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. 
Benoit Pausader, Nikolay Tzvetkov, and Xuecheng Wang
math.AP