Full flexibility of the Monge-Ampère system in codimension $d_*-d+1$

1. Problem Statement

The paper addresses the flexibility of solutions to the Monge-Ampère system and the isometric immersion problem in high dimensions.

• The Monge-Ampère System: The authors study the system of equations for a map $v: \omega \subset \mathbb{R}^d \to \mathbb{R}^k$ and a vector field $w: \omega \to \mathbb{R}^d$:
  $$\frac{1}{2}(\nabla v)^T \nabla v + \text{sym}(\nabla w) = A$$
  where $A$ is a given symmetric matrix field. This system is the weak formulation of the Monge-Ampère system (which involves the determinant of the Hessian in higher codimensions) and arises naturally in the study of thin films and elasticity.
• The Isometric Immersion Problem: The system is intrinsically linked to finding an isometric immersion $u: \omega \to \mathbb{R}^{d+k}$ such that $(\nabla u)^T \nabla u = g$, where $g$ is a Riemannian metric.
• The Core Question: Under what conditions on the dimension $d$ and the target codimension $k$ do $C^{1,\alpha}$ solutions exist and are they dense in the space of continuous functions (i.e., is the system "fully flexible")?
• Previous State of the Art:
  • For $d=2, k=1$ (classical Monge-Ampère), flexibility was established for $\alpha < 1/7$, later improved to $\alpha < 1/3$.
  • For general $d$, previous results (e.g., Lewicka [24]) required a large codimension $k \ge 2d^*$ (where $d^* = d(d+1)/2$ is the Janet dimension) to achieve full flexibility for any $\alpha < 1$.
  • The gap between the "rigid" regime (low codimension) and the "flexible" regime (high codimension) was significant.

2. Methodology

The authors employ the Convex Integration technique, specifically the Nash-Kuiper scheme, which involves an iterative construction of solutions by adding high-frequency oscillations (corrugations) to a subsolution.

Key Technical Innovations:

1. Reduced Codimension Target: The main goal is to lower the required codimension from $k \ge 2d^*$ to $k = d^* - d + 1$.
   • Note: $d^* = \frac{d(d+1)}{2}$. Thus, the target codimension is $\bar{d} = \frac{d^2 - d + 2}{2}$.
2. Multi-Stage Corrugation Strategy:
   • The proof constructs a sequence of intermediate subsolutions $(v_k, w_k)$ by iteratively canceling "defects" (the error term $D = A - (\frac{1}{2}(\nabla v)^T \nabla v + \text{sym}\nabla w)$).
   • The defect is decomposed into rank-one matrices of the form $a^2 \eta \otimes \eta$.
   • Step 1 (Codimension 1): The authors use the first codimension direction to cancel defects associated with specific vectors $\eta_{1j}$. This involves a higher-tier induction using $N$ iterations of an integration-by-parts argument (Lemma 2.4) to reduce error orders.
   • Step 2 (Remaining Codimensions): The remaining codimensions ($n=2, \dots, \bar{d}$) are used to cancel the remaining rank-one defects.
3. The "Main Observation" (New Ingredient):
   • In previous works, the error terms generated by adding corrugations often involved second derivatives of the current subsolution ($\nabla^2 v_k$), leading to a rapid blow-up of derivatives.
   • The authors observe that the leading error term involving $\nabla^2 v_k$ can be rewritten as a sum of terms involving only the initial second derivatives ($\nabla^2 v_0$) plus oscillatory terms from previous steps.
   • Because these terms oscillate in fixed directions, they can be further decomposed and reduced using arguments similar to Lemma 2.4, but without the $P_i^j$ components (which usually carry the worst error growth).
   • This allows for a smaller blow-up rate of the second derivatives relative to the decay rate of the defect.
4. Iterative Scheme (Nash-Kuiper):
   • The construction is repeated $K$ times.
   • By choosing parameters $N$ and $K$ sufficiently large, the ratio of the blow-up rate of the second derivatives to the decay rate of the defect can be made arbitrarily small.
   • This ensures that the sequence of approximate solutions converges in the $C^{1,\alpha}$ norm for any $\alpha < 1$.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

Let $\omega \subset \mathbb{R}^d$ be a bounded domain. Let the target codimension be $k = d^* - d + 1$.
Given a subsolution $(v, w)$ satisfying the strict inequality $\frac{1}{2}(\nabla v)^T \nabla v + \text{sym}\nabla w < A$, and for any $\epsilon > 0$ and any Hölder exponent $\alpha < 1$, there exist $C^{1,\alpha}$ functions $\tilde{v}, \tilde{w}$ such that:

1. $\|\tilde{v} - v\|_0 \le \epsilon$ and $\|\tilde{w} - w\|_0 \le \epsilon$.
2. $\frac{1}{2}(\nabla \tilde{v})^T \nabla \tilde{v} + \text{sym}(\nabla \tilde{w}) = A$ exactly.

Significance: This establishes full flexibility (density of solutions) for the Monge-Ampère system in the optimal codimension $k = d^* - d + 1$ for all $\alpha < 1$. This significantly improves upon the previous requirement of $k \ge 2d^*$.

Corollary 1.2 (Weak Solutions)

The set of $C^{1,\alpha}$ weak solutions to the Monge-Ampère system (in the sense of distributions) is dense in $C^0$ for any $\alpha < 1$, provided the codimension is at least $d^* - d + 1$.

Extension to Isometric Immersions (Theorem 5.1)

The same proof scheme extends to the isometric immersion problem. The authors prove that any $C^2$ Riemannian metric $g$ on a $d$-dimensional manifold admits a $C^{1,\alpha}$ isometric immersion into $\mathbb{R}^{d + (d^* - d + 1)} = \mathbb{R}^{d^* + 1}$.

• This generalizes the result of Lewicka [27] (which was for $d=k=2$) to arbitrary dimensions.
• Using techniques from [12], this can be extended to compact manifolds in codimension $k \ge (d+1)d^* - d + 1$.

4. Significance of the Work

1. Optimization of Codimension: The paper closes a major gap in the theory of flexible isometric immersions and Monge-Ampère systems. It shows that full flexibility is achievable in a codimension that is roughly half of what was previously thought necessary ($d^* - d + 1$ vs $2d^*$).
2. Regularity Threshold: The result holds for all Hölder exponents $\alpha < 1$. This is the "full flexibility" regime, contrasting with the "rigid" regime where solutions are unique or non-existent for high regularity.
3. Methodological Advancement: The core innovation—rewriting the error terms to depend on the initial data rather than the current iterate's high derivatives—provides a new tool for convex integration. This technique could be applicable to other geometric PDEs where controlling the growth of derivatives is the primary bottleneck.
4. Unification: The work unifies the study of the Monge-Ampère system and isometric immersions, showing that the flexibility properties of the former directly imply the latter in high codimensions.

Summary

Cao, Hirsch, Inauen, and Lewicka have proven that the Monge-Ampère system and isometric immersion problems are fully flexible in the codimension $k = \frac{d^2 - d + 2}{2}$ for all $C^{1,\alpha}$ regularities with $\alpha < 1$. This is achieved by refining the convex integration scheme to control the blow-up of second derivatives more efficiently than previous methods, thereby lowering the dimensional threshold required for flexibility.