On a theorem of Lyapunov-Poincar\'e in higher dimension

TL;DR

This paper generalizes the Lyapunov-Poincaré center theorem to higher-dimensional real and complex foliations, providing criteria for the existence of analytic first integrals and revisiting classical results in the context of singularities.

Contribution

It extends the classical center theorem to higher dimensions and singular holomorphic foliations, offering new criteria for first integrals using foliation theory.

Findings

Generalization of the center theorem to higher dimensions

Criteria for the existence of analytic first integrals

Revisiting classical results on integrable perturbations

Abstract

The classical Lyapunov-Poincar\'e center theorem assures the existence of a first integral for an analytic one-form near a center singularity in dimension two, provided that the first jet of the one-form is nondegenerate. The basic point is the existence of an analytic first integral for the given one-form. In this paper we consider generalizations for two main frameworks: (i) real analytic foliations of codimension one in higher dimension and (ii) singular holomorphic foliations in dimension two. All this is related to the problem of finding criteria assuring the existence of analytic first integrals for a given codimension one germ with a suitable first jet. Our approach consists in giving an interpretation of the center theorem in terms of holomorphic foliations and, following an idea of Moussu, apply the holomorphic foliations arsenal in the obtaining the required first integral. As a consequence we are able to revisit some of Reeb classical results on integrable perturbations of exact homogeneous one-forms, and prove some versions of these to the framework of non-isolated (perturbations of transversely Morse type) singularities.