The Monge-Ampère system in dimension two: A regularity improvement

Abstract

We prove a convex integration result for the Monge-Ampère system introduced in [7], in case of dimension $d=2$ and arbitrary codimension $k\ge 1$. Our prior result [8] stated flexibility up to the Hölder regularity $C^{1,\frac{1}{1+4/k}}$, whereas presently we achieve flexibility up to $C^{1,1}$ when $k\ge 4$ and up to $C^{1,\frac{2^k-1}{2^{k+1}-1}}$ for any k. This first result uses the approach closest to that of Källen [6] in the context of the isometric immersion problem, while the second result uses the double iteration procedure from [7] combined with the approach of Cao-Hirsch-Inauen [1], agreeing with it for $k=1$ at the Hölder regularity up to $C^{1,1/3}$.