  We discuss strong local and global well-posedness for the three-dimensional
NLS equation with nonlinearity concentrated on $\mathbb{S}^2$. Precisely, local
well-posedness is proved for any $C^2$ power-nonlinearity, while global
well-posedness is obtained either for small data or in the defocusing case
under some growth assumptions. With respect to point-concentrated NLS models,
widely studied in the literature, here the dimension of the support of the
nonlinearity does not allow a direct extension of the known techniques and
calls for new ideas.
