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A piece of paper that has been bent can be modeled as follows.
A bounded simply connected domain \(S \subset \mathbb{R}^2\)
models
the piece of paper in its reference configuration,
before any bending has been applied to it.
The possible shapes obtained after bending are
\(W^{2,2}\)-isometric immersions from \(S\) to \(\mathbb{R}^3\).
The isometry constraint reflects
our intuition that local distances between points along the piece of paper
remain unchanged by bendings.
It is well-known that flat surfaces, i.e. the possible shapes that a piece a paper has after bending,
are developable. Further, the deformations minimising Kirchhoff's nonlinear bending energy (or Willmore energy) are known to
satisfy a partial regularity result. In the talk, we present some results on the geometry of the main part of the singular set, i.e. of the planar regions in the deformed configuration.
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