Model rigid CR submanifolds of CR dimension 1

Abstract

The following three classes of models of rigid submanifolds of higher type with CR dimension one are discussed: 1) A tube-like model that only depends on the real part of the holomorphic tangent coordinate; 2) a radial model that depends on the modulus of the holomorphic tangent coordinate and 3) a free model. The first and third models have a Lie group structure which is analyzed. A characterization of the hull of holomorphy of the first two models is presented along with a partial result on the hull of holomorphy of the third.

Failure of hypoellipticity for the canonical solution to $${\bar\partial_b}$$ on some nilpotent Lie groups

“…2 The group structure and L 2 analysis on Σ n As shown by Boggess et al in [1], Σ n admits a nilpotent Lie group structure compatible with the holomorphic structure of C n , in much the very same way that the classical case of the Heisenberg group. To be more precise, it follows from [1] that for each z ∈ C n there is an affine holomorphic self map T z of C n such that, with the product z · w = T z w, C n becomes a Lie group having Σ n as a closed Lie subgroup for which the C R tangential vector fieldL is left invariant.…”

“…As shown in [1] Σ n has type n, that is to say, the tangent space to Σ n at 0 is spanned by iterated n th order commutators of the real and imaginary part ofL and commutators of length less than n fail to span.…”

“…we seek an estimate like ||Lu|| L 2 ≤ C||Lu|| L 2 (1) for all u ⊥ NL . The case n = 2 is well known for Σ 2 is just the 1-dimensional Heisenberg group in its polarized form [6], where even a local version of (1) holds, i.e.…”

Analytic regularity of CR-mappings

“…an example of Boggess-Glenn-Nagel [BGN98] (see Example 4.1) shows. This difficulty becomes visible when studying extension properties of CR-mappings between real-analytic generic submanifolds.…”

“…In the case when M ⊂ C N is a real hypersurface, Theorem 1.1 is due to Coupet-Pinchuk-Sukhov [CPS99]. One additional difficulty for a source manifold M of higher codimension is that a one-sided extension for CR-functions on M has to be replaced by a wedge extension, where the additional directions of the wedges may change as M is shrinked (see [BGN98] and Example 4.1 below). As a consequence, we need to establish Theorem 2.4 and 2.6 below whose proof requires the full machinery of Tumanov's CR-extension theory.…”