Submanifolds and the Hofer norm

TL;DR

This paper investigates the conditions under which the Chekanov-Hofer pseudometric becomes a true metric on submanifold orbits, revealing that coisotropic submanifolds are essential and identifying classes with vanishing pseudometric.

Contribution

It establishes necessary and sufficient conditions for the Chekanov-Hofer pseudometric to be a genuine metric on submanifold orbits, extending Chekanov's work beyond Lagrangian submanifolds.

Findings

Coisotropic submanifolds induce a genuine Chekanov-Hofer metric.

Generic embeddings of codimension > 1 have zero pseudometric.

Examples of submanifolds with zero displacement energy but non-displaceability.

Abstract

Chekanov showed that the Hofer norm on the Hamiltonian diffeomorphism group of a geometrically bounded symplectic manifold induces a nondegenerate metric on the orbit of any compact Lagrangian submanifold under the group. In this paper we consider the orbits of more general submanifolds. We show that, for the Chekanov-Hofer pseudometric on the orbit of a closed submanifold to be a genuine metric, it is necessary for the submanifold to be coisotropic, and we show that this condition is sufficient under various additional geometric assumptions. At the other extreme, we show that the image of a generic closed embedding with any codimension larger than one is "weightless," in the sense that the Chekanov-Hofer pseudometric on its orbit vanishes identically. In particular this yields examples of submanifolds which have zero displacement energy but are not infinitesimally displaceable.