We prove a convex integration result for the Monge-Ampere system in dimension $d=2$ and arbitrary codimension $k\geq 1$. We achieve flexibility up to the Holder regularity $\mathcal{ C}^{ 1,\frac{ 1}{ 1+ 4/k}}$, improving hence the previous $\mathcal{ C}^{ 1,\frac{ 1}{ 1+ 6/k}}$ regularity that followed from flexibility up to $\mathcal{ C}^{ 1,\frac{ 1}{ 1+d(d+1)/k}}$ in our previous work, valid for any $d,k\geq 1$. 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 $k\to\infty$, due to Kallen.
Categories: math.AP
