We consider local CR-immersions of a strictly pseudoconvex real hypersurface M ⊂ C n+1 , near a point p ∈ M , into the unit sphere S ⊂ C n+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 φ • f , where φ is a biholomorphic automorphism of the unit ball B ⊂ C n+d+1 . As an application of this result, we show that an isolated singularity of an irreducible analytic variety of codimension d in C n+d+1 is uniquely determined up to affine linear transformations by the local CR geometry at a point of its Milnor link.

We present a large class of homogeneous 2 -nondegenerate CR-manifolds M , both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains U,V in M extends to a global real-analytic CR-automorphism of M . We show that this class contains G -orbits in Hermitian symmetric spaces Z of compact type, where G is a real form of the complex Lie group \mathsf{Aut}(Z)^{0} and G has an open orbit that is a bounded symmetric domain of tube type.

We show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface M in C 2 at a point p ∈ M are uniquely determined by their jets of some finite order at p if and only if M is not Levi-flat near p. This seems to be the first necessary and sufficient result on finite jet determination and the first result of this kind in the infinite type case.If M is of finite type at p, we prove a stronger assertion: the local real-analytic CR automorphisms of M fixing p are analytically parametrized (and hence uniquely determined) by their 2-jets at p. This result is optimal since the automorphisms of the unit sphere are not determined by their 1-jets at a point of the sphere. The finite type condition is necessary since otherwise the needed jet order can be arbitrarily high [Kow1,2], [Z2]. Moreover, we show, by an example, that determination by 2-jets fails for finite type hypersurfaces already in C 3 .We also give an application to the dynamics of germs of local biholomorphisms of C 2 .

We consider the class of Levi nondegenerate hypersurfaces M in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] that admit a local (CR transversal) embedding, near a point p ε M , into a standard nondegenerate hyperquadric in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] with codimension k := N - n small compared to the CR dimension n of M . We show that, for hypersurfaces in this class, there is a normal form (which is closely related to the embedding) such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric. We also show that if the signature of M and that of the standard hyperquadric in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] are the same, then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric.

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