Rigidity and flexibility for isometric embeddings in low dimension

A classical theorem in geometry says that the standard 2-sphere is
 rigid, it is unbendable.
 On the other hand, the celebrated result of Nash and Kuiper shows that
 one can nevertheless
 "crumple" the sphere into an arbitrarily small volume, without
 creating creases! In more precise terms, whilst there is essentially a unique isometric embedding of the sphere of class C2,
 there is a huge set of embeddings of class C1.
 The difference of course lies in the curvature. Borisov proved in the
 1950s that the rigidity result can be extended
 to embeddings with Hölder continuous derivative of order 2/3+, whereas
 in 1965 he announced that the Nash-Kuiper construction - leading to
 flexibility- still works with Hölder contiuous derivatives of order
 1/7-.  In the talk I will revisit these results with a more modern
 "PDE-approach", based on joint work with Sergio Conti and Camillo De
 Lellis. At the end of the talk I will comment on the relevance of the
 latter construction to turbulence.