Rigidity of CR-immersions into Spheres

We consider local CR-immersions of a strictly pseudoconvex real hypersurface $M \subset \mathbb{C}^{n+1}$, near a point $p \in M$, into the unit sphere $\mathbb{S} \subset \mathbb{C}^{n+d+1}$ with $d \gt 0$. Our main result is that if there is such an immersion $f : (M,p) \to \mathbb{S}$ and $d \lt n / 2$, then $f$ is 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 $B \subset \mathbb{C}^{n+d+1}$. As an application of this result, we show that an isolated singulary of an irreducible analytic variety of codimension $d$ in $\mathbb{C}^{n+d+1}$ is uniquely determined up to affine linear transformations by the local CR geometry at a point of its Milnor link.

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Published 1 January 2004