Geometric variants of the Hofer norm

By D. Mcduff. 2001

56 citations recorded by the source corpus.

This note discusses some geometrically defined seminorms on the group $\Ham(M, \omega)$ of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M, \omega)$, giving conditions under which they are nondegenerate and explaining their relation to the Hofer norm. As a consequence we show that if an element in $\Ham(M, \omega)$ is sufficiently close to the identity in the $C^{2}$-topology then it may be joined to the identity by a path whose Hofer length is minimal among all paths, not just among paths in the same homotopy class relative to endpoints. Thus, true geodesics always exist for the Hofer norm. The main step in the proof is to show that a "weighted" version of the nonsqueezing theorem holds for all fibrations over $S^2$ generated by sufficiently short loops.Further, an example is given showing that the Hofer norm may differ from the sum of the one sided seminorms.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d5f5d908833c440b7475e30bfe16be97e32e1e1b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. F. Schlenk
2. Michael Usher
3. Yong-Geun Oh
4. Y. Ostrover
5. D. Mcduff
6. Michael Entov
7. L. Polterovich
8. S. Tolman

In Mathematical Physics

1. Y. Ostrover
2. Yong-Geun Oh
3. Michael Entov
4. L. Polterovich
5. Michael Usher
6. D. Mcduff
7. F. Schlenk
8. S. Tolman

Citations in the corpus

• Hofer–Zehnder capacity and length minimizing Hamiltonian paths (2001, background): “(For a discussion of their properties, see [17].”
• Calabi quasimorphism and quantum homology (2002, background): “4 Example: CP 2 blown up at one point LetM be the complex blow up of CP 2 at one point equipped with a monotone symplectic form (see [23],[34]).”
• Calabi quasi-morphisms for some non-monotone symplectic manifolds (2005, methodology): “[14], [21]), where ωμ takes the value μ on the exceptional divisor, and 1 on the class of the line [CP 1].”
• Topological properties of Hamiltonian circle actions (2004, methodology): “The above results also agree with the formulas found in [15] §4 for rotations of the one point blow up of CP(2): see Example 5.”
• The Gromov width of 4-dimensional tori (2011, background): “Computations and estimates of the Gromov width for various examples can be found in [4, 5, 6, 15, 18, 19, 21, 22, 29, 34, 38, 41, 42].”