The Monge–Ampère system in dimension two and codimension three

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.