Finite jet determination of holomorphic mappings at the boundary

TL;DR

This paper proves that holomorphic mappings between certain real hypersurfaces are uniquely determined by finite jets at a boundary point, and provides conditions for the finiteness of automorphism groups.

Contribution

It establishes a finite jet determination result for boundary mappings and characterizes conditions for finite-dimensional automorphism groups of hypersurfaces.

Findings

Holomorphic mappings are determined by their 2k-jet at a point.

Sufficient conditions are provided for the automorphism group to be finite dimensional.

The results apply to smooth real hypersurfaces with k-nondegeneracy.

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.