Introduction to Symplectic and Contact Geometry and the Study of Periodic Orbits

Abstract

The objective during the first two sections of the present Master Final Thesis is to provide an introduction as self-contained as possible to the fields of Symplectic and Contact Geometry and, ultimately, to devote the last section to prove a main theorem due to Hofer regarding existence of periodic orbits of the Reeb vector field on overtwisted contact $3$-manifolds. This theorem proves a particular case of the much more general Weinstein conjecture, which still remains open at the moment of writing this Thesis.

Some more concepts studied in this Thesis include Darboux’s Theorem, an overview of the interplay between contact and Symplectic Geometry and a theorem on the existence of periodic orbits for Hamiltonian vector fields along convex energy surfaces inside $\mathbb{R}^{2n}$, another instance where the Weinstein conjecture is resolved positively. In the final section, special attention is given to the study of pseudoholomorphic curves and their properties, since they constitute the main tool to prove Hofer’s Theorem, and also to related ideas such as Bishop Families and pseudoconvexity. The proof of Hofer’s Theorem is finally given by combining the machinery developed throughout the work.