  The state-following technique allows the study of metastable glassy states
under external perturbations. Here we show how this construction can be used to
study the behavior of glassy states of Hard Spheres in infinite dimensions
under compression or shear strain. In a preceding work it has been shown that
in both cases, when the external perturbation is sufficiently strong, glassy
states undergo a second-order transition, called the Gardner transition,
whereupon a hierarchical structure of marginal micro-states manifests within
the original glass state. The purpose of this work is to study the solution of
the state-following construction in this marginal phase. We show that upon
compression, close to the jamming transition, the metastable states are
described by a scaling solution characterized by a set of non-trivial critical
exponents that agree with the results obtained in a preceding work, and we
compute the value of the final jamming density for various glassy states.
Moreover we show that under the action of the shear strain, beyond the Gardner
point, the metastable states can be followed in the marginal phase and we
detect an overshoot in the stress-strain curve in agreement with numerical and
experimental observations. Finally we further characterize the Gardner
transition point by computing both the critical 4-point susceptibility and the
exponents that characterize the critical slowing down of the dynamics within a
glassy state close to the transition.
