[Paper Review] Finite jet determination of holomorphic mappings at the boundary

Peter Ebenfelt|ArXiv.org|Jan 21, 2000

Holomorphic and Operator Theory18 references3 citations

TL;DR

This paper establishes finite jet determination for holomorphic mappings between smooth real hypersurfaces in ℂ^N at the boundary, proving that such mappings are completely determined by their 2k₀-jets at a point where the source hypersurface is k₀-nondegenerate. The result extends Chern–Moser's finite jet determination to smooth, non-analytic settings and provides sufficient conditions for the space of infinitesimal CR automorphisms to be finite-dimensional.

ABSTRACT

Let M,M' be smooth real hypersurfaces in N-dimensional space and assume that M is k-nondegenerate at a point p in M. We prove that holomorphic mappings that extend smoothly to M, sending a neighborhood of p in M diffeomorphically into M' are completely determined by their 2k-jet at p. As an application of this result, we also give sufficient conditions on a smooth real hypersurface which guarantee that the space of infinitesimal CR automorphisms is finite dimensional.

Motivation & Objective

• To extend finite jet determination results from real-analytic to smooth real hypersurfaces in ℂ^N.
• To establish that holomorphic mappings sending a smooth hypersurface diffeomorphically into another are determined by a finite jet at the boundary.
• To provide sufficient conditions guaranteeing that the space of infinitesimal CR automorphisms is finite-dimensional.
• To generalize the Chern–Moser jet determination result beyond Levi nondegenerate and real-analytic settings.

Proposed method

• The proof relies on a system of differential equations derived from the geometry of CR structures on the hypersurfaces.
• It uses the concept of k₀-nondegeneracy, defined via the span of iterated CR vector fields acting on the defining function’s gradient.
• The argument leverages a result from [BER4] stating that all jets of a mapping are determined by its 2k₀-jet under the k₀-nondegeneracy condition.
• The method avoids Cartan connections by working directly with jet prolongations and CR geometry in the smooth category.
• It establishes reflection formulae and structural equations for the jet prolongation of mappings between hypersurfaces.
• The analysis is carried out in a local coordinate system using the tangential Cauchy–Riemann operator and dual coframes.

Experimental results

Research questions

• RQ1 Can holomorphic mappings between smooth real hypersurfaces be determined by a finite jet at the boundary, even when the hypersurfaces are not real-analytic?
• RQ2 What is the minimal jet order required to determine such mappings, and how does it depend on the nondegeneracy type of the hypersurface?
• RQ3 Under what conditions is the space of infinitesimal CR automorphisms of a smooth hypersurface finite-dimensional?
• RQ4 Is there a generalization of the Chern–Moser jet determination theorem to smooth, non-Levi nondegenerate hypersurfaces?
• RQ5 Can the existence of CR holomorphic vector fields be linked to the solvability of the tangential Cauchy–Riemann complex on non-finitely nondegenerate hypersurfaces?

Key findings

• Holomorphic mappings extending smoothly to a smooth real hypersurface M are completely determined by their 2k₀-jets at a point p₀ ∈ M if M is k₀-nondegenerate at p₀.
• The result holds for smooth (C^∞) hypersurfaces and smooth mappings, extending previous results that required real-analyticity.
• The space of infinitesimal CR automorphisms of a smooth hypersurface is finite-dimensional if the hypersurface is finitely nondegenerate at some point.
• The proof shows that all higher-order jets of the mapping are determined by the 2k₀-jet, via a system of differential equations derived from CR geometry.
• A necessary condition for infinite-dimensional automorphism groups is the existence of nontrivial CR holomorphic vector fields, which may arise when the hypersurface fails to be finitely nondegenerate.
• Solvability of the tangential ∂̄_b complex at level (0,1) implies the existence of CR holomorphic vector fields on non-finitely nondegenerate hypersurfaces under certain geometric conditions.