Construction of an isometric immersion of a bounded, planar region from a framed curve

Abstract

We develop a framework for characterizing isometric immersions of simply connected, bounded, planar regions with piecewise smooth boundaries into three-dimensional space. Each immersion is associated with a framed curve along the boundary of the image surface, comprised by a parametrized curve and a unit normal vector. We identify a set of compatibility and regularity conditions on this framed curve that ensure the existence of a $C^1$ isometric immersion that is $C^2$ almost everywhere and possesses finite bending energy. Under these conditions, we derive an exact dimensional reduction of the bending energy to a line integral over the boundary curve, without relying on asymptotic assumptions or approximations. By analyzing the behavior of the unit normal vector along the framed boundary, we distinguish between planar and curved regions of the immersed surface. We identify the geometric conditions under which global $C^2$ regularity is potentially lost, in which case the associated immersion belongs to $W^{2,2}$ -- a Sobolev space that arises naturally in variational models of unstretchable elastic surfaces.