The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics

Abstract

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.

Transversality of holomorphic mappings between real hypersurfaces in different dimensions

“…Having tranversality at most points is crucial in the study of rigidity of embeddings into hyperquadrics. See e.g., [4,[11][12][13][14]16,19,21,25]. See also [1,26] for recent related work on transversality of holomorphic Segre mappings.…”

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“…We remark that H n is linearly equivalent to H When N < 2n − 1, Theorem 4.1 (I) already follows from a recent work of Ebenfelt-Huang-Zaitsev [EHZ2].…”

Local Cauchy-Riemann embeddability of real hyperboloids into spheres

“…, [3]). All general real ellipsoids of the class (1.3) are locally CR-embeddable into the sphere with CR co-dimension 1.…”