The Monge-Ampere system: convex integration with improved regularity in dimension two and arbitrary codimension

We prove a convex integration result for the Monge-Ampere system in dimension $d=2$ and arbitrary codimension $k\ge 1$. We achieve flexibility up to the Holder regularity ${\mathcal{C}}^{1,\frac{1}{1+4\mathrm{/}k}}$, improving hence the previous ${\mathcal{C}}^{1,\frac{1}{1+6\mathrm{/}k}}$ regularity that followed from flexibility up to ${\mathcal{C}}^{1,\frac{1}{1+d(d+1)\mathrm{/}k}}$ in our previous work, valid for any $d,k\ge 1$. The present result agrees with flexibility up to ${\mathcal{C}}^{1,\frac{1}{5}}$ for $d=2,k=1$ obtained by Conti, Delellis, Szekelyhidi, as well as with the ${\mathcal{C}}^{1,\alpha }$ result where $\alpha \to 1$ as $k\to \mathrm{\infty }$, due to Kallen.