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

We prove a convex integration result for the Monge-Amp\`ere system, in case of dimension $d=2$ and arbitrary codimension k≥ 1. Our prior result stated flexibility up to the H\"older regularity C1,1/1+ 4/k, whereas presently we achieve flexibility up to C1,1 when k≥ 4 and up to C^1,2ᵏ-12ᵏ⁺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 C1,1/3.