Submanifolds and the Hofer norm

  • Michael Usher

    University of Georgia, Athens, United States


In [Ch00], 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.

Cite this article

Michael Usher, Submanifolds and the Hofer norm. J. Eur. Math. Soc. 16 (2014), no. 8, pp. 1571–1616

DOI 10.4171/JEMS/470