Finite energy foliations and surgery on transverse links

Pseudoholomorphic curves have become an essential tool in symplectic topology since their introduction by Gromov in 1985. They have also found application in contact topology, where the existence of punctured holomorphic curves is closely related to the dynamics of the Reeb vector field, and can be used to define contact invariants via symplectic field theory.; Punctured holomorphic curves have particularly nice properties in contact three-manifolds and their four-dimensional symplectizations, where intersections and transversality can be controlled algebraically. This leads to the existence of two-dimensional foliations by embedded holomorphic curves in the symplectization, which project onto the contact manifold as one-dimensional singular foliations transverse to the Reeb orbits. The existence of such foliations has been established previously for generic tight three-spheres by Hofer, Wysocki and Zehnder, with powerful consequences for the Reeb dynamics.; The present thesis aims at extending this existence result to more general three-manifolds and contact structures. To accomplish this, we develop a method for preserving families of holomorphic curves under surgery along knots which cut transversely through both the contact structure and the holomorphic curves. This is done by considering a mixed boundary value problem on punctured Riemann surfaces with boundary, then using the compactness properties of holomorphic curves to degenerate each boundary component to a puncture. The result is that a holomorphic foliation on the tight three-sphere can be used to construct a similar foliation for every closed three-manifold with any overtwisted contact structure. This is the first step in a program suggested by Hofer to prove the Weinstein conjecture in dimension three by constructing foliations or related objects for generic contact manifolds. The constructions here also lead to some concrete examples of an algebraic theory in the spirit of Floer homology and symplectic field theory, which is conjectured to be of fundamental significance in the theory of holomorphic foliations, and may turn out to have broader applications in three-dimensional topology.