Rigidity of CR-immersions into Spheres

Abstract

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.

Local Cauchy-Riemann embeddability of real hyperboloids into spheres

“…Then we derive a CR analogue of the Gauss equation by means of a pseudohermitian embedding system: the CR-Gauss equation was used for rigidity problems in Webster [12] and in Ebenfelt, Huang and Zaitsev [2]. We solve the CR-Gauss equation by means of Huang's lemma and conclude Theorem 1.2.…”

“…The adapted coframe onM 2N +1 in Lemma 3.1 satisfieŝ −ω a β ν ωᾱ aμ +R βᾱνμ − R βᾱνμ = 0. Substitute (3.7) into (2.7) and obtain the CR analogue of the Gauss equation (see [2] and [12]). IfM 2N +1 is a sphere S 2N +1 , the CR curvature tensorŜ BĀCD vanishes and the Gauss equation reads…”

The $X$-varieties for CR mappings between hyperquadrics

“…Again we observe the complexity issue. See [BH] and [EHZ1] for what happens when one restricts the number of negative eigenvalues. The homogeneous polynomial mappings taking spheres to spheres play a key role in the classification problem and have many additional interesting properties.…”

“…[DKR]. See also [D3], [F2], [H], and [HJ] for related results concerning mappings between spheres and see [BH], [EHZ1], and [EHZ2] for generalizations to mappings between hyperquadrics.…”

Regularity of rigid CR hypersurfaces in a sphere

“…The idea of the proof is to show that any nowhere-vanishing infinitesimal CR automorphism of M in S extends locally to an infinitesimal CR automorphism of S and therefore is real analytically extendable. To show this, we use the rigidity of CR embeddings into spheres proved in [Web79] and [EHZ02]. That is, any two germs of CR hypersurfaces in S of sufficiently small codimension with equivalent CR structure can be mapped to each other by a CR automorphism of S. Since a nowhere-vanishing CR automorphism of M extends real analytically to a neighborhood, M is foliated by real analytic curves.…”