  This paper unites the gauge-theoretic and hyperbolic-geometric perspectives
on the asymptotic geometry of the character variety of SL(2,C) representations
of a surface group. Specifically, we find an asymptotic correspondence between
the analytically defined limiting configuration of a sequence of solutions to
the SU(2) self-duality equations on a closed Riemann surface constructed by
Mazzeo-Swoboda-Weiss-Witt, and the geometric topological shear-bend parameters
of equivariant pleated surfaces in hyperbolic three-space due to Bonahon and
Thurston. The geometric link comes from the nonabelian Hodge correspondence and
a study of high energy degenerations of harmonic maps. Our result has several
applications. We prove: (1) the local invariance of the partial
compactification of the moduli space of solutions to the self-duality equations
by limiting configurations; (2) a refinement of the harmonic maps
characterization of the Morgan-Shalen compactification of the character
variety; and (3) a comparison between the family of complex projective
structures defined by a quadratic differential and the realizations of the
corresponding flat connections as Higgs bundles, as well as a determination of
the asymptotic shear-bend cocycle of Thurston's pleated surface.
