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The Monge-Ampere system in dimension two and codimension three

Dominik Inauen, Marta Lewicka

TL;DR

This work advances the convex-integration approach to the Monge-Ampère system in dimension two with codimension three by synthesizing Kuiper’s corrugation and Nash’s spirals through a Fibonacci-based stage construction. The authors prove flexibility up to the Hölder regularity $\mathcal{C}^{1,1-1/\sqrt{5}}$, establishing a precise interpolation between prior exponent regimes and surpassing the $1/2$ barrier for this codimension. A two-codimension, cyclic-stage scheme driven by Fibonacci frequencies controls the defect decay and Hessian growth, with rigorous stage bounds that culminate in the final $\mathcal{C}^{1,\alpha}$ solutions for any $\alpha<1-1/\sqrt{5}$. The paper also derives density results for $\mathcal{C}^{1,\alpha}$ weak solutions and relates the findings to isometric immersion problems via the Gauss-curvature correspondence. Overall, the results significantly extend the flexibility landscape for the Monge-Ampère system in low dimensions.

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.

The Monge-Ampere system in dimension two and codimension three

TL;DR

This work advances the convex-integration approach to the Monge-Ampère system in dimension two with codimension three by synthesizing Kuiper’s corrugation and Nash’s spirals through a Fibonacci-based stage construction. The authors prove flexibility up to the Hölder regularity , establishing a precise interpolation between prior exponent regimes and surpassing the barrier for this codimension. A two-codimension, cyclic-stage scheme driven by Fibonacci frequencies controls the defect decay and Hessian growth, with rigorous stage bounds that culminate in the final solutions for any . The paper also derives density results for weak solutions and relates the findings to isometric immersion problems via the Gauss-curvature correspondence. Overall, the results significantly extend the flexibility landscape for the Monge-Ampère system in low dimensions.

Abstract

We revisit the convex integration constructions for the Monge-Ampère system and prove its flexibility in dimension and codimension , up to . To our knowledge, it is the first result in which the obtained Hölder exponent is larger than but it is not contained in the full flexibility up to result. Previous various approaches, based on Kuiper's corrugations, always led to the Hölder regularity not exceeding , while constructions based on the Nash spirals (when applicable) led to the regularity . Combining the two approaches towards an interpolation between their corresponding exponent ranges has been so far an open problem.
Paper Structure (12 sections, 14 theorems, 147 equations)

This paper contains 12 sections, 14 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. An introduction of the method and the proofs in lew_conv.
  3. The improved method in CSlew_improved.
  4. A further improvement and proofs in CHIlew_improved2.
  5. Proofs in the present paper.
  6. The organization of the paper.
  7. Notation.
  8. Preparatory statements
  9. The Källen iteration procedure
  10. The double step construction and its iteration
  11. The Fibonacci frequencies and a proof of Theorem \ref{['thm_stage']}
  12. The Nash-Kuiper scheme and a proof of Theorem \ref{['th_final']}

Key Result

Theorem 1.1

Let $\omega\subset\mathbb{R}^2$ be an open, bounded domain. Given the fields $v\in\mathcal{C}^1(\bar{\omega},\mathbb{R}^3)$, $w\in\mathcal{C}^1(\bar{\omega},\mathbb{R}^2)$ and $A\in\mathcal{C}^{0,\beta}(\bar{\omega},\mathbb{R}^{2\times 2}_\mathrm{sym})$, assume that: in the sense of matrix inequalities. Then, for every exponent $\alpha$ with: and for every $\epsilon>0$, there exists $\tilde{v}\i

Theorems & Definitions (19)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Proposition 4.1
  • ...and 9 more