The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds

Let (M,g) be a three-dimensional smooth compact Riemannian manifold such that all geodesics are simple and closed with a common minimal period, such as the 3-sphere S^3 with canonical metric. In this work the global well-posedness problem for the quintic nonlinear Schrödinger equation i\partial_t u+Δu=\pm|u|^4u, u|_{t=0}=u_0 is solved for small initial data u_0 in the energy space H^1(M), which is the scaling-critical space. Further, local well-posedness for large data, as well as persistence of higher initial Sobolev regularity is obtained. This extends previous results of Burq-Gérard-Tzvetkov to the endpoint case.

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