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# Mathematical Research Letters

## Volume 22 (2015)

### Number 3

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

Pages: 741 – 761

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n3.a7

#### Authors

#### 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.

#### Keywords

nonlinear Schrödinger equation, compact manifold, wellposedness

#### 2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35R01.