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- Title
On the Hofer Geometry Injectivity Radius Conjecture.
- Authors
Savelyev, Yasha
- Abstract
We verify here some variants of topological and dynamical flavor of the injectivity radius conjecture in Hofer geometry, Lalonde-Savelyev [4] in the case of Ham(S², ω) and Ham(∑, ω), for ∑ a closed positive genus surface. In particular we show that any loop in Ham(S², ω), respectively, Ham(∑, ω) with L+ Hofer length less than area(S²)/2, respectively, any L+ length is contractible through (L+) Hofer shorter loops, in the C∞ topology. We also prove some stronger variants of this statement on the loop space level. One dynamical-type corollary is that there are no smooth, positive Morse index (Ustilovsky) geodesics, in Ham(S², ω), respectively, in Ham(∑, ω) with L+ Hofer length less than area(S²)/2, respectively, any length. The above condition on the geodesics can be expanded as an explicit and elementary dynamical condition on the associated Hamiltonian flow. We also give some speculations on connections of this later result with curvature properties of the Hamiltonian diffeomorphism group of surfaces.
- Subjects
INJECTIVE functions; RADIUS (Geometry); LOOP spaces; GEODESICS; HAMILTONIAN systems; CURVATURE; DIFFEOMORPHISMS
- Publication
IMRN: International Mathematics Research Notices, 2016, Vol 2016, Issue 23, p7253
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnw023