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

Abstract

We prove the existence of an elliptic Reeb orbit for some contact forms on the real projective three space $\mathbb{R}{P}^3$. 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.