The Monge-Ampère system: Convex integration with improved regularity in dimension two and arbitrary codimension

We prove a convex integration result for the Monge-Ampère system in dimension $d=2$ and arbitrary codimension $k\geq 1$. We achieve flexibility up to the Hölder regularity $\mathcal{C}^{1,\frac{1}{1+4/k}}$, improving the previous $\mathcal{C}^{1,\frac{1}{1+ 6/k}}$ regularity threshold that followed from flexibility up to $\mathcal{C}^{1,\frac{1}{1+d(d+1)/k}}$ in [10], valid for any $d,k\geq 1$. Our result agrees with flexibility up to $\mathcal{C}^{1,\frac{1}{5}}$ for $d=2, k=1$ obtained in [2], and it fills the previously uncharted codimension interval $k\in \big ( 1, 6\big)$, whereas flexibility up to $\mathcal{C}^{1,1}$ has been showed to hold in [10] at any $k\geq 6$. As an application, we state an energy scaling bound in the quantitative immersability of Riemannian metrics, for the family of nonlinear energy functionals modeled on the energies of deformations of thin prestrained films in nonlinear elasticity [9].