Rigidity of CR-immersions into spheres

Abstract

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