A distance expanding flow on exact Lagrangian cobordism classes

TL;DR

This paper demonstrates that the Liouville flow induces a distance-expanding flow on the exact Lagrangian cobordism classes, implying infinite diameter under certain conditions, with elementary geometric proofs.

Contribution

It shows that the Liouville flow induces a distance-expanding flow on Lagrangian cobordism classes, revealing infinite diameter in certain cases.

Findings

Liouville flow induces a flow that expands cobordism distances

The cobordism metric space has infinite diameter under complete Liouville flow

Elementary differential geometry suffices for the proof

Abstract

Given an exact Lagrangian $L$ of an exact symplectic manifold $(M,d\lambda )$, Cornea and Shelukhin recently introduced a remarkable "cobordism metric" $d_c$ on the exact Lagrangian cobordism class $\mathcal{L}(L)$ of $L$. In this note we show that the Liouville flow of $(M,d\lambda )$ induces a flow on $(\mathcal{L}(L),d_c)$ which expands cobordism distances. In particular we deduce that $(\mathcal{L}(L),d_c)$ has infinite diameter whenever the Liouville flow is complete. We also discuss (Hamiltonian and Lagrangian) Hofer-geometric versions of our result. The proof only uses elementary differential geometry.