The Monge-Ampère system in dimension two is fully flexible in codimension two

We prove that every ${\mathcal{C}}^1(\bar{\omega })$-regular subsolution of the Monge-Ampère system posed on a $2$-dimensional domain $\omega$ and with target codimension $2$, can be uniformly approximated by its exact solutions with regularity ${\mathcal{C}}^{1,\alpha }(\bar{\omega })$ for any $\alpha <\min \{1,\frac{s+\beta }{2}\}$, where ${\mathcal{C}}^{s,\beta }$ is the assumed regularity of the system’s right hand side. This result suggests the full flexibility of Poznyak’s theorem for isometric immersions of $2$d Riemannian manifolds into ${\mathbb{R}}^4$, and asserts it in the parallel setting of the Monge-Ampère system.