Developable surfaces with prescribed boundary

TL;DR

This paper proves that a generic closed space curve can bound only finitely many developable surfaces with nonzero mean curvature, highlighting limitations in the surface configurations for given boundaries.

Contribution

It establishes a finiteness result for developable surfaces bounded by a given generic curve, advancing understanding of their geometric constraints.

Findings

Finitely many developable surfaces can bound a generic closed curve.

The result impacts the study of developable surface dynamics.

Provides a theoretical foundation for boundary-surface relationships.

Abstract

It is proved that a generic simple, closed, piecewise regular curve in space can be the boundary of only finitely many developable surfaces with nonvanishing mean curvature. The relevance of this result in the context of the dynamics of developable surfaces is discussed.