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In this paper, we prove (1): for any closed contact three-manifold with a $C^\infty$-generic contact form, the union of periodic Reeb orbits is dense, (2): for any closed surface with a $C^\infty$-generic Riemannian metric, the union of closed geodesics is dense. The key observation is $C^\infty$-closing lemma for 3D Reeb flows, which follows from the fact that the embedded contact homology (ECH) spectral invariants recover the volume.

Dense existence of periodic Reeb orbits and ECH spectral invariants