[Paper Review] Regularity of CR-mappings into Levi-degenerate hypersurfaces

Ilya Kossovskiy, Bernhard Lamel+1 more|arXiv (Cornell University)|Sep 2, 2016

Holomorphic and Operator Theory19 references3 citations

TL;DR

This paper establishes regularity results for CR-maps from strictly pseudoconvex hypersurfaces in $\mathbb{C}^n$ to Levi-degenerate hypersurfaces in $\mathbb{C}^{n+1}$, proving that such maps are real-analytic or smooth on a dense open subset under mild nondegeneracy assumptions on the target. The key contribution is a smooth and real-analytic reflection principle for codimension-one CR-maps into Levi-degenerate targets, extending known results from the Levi-nondegenerate case.

ABSTRACT

We provide regularity results for CR-maps between real hypersurfaces in complex spaces of different dimension with a Levi-degenerate target. We address both the real-analytic and the smooth case. Our results allow immediate applications to the study of proper holomorphic maps between Bounded Symmetric Domains.

Motivation & Objective

• To extend the Schwarz reflection principle to CR-maps between real hypersurfaces of different dimensions when the target is Levi-degenerate.
• To address the regularity problem in both the real-analytic and smooth categories for CR-maps into Levi-degenerate targets, where prior results were limited.
• To establish generic analyticity and smoothness of CR-maps under minimal assumptions on the target hypersurface, such as holomorphic nondegeneracy or finite nondegeneracy.
• To provide foundational tools for studying proper holomorphic maps between bounded symmetric domains, which are naturally bounded by uniformly 2-non-degenerate hypersurfaces.

Proposed method

• Uses the notion of $k$-nondegeneracy of CR-maps as a central tool to analyze the differential systems associated with the mapping jet prolongation.
• Applies the implicit function theorem to systems of equations derived from applying CR vector fields to the defining equation of the target hypersurface.
• Employs a normalization of the CR-map to express components in terms of $\bar{F}$, $L\bar{F}$, and $F_{n+1}$, enabling the construction of a holomorphic extension via implicit function theory.
• Analyzes the vanishing of Lie derivatives of characteristic forms to detect nondegeneracy conditions, defining the set of points where the map fails to be $\ell$-nondegenerate via determinantal ideals.
• Utilizes the concept of holomorphic nondegeneracy and finite nondegeneracy of the target to control the growth of jet prolongations and ensure regularity.
• Adapts techniques from Mir [Mi1] and extends them to the Levi-degenerate setting by distinguishing cases based on the non-vanishing of Lie derivatives of the extended map.

Experimental results

Research questions

• RQ1 Under what conditions does a $C^2$ CR-map from a strictly pseudoconvex hypersurface into a Levi-degenerate hypersurface extend analytically to a neighborhood of the source?
• RQ2 Can the smooth regularity of CR-maps into Levi-degenerate targets be improved from $C^2$ to $C^\infty$ under suitable nondegeneracy assumptions?
• RQ3 Is there a reflection principle for CR-maps into Levi-degenerate hypersurfaces analogous to the known results in the Levi-nondegenerate case?
• RQ4 What is the role of holomorphic nondegeneracy and finite nondegeneracy in ensuring regularity of CR-maps into codimension-one Levi-degenerate targets?
• RQ5 Can the regularity of CR-maps be established uniformly across the source manifold when the target is uniformly 2-non-degenerate?

Key findings

• A $C^2$ CR-map $F: M \to M'$ from a real-analytic strictly pseudoconvex hypersurface $M \subset \mathbb{C}^n$ to a real-analytic Levi-degenerate hypersurface $M' \subset \mathbb{C}^{n+1}$ is real-analytic on a dense open subset of $M$ if $M'$ is holomorphically nondegenerate.
• A $C^2$ CR-map $F: M \to M'$ from a smooth strictly pseudoconvex hypersurface $M \subset \mathbb{C}^n$ to a smooth Levi-degenerate hypersurface $M' \subset \mathbb{C}^{n+1}$ is smooth on a dense open subset of $M$ if $M'$ is finitely nondegenerate.
• If the target $M'$ is uniformly 2-non-degenerate, then any CR-map $F: M \to M'$ is everywhere real-analytic (resp. smooth) on $M$ in the real-analytic (resp. smooth) category.
• The proof relies on constructing a holomorphic extension $\Psi$ of the map components via the implicit function theorem applied to the system of equations derived from CR vector fields.
• The case distinction based on the non-vanishing of Lie derivatives of $\Psi$ ensures that either the map is real-analytic on a dense open set or a contradiction arises if the map were to be non-analytic everywhere.
• The absence of Case 2B (where the extended map is a biholomorphism onto the target) is ruled out by the holomorphic nondegeneracy of $M'$, ensuring that analyticity holds generically.