Holomorphic Legendrian curves

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on$\mathbb{C}^{2n+1}$for any$n\in \mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface$M$admits a proper holomorphic Legendrian embedding$M{\hookrightarrow}\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface$M={M\unicode[STIX]{x0030A}}\,\cup \,bM$there exists a topological embedding$M{\hookrightarrow}\mathbb{C}^{2n+1}$whose restriction to the interior is a complete holomorphic Legendrian embedding${M\unicode[STIX]{x0030A}}{\hookrightarrow}\mathbb{C}^{2n+1}$. As a consequence, we infer that every complex contact manifold$W$carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on$W$.

A Linking/S1-equivariant Variational Argument in the Space of Dual Legendrian Curves and the Proof of the Weinstein Conjecture on S3 ”in the Large”

Let α be a contact form on SWe establish in this paper that some cycles (an infinite number of them, indexed by odd integers, tending to ∞) in the S 1The arguments hold under the basic assumption that no periodic orbit of index 1 connects LTherefore, to a certain extent, the present result runs, especially in the case of threedimensional over-twisted [8] contact forms, against the existence of non-trivial algebraic invariants defined by the periodic orbits of ξ and independent of what ker α and/or α are.

The Topology of a Subspace of the Legendrian Curves on a Closed Contact 3-Manifold

In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S