  We use a neck stretching argument for holomorphic curves to produce
symplectic disks of small area and Maslov class with boundary on Lagrangian
submanifolds of nonpositive curvature. Applications include the proof of
Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic
space, the construction of a new symplectic capacity, obstructions to
Lagrangian embeddings into uniruled symplectic manifolds, a quantitative
version of Arnold's chord conjecture, and estimates on the size of Weinstein
neighbourhoods. The main technical ingredient is transversality for the
relevant moduli spaces of punctured holomorphic curves with tangency
conditions.
