Existence of flexible closed surfaces (non-convex rigidity)

Determine whether there exists a non-convex flexible closed surface in Euclidean three-dimensional space; specifically, ascertain whether a compact boundaryless surface can admit a continuous isometric deformation that preserves both its intrinsic metric and smoothness.

Background

The paper introduces flexible surfaces as those admitting continuous deformations that preserve both metric and smoothness, and recalls the classical rigidity theorem for convex closed surfaces (Cauchy–Alexandrov–Pogorelov). The authors describe the historical question of whether non-convex closed surfaces can be flexible as an open problem.

They then review known progress in the polyhedral setting, including Connelly’s construction of a non-self-intersecting flexible closed polyhedral surface and subsequent examples by Steffen, Kuiper, and others, indicating that polyhedral instances exist. The statement here reflects the formulation of the broader historical question in the Introduction.