The Monge–Ampère 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 the regularity ${\mathcal{C}}^{1,1-1/\sqrt{5}}$. To our knowledge, it is the first result in which the obtained Hölder exponent $1-1/\sqrt{5}$ exceeds $1/2$ while not falling within the regime of full ${\mathcal{C}}^{1,1}$ flexibility. Previous approaches based on Kuiper’s corrugations consistently led to Hölder regularity not exceeding ${\mathcal{C}}^{1,1/2}$, whereas constructions based on Nash’s spirals (when applicable) achieved ${\mathcal{C}}^{1,1}$ regularity. Bridging these two approaches to interpolate between their respective exponent ranges has, until now, remained an open problem.