The actin cortex is a complex cytoskeletal machinery that drives and responds to changes in cell shape. It must generate or adapt to plasma membrane curvature to facilitate diverse functions such as cell division, migration, and phagocytosis. Due to the complex molecular makeup of the actin cortex, it remains unclear whether actin networks are inherently able to sense and generate membrane curvature, or whether they rely on their diverse binding partners to accomplish this. Here, we show that curvature sensing is an inherent capability of branched actin networks nucleated by Arp2/3 and VCA. We develop a robust method to encapsulate actin inside giant unilamellar vesicles (GUVs) and assemble an actin cortex at the inner surface of the GUV membrane. We show that actin forms a uniform and thin cortical layer when present at high concentration and distinct patches associated with negative membrane curvature at low concentration. Serendipitously, we find that the GUV production method also produces dumbbell-shaped GUVs, which we explain using mathematical modeling in terms of membrane hemifusion of nested GUVs. We find that branched actin networks preferentially assemble at the neck of the dumbbells, which possess a micrometer-range convex curvature comparable with the curvature of the actin patches found in spherical GUVs. Minimal branched actin networks can thus sense membrane curvature, which may help mammalian cells to robustly recruit actin to curved membranes to facilitate diverse cellular functions such as cytokinesis and migration.