While many smartphones now include a waterfall (or curved edge) display, this edge is rarely exploited for input. In this paper, we introduce EdgeMark, a thumb-operated, one-handed, gestural menu to provide rapid and subtle access to commands. In a first study, we assess the movement range of the user during uni-manual input to define a spatial range for our EdgeMark menu. In a second study, we compare three potential EdgeMark designs to a marking menu variant to assess accuracy and performance. Our findings indicate that EdgeMark menus can serve as a reliable, effective mechanism for subtle, one-hand command invocation in modern, waterfall-edged smartphones.