Minimal mass blow-up solutions for the $L^2$ critical NLS with inverse-square potential

We study minimal mass blow-up solutions of the focusing $L^2$ critical nonlinear Schr\"odinger equation with inverse-square potential, $$i\partial_t u + \Delta u + \frac{c}{|x|^2}u+|u|^{\frac{4}{N}}u = 0,$$ with $N\geqslant 3$ and $0<c<\frac{(N-2)^2}{4}$. We first prove a sharp global well-posedness result: all $H^1$ solutions with a mass (i.e. $L^2$ norm) strictly below that of the ground states are global. Note that, unlike the equation in free space, we do not know if the ground state is unique in the presence of the inverse-square potential. Nevertheless, all ground states have the same, minimal, mass. We then construct and classify finite time blow-up solutions at the minimal mass threshold. Up to the symmetries of the equation, every such solution is a pseudo-conformal transformation of a ground state solution.