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

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