| Pseudoholomorphic curves were introduced in symplectic geometry by M. Gromov [5] and soon proved to be a powerful tool to address different problems within the subject. In 1993 Hofer introduced the pseudoholomorphic curves in symplectisations of contact manifolds [6] and used them to prove many cases of the Weinstein Conjecture concerning periodic orbits for Reeb vector fields. Since then, many aspects of this theory have been developed by H. Hofer, K. Wysocki and E. Zehnder [8, 9]. The interest in this theory stems from its applicability to problems in Hamiltonian dynamics and generalizations of Floer's theory, namely symplectic field theory. The theory of pseudoholomorphic curves in symplectisations is the theory of holomorphic curves on punctured Riemann surfaces. Therefore it is important to have precise understanding of the behavior of these curves near the punctures. In this thesis we study two analytical aspects of this theory. In a first contribution we give a precise asymptotic formula for the behavior of a pseudoholomorphic curve near a puncture. This formula is an important ingredient in many application, for instance the construction of invariants and Fredholm theory. The second result concerns the regularity of the compactification obtained after adding the asymptotic limit to the image of the cylinder. |
