The Gradient flow equation of Rabinowitz action functional in a symplectization

TL;DR

This paper establishes a correspondence between gradient flow lines of the Rabinowitz action functional and those of a restricted negative area functional on symplectizations, linking their properties and motivations.

Contribution

It demonstrates a one-to-one correspondence between the gradient flows of these two functionals on symplectizations, clarifying their relationship and properties.

Findings

Gradient flow lines correspond bijectively between the two functionals.

The restricted functional satisfies Chas-Sullivan additivity.

The result clarifies the functional's behavior in symplectizations.

Abstract

Rabinowitz action functional is the Lagrange multiplier functional of the negative area functional to a constraint given by the mean value of a Hamiltonian. In this note we show that on a symplectization there is a one-to-one correspondence between gradient flow lines of Rabinowitz action functional and gradient flow lines of the restriction of the negative area functional to the constraint. In the appendix we explain the motivation behind this result. Namely that the restricted functional satisfies Chas-Sullivan additivity for concatenation of loops which the Rabinowitz action functional does in general not do.