Convergence of Normal Form Power Series for Infinite-Dimensional Lie Pseudo-Group Actions

TL;DR

This paper proves the convergence of normal form power series for analytic submanifolds under broad infinite-dimensional Lie pseudo-group actions, extending classical results with a new analytic approach.

Contribution

It introduces a convergence proof for normal forms using the equivariant moving frame method and initial value problems, generalizing Chern and Moser's theorem.

Findings

Normal form power series converge under broad conditions

The method applies to infinite-dimensional Lie pseudo-group actions

Includes classical convergence results as special cases

Abstract

We prove the convergence of normal form power series for suitably nonsingular analytic submanifolds under a broad class of infinite-dimensional Lie pseudo-group actions. Our theorem is illustrated by a number of examples, and includes, as a particular case, Chern and Moser's celebrated convergence theorem for normal forms of real hypersurfaces. The construction of normal forms relies on the equivariant moving frame method, while the convergence proof is based on the realization that the normal form can be recovered as part of the solution to an initial value problem for an involutive system of differential equations, whose analyticity is guaranteed by the Cartan-K\"ahler Theorem.