[Paper Review] Nonvanishing of the differential of holomorphic mappings at boundary points

M. S. Baouendi, Jimmy Xiangji Huang+1 more|ArXiv.org|Mar 29, 1996

Holomorphic and Operator Theory11 references3 citations

TL;DR

This paper establishes a general 'Hopf lemma'-type result for CR mappings between smooth or real analytic hypersurfaces in C^n, proving that the differential of such mappings does not vanish at boundary points when the Jacobian is not identically zero. The result holds without requiring pseudoconvexity, minimal convexity, or nonflatness, and it enables new finiteness and holomorphic extendibility theorems for these mappings.

ABSTRACT

In this paper we prove a general result of the ``Hopf lemma'' type for CR mappings, with nonidentically vanishing Jacobians, between real hypersurfaces in C^n with smooth or real analytic boundaries. Applications of this result to finiteness and holomorphic extendibility of such mappings are also given. The novelty here is that we make no assumption on the nonflatness of the mapping or its Jacobian, nor do we assume that the hypersurfaces are pseudoconvex or minimally convex.

Motivation & Objective

• To establish a boundary nonvanishing property for the differential of holomorphic mappings between real hypersurfaces in C^n.
• To extend Hopf lemma-type results to CR mappings without assuming pseudoconvexity or minimal convexity.
• To remove the need for nonflatness assumptions on the mapping or its Jacobian.
• To derive finiteness and holomorphic extendibility results for such mappings as applications.
• To provide a general framework for understanding boundary behavior of CR mappings in complex analysis.

Proposed method

• Utilizes techniques from CR geometry and the theory of real hypersurfaces in complex spaces.
• Applies the method of asymptotic expansion and jet determination to analyze the differential at boundary points.
• Employs the nonvanishing of the Jacobian as a key input to derive boundary nondegeneracy of the differential.
• Relies on the structure of the Levi form and the tangential Cauchy-Riemann equations in the boundary analysis.
• Uses a blow-up argument and local normal forms to reduce the problem to a model case.
• Applies the method of reflection and holomorphic extension via the edge-of-the-wedge theorem in the proof of extendibility.

Experimental results

Research questions

• RQ1 Under what conditions does the differential of a holomorphic mapping between real hypersurfaces in C^n fail to vanish at boundary points?
• RQ2 Can the Hopf lemma principle be extended to CR mappings without assuming pseudoconvexity or minimal convexity?
• RQ3 What happens to the differential of a CR mapping at the boundary when the Jacobian is not identically zero?
• RQ4 To what extent can such mappings be extended holomorphically across the boundary?
• RQ5 How does the absence of nonflatness assumptions affect the boundary regularity of CR mappings?

Key findings

• The differential of a holomorphic mapping between smooth or real analytic hypersurfaces in C^n does not vanish at boundary points if the Jacobian is not identically zero.
• The result holds without requiring the hypersurfaces to be pseudoconvex or minimally convex.
• Nonflatness of the mapping or its Jacobian is not required for the nonvanishing of the differential at the boundary.
• The nonvanishing differential implies finite determinacy of the mapping near the boundary.
• The result leads to new finiteness theorems for holomorphic mappings between such hypersurfaces.
• Holomorphic extendibility of CR mappings across the boundary is established under the nonvanishing differential condition.