Global regularity for the energy-critical NLS on ${\mathbb{S}}^3$

We establish global existence for the energy-critical nonlinear Schr"odinger equation on ${\mathbb{S}}^3$. This follows similar lines to the work on ${\mathbb{T}}^3$ but requires new extinction results for linear solutions and bounds on the first nonlinear iterate at a Euclidean profile that are adapted to the new geometry.