The energy-critical nonlinear Schrödinger equation on a product of spheres

Abstract

Let $(M,g)$ be a compact smooth $3$-dimensional Riemannian manifold without boundary. It is proved that the energy-critical nonlinear Schrödinger equation is globally well-posed for small initial data in $H^1(M)$, provided that a certain tri-linear estimate for free solutions holds true. This estimate is known to hold true on the sphere and tori in $3d$ and verified here in the case $\mathbb{S}\times {\mathbb{S}}^2$. The necessity of a weak form of this tri-linear estimate is also discussed.

v2: 18 pages, minor revision, Section 4 added

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