Some regularity theorems for CR mappings

Abstract

The purpose of this article is to study Lipschitz CR mappings from an h-extendible (or semi-regular) hypersurface in C n . Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved.Abstract. The purpose of this article is to study Lipschitz CR mappings from an h-extendible (or semi-regular) hypersurface in C n . Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved.

Bounds for invariant distances on pseudoconvex Levi corank one domains and applications

“…Suppose that there is a sequence of points q j ∈ B D j (z j , R) such that q j → q 0 ∈ ∂D ∞ where q 0 is any finite boundary point. Now, Theorem 2.3 of [5] assures us that the estimates for the infinitesimal Kobayashi metric due to Thai-Thu remain stable for the family of the scaled domains D j , i.e., there is a neighbourhood U of q 0 in C n such that…”

“…Said differently, there are holomorphic coordinates at p 0 in which a tiny neighbourhood around p 0 can be written as z ∈ C n : 2ℜz n + |z 1 | 2 + |z 2 | 2 + · · · + |z n−1 | 2 + higher order terms < 0 , which violates the assumption that the Levi rank at p 0 ∈ ∂D 1 is exactly n − 2. Alternatively, one can use Theorem 1.4 or results from [5] to arrive at a contradiction. Hence the theorem.…”

“…higher order terms < 0 , which violates the assumption that the Levi rank at p 0 ∈ ∂D 1 is exactly n − 2. Alternatively, one can use Theorem 1.4 or results from [5] to arrive at a contradiction. Hence the theorem.…”

“…Here is an example to illustrate this point of view and we refer the reader to [5] for an alternative proof that works for proper holomorphic mappings as well.…”