The Monge-Ampere system in dimension two: a further regularity improvement

Abstract

We prove a convex integration result for the Monge-Amp\`ere system, in case of dimension $d=2$ and arbitrary codimension $k\geq 1$. Our prior result stated flexibility up to the H\"older regularity $\mathcal{C}^{1,\frac{1}{1+ 4/k}}$, whereas presently we achieve flexibility up to $\mathcal{C}^{1,1}$ when $k\geq 4$ and up to $\mathcal{C}^{1,\frac{2^k-1}{2^{k+1}-1}}$ for any $k$. This first result uses the approach of K\"allen, while the second result iterates on the approach of Cao-Hirsch-Inauen and agrees with it for $k=1$ at the H\"older regularity up to $\mathcal{C}^{1,1/3}$.