Elliptic Reeb orbit on some real projective three-spaces via ECH. (English) Zbl 07927755
Summary: 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.
MSC:
| 53D10 | Contact manifolds (general theory) |
| 53D25 | Geodesic flows in symplectic geometry and contact geometry |
| 53D42 | Symplectic field theory; contact homology |
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