On the Hofer Geometry Injectivity Radius Conjecture

DOI10.1093/IMRN/RNW023zbMATH Open1404.53114arXiv1501.02740OpenAlexW2963825443WikidataQ123302773 ScholiaQ123302773MaRDI QIDQ4560564

Yasha Savelyev

Publication date: 12 December 2018

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: We verify here some variants of topological and dynamical flavor of the injectivity radius conjecture in Hofer geometry, Lalonde-Savelyev cite{citeLalondeSavelyevOntheinjectivityradiusinHofergeometry} in the case of

H a m (S^{2})

and

H a m (S i g m a, o m e g a)

, for

S i g m a

a closed positive genus surface. In particular we show that any loop in

H a m (S^{2})

, respectively

H a m (S i g m a, o m e g a)

with

L^{+}

Hofer length less than

a r e a (S^{2}) / 2

, respectively any

L^{+}

length is contractible through (

L^{+}

) Hofer shorter loops, in the

C^{i n f t y}

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

H a m (S^{2})

, respectively in

H a m (S i g m a, o m e g a)

with

L^{+}

Hofer length less than

a r e a (S^{2}) / 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.

Full work available at URL: https://arxiv.org/abs/1501.02740

Mathematics Subject Classification ID

Groups of diffeomorphisms and homeomorphisms as manifolds (58D05) Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds (58B20) Geodesics in global differential geometry (53C22) Global theory of symplectic and contact manifolds (53D35) Topological properties of groups of homeomorphisms or diffeomorphisms (57S05)