Curvature tunes wrinkling in shells

•We have developed a differential-geometry based finite strain shell model for large deformation of soft films.•Curvature can quantitatively tune the wrinkling localization and amplitude.•Wrinkles can be remotely smoothed or even suppressed with increasing curvature.•A 3D phase diagram on stability boundaries was drawn.•We have provided a way for wrinkle-free or wrinkle-resistant surface design via curvature.