$L^2$-bounds for drilling short geodesics in convex co-compact hyperbolic 3-manifolds

TL;DR

This paper establishes universal $L^2$-bounds on how much the complex projective structure on the boundary of a hyperbolic 3-manifold changes after drilling short geodesics, with bounds proportional to the square root of the geodesics' length.

Contribution

It provides new $L^2$-bounds that are independent of the boundary's injectivity radius, extending previous $L^ Infinity$-bounds to a more general setting.

Findings

Bounded the change in complex projective structure by a universal constant times the square root of geodesic length.

Bounds do not depend on the injectivity radius of the boundary.

Extends previous $L^ Infinity$-bounds to $L^2$-bounds.

Abstract

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.