The quintic nonlinear Schr\"odinger equation on three-dimensional Zoll manifolds

TL;DR

This paper proves global well-posedness for the quintic nonlinear Schrödinger equation on certain three-dimensional manifolds with all geodesics closed, extending previous results to the critical energy space.

Contribution

It establishes global well-posedness for small initial data and local results for large data on Zoll manifolds, extending prior work to the energy-critical case.

Findings

Global well-posedness for small data in H^1

Local well-posedness for large data

Persistence of higher Sobolev regularity

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 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.