Sharp bilinear eigenfunction estimate, $L^\infty_{x_2}L^p_{t,x_1}$-type Strichartz estimate, and energy-critical NLS

We establish sharp bilinear eigenfunction estimates for the Laplace-Beltrami operator on the standard three-sphere $\mathbb{S}^3$, eliminating the logarithmic loss that has persisted in the literature since the pioneering work of Burq, G\'erard, and Tzvetkov over twenty years ago. This completes the theory of multilinear eigenfunction estimates on the standard spheres. Our approach relies on viewing $\mathbb{S}^3$ as the compact Lie group $\mathrm{SU}(2)$ and exploiting its representation theory. Motivated by applications to the energy-critical nonlinear Schr\"odinger equation (NLS) on $\mathbb{R} \times \mathbb{S}^3$, we also prove a refined anisotropic Strichartz estimate on the cylindrical space $\mathbb{R}_{x_1} \times \mathbb{T}_{x_2}$ of $L^\infty_{x_2}L^4_{t,x_1}$-type, adapted to certain spectrally localized functions. The argument relies on multiple sharp measure estimates and a robust kernel decomposition method. Combining these two key ingredients, we derive a refined bilinear Strichartz estimate on $\mathbb{R} \times \mathbb{S}^3$, which in turn yields small-data global well-posedness for the above mentioned NLS in the energy space.