Convex integration for the system of isometric immersions and the Monge-Ampere system

In 1956, N. Kuiper showed that every C^1-regular short immersion of a 2-dimensional Riemannian metric into R^3 can be uniformly approximated by its exact isometric immersions with regularity C^1. Kuiper's result improved upon a prior construction by J. Nash (thus nowadays known as the Nash-Kuiper algorithm) valid for immersions into R^4. In that fundamental work, Nash predicted that: "apparently rigidity disappears completely when the imbedding space has enough dimensions". 

 

We show that for two-dimensional metrics, rigidity indeed disappears completely already in R^4, in the sense that every C^1-regular short immersion of such a metric into R^4 can be uniformly approximated by its exact isometric immersions with regularity C^{1,\alpha} for any \alpha<1. We obtain analogous flexibility result for the Monge-Ampere system, which is a multidimensional version of the two-dimensional Monge-Ampere equation, and extend both results to arbitrary dimension d and the respective codimension k(d)=d_*-d+1 (where d_*=d(d+1)/2 is the Janet dimension). When d=2, we get k(d)=2, so k+d=4.

 

We will also present a unified flexibility statement for arbitrary dimensions d and codimensions k, recovering several previously known results as special cases, while also treating the so far uncharted range k\in (1, d_*-d+1), where no corresponding general result was previously available.