  In this paper, we study the low temperature limit of the spherical
Crisanti-Sommers variational problem. We identify the $\Gamma$-limit of the
Crisanti-Sommers functionals, thereby establishing a rigorous variational
problem for the ground state energy of spherical mixed $p$-spin glasses. As an
application, we compute moderate deviations of the corresponding minimizers in
the low temperature limit. In particular, for a large class of models this
yields moderate deviations for the overlap distribution. We then analyze the
ground state energy problem. We show that this variational problem is dual to
an obstacle-type problem. This duality is at the heart of our analysis. We
present the regularity theory of the optimizers of the primal and dual
problems. This culminates in a simple method for constructing a finite
dimensional space in which these optimizers live for any model. As a
consequence of these results, we unify independent predictions of
Crisanti-Leuzzi and Auffinger-Ben Arous regarding the 1RSB phase in this limit.
We find that the "positive replicon eigenvalue" and "pure-like" conditions are
together necessary for optimality, but that neither are themselves sufficient,
answering a question of Auffinger and Ben Arous in the negative. We end by
proving that these conditions completely characterize the 1RSB phase in
$2+p$-spin models.
