The Monge-Ampere system in dimension two and codimension three

TL;DR

This paper demonstrates the flexibility of the Monge-Ampère system in dimension two and codimension three, achieving a H"older regularity exponent between 1/2 and 1, which surpasses previous limitations.

Contribution

It introduces a novel convex integration construction that improves the H"older regularity for solutions of the Monge-Ampère system beyond the classical 1/2 barrier.

Findings

Achieves H"older exponent 1 - 1/√5, larger than 1/2.

First construction surpassing the 1/2 regularity barrier.

Combines approaches based on Kuiper's corrugations and Nash spirals.

Abstract

We revisit the convex integration constructions for the Monge-Amp\`ere 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 H\"older 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 H\"older 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.