The endosomal sorting complexes required for transport (ESCRT) machinery functions in HIV-1 budding, cytokinesis, multivesicular body biogenesis, and other pathways, in the course of which it interacts with concave membrane necks and bud rims. To test the role of membrane shape in regulating ESCRT assembly, we nanofabricated templates for invaginated supported lipid bilayers. The assembly of the core ESCRT-III subunit CHMP4B/Snf7 is preferentially nucleated in the resulting 100-nm-deep membrane concavities. ESCRT-II and CHMP6 accelerate CHMP4B assembly by increasing the concentration of nucleation seeds. Superresolution imaging was used to visualize CHMP4B/Snf7 concentration in a negatively curved annulus at the rim of the invagination. Although Snf7 assemblies nucleate slowly on flat membranes, outward growth onto the flat membrane is efficiently nucleated at invaginations. The nucleation behavior provides a biophysical explanation for the timing of ESCRT-III recruitment and membrane scission in HIV-1 budding.membrane bending | HIV-1 | nanofabrication | superresolution imaging