Existence of flexible closed smooth surfaces

Determine whether flexible closed smooth surfaces exist in Euclidean three-dimensional space; concretely, establish the existence or nonexistence of a smooth compact boundaryless surface that admits a continuous isometric deformation preserving both its intrinsic metric and smoothness.

Background

In the Discussion, the authors connect their "base + crinkle" construction to the outstanding smooth-category problem. While flexible closed polyhedral surfaces are known, they emphasize that no flexible closed smooth surface has been constructed.

They suggest that sequences of increasingly refined flexible polyhedral surfaces converging to a smooth limit might offer a constructive path toward resolving this question, underscoring the significance of the problem for both geometry and potential applications.