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

Brayan Ferreira

• 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.