  I present the first analytical study of gravitational collapse in a compact
CMC foliation with $S^3$ spatial topology. The solutions I find, in this
context, will be both solutions of Shape Dynamics and General Relativity. The
aim is to describe a system undergoing gravitational collapse in Shape
Dynamics, so a well-justified and useful simplification is to assume spherical
symmetry. This kills all the local gravitational degrees of freedom, but some
nontrivial degrees of freedom are recovered by introducing matter. The simplest
form of matter is infinitely thin spherical shells of dust, of which I need at
least two in order to have a nontrivial dynamics. With a single shell the
system is dynamically trivial, but it nevertheless admits a solution which
represents a `frozen' shell at equilibrium in a globally de Sitter universe.
Such a solution is, to my knowledge, new. I am able to solve analytically also
the case with two shells, which has a nontrivial dynamics. When the rest mass
of one shell is much smaller than the other, the system is suitable to model a
compact universe in which one subsystem (the `light' shell) undergoes
gravitational collapse while the rest of the matter (the `heavy' shell) plays
the role of spectator. It turns out that, if the cosmological constant is zero
or positive but small, and the rest mass of the two shells are sufficiently
different, when the `light' shells collapses the ADM equations become
ill-defined and cease to admit a solution. The shape-dynamical description,
however, seems still well defined and can be continued past this point,
possibly signalling a departure of Shape Dynamics from exact equivalence with
General Relativity.

General Relativity and Quantum Cosmology