The Monge-Ampere system in dimension two and codimension three

Abstract

We revisit the convex integration constructions for the Monge-Ampère system and prove its flexibility in dimension $d=2$ and codimension $k=3$, up to ${\mathcal{C}}^{1,1-1/\sqrt{5}}$. To our knowledge, it is the first result in which the obtained Hölder exponent $1-\frac{1}{\sqrt{5}}$ is larger than $1/2$ but it is not contained in the full flexibility up to ${\mathcal{C}}^{1,1}$ result. Previous various approaches, based on Kuiper's corrugations, always led to the Hölder regularity not exceeding ${\mathcal{C}}^{1,1/2}$, while constructions based on the Nash spirals (when applicable) led to the regularity ${\mathcal{C}}^{1,1}$. Combining the two approaches towards an interpolation between their corresponding exponent ranges has been so far an open problem.