J-holomorphic curves with boundary in bounded geometry

The fundamental properties of J-holomorphic curves depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a J-holomorphic map in terms of its energy. The cylinder inequality stipulates and quantifies the exponential decay of energy along cylinders of small total energy.We show these inequalities hold uniformly if the geometry of the target symplectic manifold and Lagrangian boundary condition is appropriately bounded.