Elliptic Reeb orbit on some real projective three-spaces via ECH

• Brayan Ferreira ^[1]
  1. [1] Universidade Federal do Espírito Santo

     Universidade Federal do Espírito Santo

     Brasil

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 5, 2024, págs. 1833-1862
• Idioma: inglés
• DOI: 10.4171/RMI/1480
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• Resumen

  • We prove the existence of an elliptic Reeb orbit for some contact forms on the real projective three space RP3. The main ingredient of the proof is the existence of a distinguished pseudoholomorphic curve in the symplectization given by the U map on ECH. Also, we check that the first value on the ECH spectrum coincides with the smallest action of null-homologous orbit sets for 1/4-pinched Riemannian metrics. This action coincides with twice the length of a shortest closed geodesic. In addition, we compute the ECH spectrum for the irrational Katok metric example.