $\varphi \circ f$, where $\varphi$ is a biholomorphic automorphism of the unit ball $\mathbb B\subset\bC^{n+d+1}$. As an application of this result, we show that an isolated singularity of an irreducible analytic variety of codimension $d$ in $\bC^{n+d+1}$ is uniquely determined up to affine linear transformations by the local CR geometry at a point of its Milnor link.