The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds

Abstract

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 ${\Bbb S}^3$ with canonical metric. In this work the global well-posedness problem for the quintic nonlinear Schr\"odinger equation $i\partial_t u+\Delta u= \pm |u|^4u$, $u|_{t=0}=u_0$ 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-G\'erard-Tzvetkov to the endpoint case.