  We give $L^2$-bounds on the change in the complex projective structure on the
boundary of conformally compact hyperbolic 3-manifold with incompressible
boundary after drilling short geodesics. We show that the change is bounded by
a universal constant times the square root of the length of the drilled
geodesics. While $L^\infty$-bounds of this type where obtained by the second
author (2004), our bounds here do not depend on the injectivity radius of the
boundary.
