In vivo lipid membranes interact with rough supramolecular structures such as protein clusters and fibrils. How these features whose size ranges from a few nanometers to a few tens of nanometers impact lipid and protein mobility is still being investigated. Here, we study supported phospholipid bilayers, a unique biomimetic model, deposited on etched surfaces bearing nanometric corrugations. The surface roughness and mean curvature are carefully characterized by AFM imaging using ultrasharp tips. Neutron specular reflectivity supplements this surface characterization and indicates that the bilayers follow the large-scale corrugations of the substrate. We measure the lateral mobility of lipids in both the fluid and gel phases by fluorescence recovery after patterned photobleaching. Although the mobility is independent of the roughness in the gel phase, it exhibits a 5-fold decrease in the fluid phase when the roughness increases from 0.2 to 10 nm. These results are interpreted with a two-phase model allowing for a strong decrease in the lipid mobility in highly curved or defect-induced gel-like nanoscale regions. This suggests a strong link between membrane curvature and fluidity, which is a key property for various cell functions such as signaling and adhesion.