Positive loops and L∞-contact systolic inequalities

• Peter Albers ^[1] ; Urs Fuchs ^[1] ; Will J. Merry ^[2]
  1. [1] Heidelberg University

     Heidelberg University

     Stadtkreis Heidelberg, Alemania

     
  2. [2] Swiss Federal Institute of Technology in Zurich

     Swiss Federal Institute of Technology in Zurich

     Zürich, Suiza

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 23, Nº. 4, 2017, págs. 2491-2521
• Idioma: inglés
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• Resumen

  • We prove an inequality between the L∞-norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds was proved by Casals et al. (J Symplectic Geom 14:1013–1031, 2016). As corollaries of the inequality we deduce various results. E.g. we prove that certain periodic Reeb flows are the unique minimisers of the L∞-norm. Moreover, we establish L∞-type contact systolic inequalities in the presence of a positive loop