In this note we investigate the relationship between the conformal boundary at infinity of a hyperbolic 3-manifold and the boundary of its convex core. In particular, we prove that the length of a curve in the conformal boundary gives an upper bound on the length of the corresponding curve in the boundary of the convex core.
R. D. Canary. "The conformal boundary and the boundary of the convex core." Duke Math. J. 106 (1) 193 - 207, 15 January 2001. https://doi.org/10.1215/S0012-7094-01-10616-9