We study some analytic properties of distance decreasing maps on the complement of a smooth curve in the unit sphere. Let g be a complete Riemannian metric on the complement of a smooth closed curve Z in a sphere. We prove that the infimum of the scalar curvature of g is less than n(n -1) for a smooth closed curve Σ with non-trivial holonomy under certain conditions. This work is related to a recent question of Gromov.