We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a degree-theoretic analysis of the evaluation map on a suitable moduli space of holomorphic spheres. Applications of this homological result include a proof that compositions of right-handed Dehn twists on Liouville domains are of infinite order in the symplectomorphism group. We also derive uniqueness results for subcritical Stein fillings up to homotopy equivalence and, under some topological assumptions on the contact manifold, up to diffeomorphism or symplectomorphism.

We study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.Comment: 19 pages, 1 figure; v2: several minor changes and correction

We give a dynamical characterisation of odd-dimensional balls within the class of all contact manifolds whose boundary is a standard even-dimensional sphere. The characterisation is in terms of the non-existence of short periodic Reeb orbits.

Abstract. We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in C n .

Inspired by Katok's examples of Finsler metrics with a small number of closed geodesics, we present two results on Reeb flows with finitely many periodic orbits. The first result is concerned with a contact-geometric description of magnetic flows on the 2-sphere found recently by Benedetti. We give a simple interpretation of that work in terms of a quaternionic symmetry. In the second part, we use Hamiltonian circle actions on symplectic manifolds to produce compact, connected contact manifolds in dimension at least five with arbitrarily large numbers of periodic Reeb orbits. This contrasts sharply with recent work by Cristofaro-Gardiner, Hutchings and Pomerleano on Reeb flows in dimension three. With the help of Hamiltonian plugs and a surgery construction due to Laudenbach we reprove a result of Cieliebak: one can produce Hamiltonian flows in dimension at least five with any number of periodic orbits; in dimension three, with any number greater than one.