  While the shape equations describing the equilibrium of an unstretchable thin
sheet that is free to bend are known, the boundary conditions that supplement
these equations on free edges have remained elusive. Intuitively,
unstretchability is captured by a constraint on the metric within the bulk.
Naively one would then guess that this constraint is enough to ensure that the
deformations determining the boundary conditions on these edges respect the
isometry constraint. If matters were this simple, unfortunately, it would imply
unbalanced torques (as well as forces) along the edge unless manifestly
unphysical constraints are met by the boundary geometry. In this paper we
identify the source of the problem: not only the local arc-length but also the
geodesic curvature need to be constrained explicitly on all free edges. We
derive the boundary conditions which follow. Contrary to conventional wisdom,
there is no need to introduce boundary layers. This framework is applied to
isolated conical defects, both with deficit as well, but more briefly, as
surplus angles. Using these boundary conditions, we show that the lateral
tension within a circular cone of fixed radius is equal but opposite to the
radial compression, and independent of the deficit angle itself. We proceed to
examine the effect of an oblique outer edge on this cone perturbatively
demonstrating that both the correction to the geometry as well as the stress
distribution in the cone kicks in at second order in the eccentricity of the
edge.
