Subduction initiation at straight passive margins can be investigated with two-dimensional (2-D) numerical models, because the geometry is purely cylindrical. However, on Earth, straight margins rarely occur. The construction of 3-D models is therefore critical in the modeling of spontaneous subduction initiation at realistic, curved passive margins. Here we report on the results obtained from gravitationally driven, 3-D thermomechanical numerical models using a visco-plastic rheology and a passive margin with a single curved section in the middle. The models show that the curvature angle β can control subduction initiation: the greater β is, the more difficult subduction initiation becomes. The 3-D thermomechanical models provide an in-depth physical understanding of the processes. Specifically, we find that pressure gradients, arising from density differences between oceanic and continental rocks, drive subduction initiation, and strongly influence the timing. The main difference between straight (cylindrical) and curved margins is that the orientation of the pressure gradient in 3-D is no longer constant, thus producing a horizontal, along-margin component of flow. We thus conclude that the reason for the impedance of subduction initiation is the result of partitioning of the vertical velocity component into a horizontal component, which therefore decreases the effective slab pull. We infer that, although favorable for subduction initiation in a 2-D model, because the estimated force balance is adequate, the pronounced curvature in the southeast Brazilian margin is a likely explanation why subduction initiation is hampered there.