The Monge-Ampere System: Convex Integration in Arbitrary Dimension and Codimension

The Monge-Ampere equation $\det\nabla^2 v =f$ posed on a $d=2$ dimensional domain $\omega$ and in which we are seeking a scalar (i.e. dimension k=1) field v on $\omega$, has a natural weak formulation that appears as the constraint condition in the $\Gamma$-limit of the dimensionally reduced non-Euclidean elastic energies. This formulation reads: $curl^2 (\nabla v\otimes \nabla v) = -2f$ and it allows, via the Nash-Kuiper scheme of convex integration, for constructing multiple solutions that are dense in $C^0(\omega)$, at the regularity $C^{1,\alpha}$ for any $\alpha<1/7$, no matter the sign of the right hand side function $f$.

Does a similar result hold in higher dimensions $d>2$ and codimensions $k>1$? Indeed it does, but one has to replace the Monge-Ampere equation by the Monge-Ampere system, by altering $curl^2$ to the corresponding operator that arises from the prescribed Riemann curvature problem, similarly to how the prescribed Gaussian curvature problem leads to the Monge-Ampere equation in 2d.

Our main result is a proof of flexibility of the Monge-Ampere system at $C^{1,\alpha}$ for any $\alpha<1/(1+d(d+1)/k)$. This finding extends our previous result where $d=2, k=1$, and stays in agreement with the known flexibility thresholds for the isometric immersion problem: with the Conti-Delellis-Szekelyhidi result for $\alpha<1/(1+d(d+1))$ when k=1, as well as with the Kallen result where $\alpha\to$ 1 as k$\to\infty$.
For $d=2$, the flexibility exponent may be even improved to $\alpha<1/(1+4/k)$, using the conformal invariance of 2d metrics to the flat metric. We will also discuss a possible improvement when $d>2$.