Global Regularity for Defocusing Critical Nonlinear Schrödinger Equations on Hyperbolic Space

We establish the global well-posedness and scattering for the defocusing, energy-critical nonlinear Schrödinger equation (NLS) on the real hyperbolic space Hd for d 3. The problem is critical with respect to the energy norm H1(Hd), corresponding to the nonlinearity power p = 1 + 4d-2. Unlike the Euclidean setting, the hyperbolic geometry induces a spectral gap and exponential volume growth, which significantly alters the dispersive properties of the linear flow. We utilize the concentration-compactness / rigidity method, adapting the profile decomposition to the non-compact Riemannian manifold setting. A key component of our analysis relies on the specific Strichartz estimates derived by Anker and Pierfelice, combined with a Morawetz-type interaction inequality tailored to the hyperbolic metric to rule out the existence of critical elements (blow-up profiles). This work extends the Euclidean theory of Colliander, Keel, Staffilani, Takaoka, and Tao to the constant negative curvature setting.