Cellular membranes exhibit a variety of controlled curvatures, with filopodia, microvilli, and mitotic cleavage furrows being only a few of many examples. Coupling between local curvature and chemical composition in membranes could provide a means of mechanically controlling the spatial organization of membrane components. Although this concept has surfaced repeatedly over the years, controlled experimental investigations have proven elusive. Here, we introduce an experimental platform, in which microfabricated surfaces impose specific curvature patterns onto lipid bilayers, that allows quantification of mechanochemical couplings in membranes. We find that, beyond a critical curvature value, membrane geometry governs the spatial ordering of phase-separated domain structures in membranes composed of cholesterol and phospholipids. The curvature-controlled ordering, a consequence of the distinct mechanical properties of the lipid phases, makes possible a determination of the bending rigidity difference between cholesterol-rich and cholesterol-poor lipid domains. These observations point to a strong coupling between mechanical bending and chemical organization that should have wide-reaching consequences for biological membranes. Curvature-mediated patterning may also be useful in controlling complex fluids other than biomembranes.