Abstract : The Gibbs measure problem consists in constructing a unique global invariant flow supported by the measure itself. We address this problem for the cubic nonlinear Schrödinger equation posed on the two-dimensional sphere \(\mathbb{S}^2\). In this setting, concentration phenomena along certain geodesics rule out the perturbative approaches that are effective on the torus. To overcome these instabilities, we develop a non-perturbative scheme based on a refined analysis of the probabilistic structure of approximate solutions.

Joint work with N. Burq, C. Sun, and N. Tzvetkov