Locked and unlocked smooth embeddings of surfaces

We study the continuous motion of smooth isometric embeddings of a planar surface in three-dimensional Euclidean space, and two related discrete analogues of these embeddings, polygonal embeddings and flat foldings without interior vertices, under continuous changes of the embedding or folding. We show that every star-shaped or spiral-shaped domain is unlocked: a continuous motion unfolds it to a flat embedding. However, disks with two holes can have locked embeddings that are topologically equivalent to a flat embedding but cannot reach a flat embedding by continuous motion.