On periodic solutions of nonlinear wave equations, including Einstein equations with a negative cosmological constant

TL;DR

This paper explores the construction of periodic solutions for nonlinear wave equations, including Einstein equations with negative cosmological constant, using analytic continuation, but notes a gap in the proof and questions the strategy's completeness.

Contribution

It introduces a method to construct infinite-dimensional families of time-periodic solutions for complex Einstein-related equations, highlighting a novel approach with limitations.

Findings

Constructed families of time-periodic solutions for Einstein equations.

Applied analytic continuation to nonlinear wave equations.

Identified a gap in the proof and discussed the strategy's limitations.

Abstract

Original abstract: "We construct periodic solutions of nonlinear wave equations using analytic continuation. The construction applies in particular to Einstein equations, leading to infinite-dimensional families of time-periodic solutions of the vacuum, or of the Einstein-Maxwell-dilaton-scalar fields-Yang-Mills-Higgs-Chern-Simons-$f(R)$ equations, with a negative cosmological constant." However, there is a gap in the proof, and it is unlikely that the strategy presented can be upgraded to a full proof.