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In this paper we present a method to study global regularity properties of solutions of large-data critical Schrödinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global Morawetz inequality adapted to the geometry of the manifold (in other words we adapt the method of Kenig and Merle to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (a theorem of Colliander, Keel, Staffilani, Takaoka and Tao).
As an application we prove global well-posedness and scattering in for the energy-critical defocusing initial-value problem
We consider a generalized Ricci flow with a given (not necessarily closed) three-form and establish higher-derivative estimates for compact manifolds. As an application, we prove the compactness theorem for this generalized Ricci flow. Similar results still hold for a more generalized Ricci flow.
In specific types of partially rectangular billiards we estimate the mass of an eigenfunction of energy in the region outside the rectangular set in the high-energy limit. We use the adiabatic ansatz to compare the Dirichlet energy form with a second quadratic form for which separation of variables applies. This allows us to use sharp one-dimensional control estimates and to derive the bound assuming that is not resonating with the Dirichlet spectrum of the rectangular part.
We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments. This result was obtained previously by Colliander, Keel, Staffilani, Takaoka, and Tao.