Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces

We prove global well-posedness and scattering for the energy-critical defocusing nonlinear Schrödinger equation {equation*} {cases} i∂_t u + Δu = |u|4/d-2 u, u(0) = u_0, {cases} {equation*} on hyperbolic space Hᵈ(d ≥ 4). We mainly utilize the concentration-compactness/rigidity method. We establish a linear profile decomposition on high-dimensional hyperbolic spaces and prove that the nonlinear profile on hyperbolic spaces can be approximated by the solutions of the energy-critical defocusing nonlinear Schrödinger equation on Euclidean spaces, then obtaining the global well-posedness and scattering of the energy-critical defocusing nonlinear Schrödinger equation on the high-dimensional hyperbolic space.