  The purpose of this article is to investigate geometric properties of the
parameter locus of the H\'enon family where the uniform hyperbolicity of a
horseshoe breaks down. As an application, we obtain a variational
characterization of equilibrium measures "at temperature zero" for the
corresponding non-uniformly hyperbolic H\'enon maps. The method of the proof
also yields that the boundary of the hyperbolic horseshoe locus in the
parameter space consists of two monotone pieces, which confirms a conjecture in
[AI]. The proofs of these results are based on the machinery developed in [AI]
which employs the complexification of both the dynamical and the parameter
spaces of the H\'enon family together with computer assistance.
