Flexibility of Codimension One $C^{1,\theta}$ Isometric Immersions

This paper improves the known threshold for the flexibility of $C^{1,\theta}$ isometric immersions of $n$-dimensional Riemannian metrics into $\mathbb{R}^{n+1}$ by proving that any short immersion can be approximated by such immersions for $\theta < 1/(1+2(n-1))$ when $n \geq 3$, utilizing a refined convex integration scheme with enhanced iterative integration by parts.

The Gist

Here is an explanation of Dominik Inauen's paper, translated from complex mathematical jargon into a story about folding, stretching, and the limits of flexibility.

The Big Picture: The "Rubber Sheet" Problem

Imagine you have a piece of fabric with a specific pattern drawn on it (this is your Riemannian metric). You want to drape this fabric over a table (the Euclidean space) without stretching, tearing, or wrinkling the pattern. You want the distances between any two points on the fabric to remain exactly the same as they were on the flat table.

In the world of smooth, perfect fabrics (mathematicians call this $C^2$ or "twice differentiable"), this is often impossible. If the fabric has a specific curvature, it might be rigid. You can't just fold it into a shape that fits the table; it will either snap or stretch. This is like trying to fold a sheet of glass; it breaks.

However, in 1954, mathematicians John Nash and Nicolaas Kuiper discovered a shocking secret: If you allow the fabric to be a little bit "rough" or "jagged" (mathematically, $C^1$), you can fold it into any shape you want, no matter how complex, without stretching it. You can crumple a map of the world into a ball the size of a grape, and the distances on the map will still be perfect.

The Catch: The "roughness" matters.

• If the fabric is too smooth, it's rigid.
• If it's very rough, it's flexible.
• The Question: Where is the tipping point? Is there a specific level of "roughness" (called a Hölder exponent, $\theta$) where the fabric suddenly stops being able to fold?

The Goal of This Paper

For decades, mathematicians have been trying to find that tipping point.

• Old Result: We knew we could fold the fabric if it was rough enough (specifically, if the roughness was below a certain threshold).
• The New Result: Inauen has pushed that threshold further. He proved that we can fold the fabric even if it is smoother than we previously thought possible.

Think of it like this: Previously, we knew you could crumple a piece of sandpaper. Inauen proved you can actually crumple a piece of fine-grit sandpaper (which is smoother) and still keep the distances perfect.

How Did He Do It? The "Convex Integration" Analogy

To solve this, mathematicians use a technique called Convex Integration. Imagine you are trying to fix a dent in a car door.

1. The Problem: The door is dented (the metric is "short" or missing some length).
2. The Fix: You don't just push it out. Instead, you add tiny, invisible ripples (corrugations) to the metal.
3. The Trick: You add these ripples in different directions. One ripple fixes the dent in the North-South direction, another in East-West.
4. The Iteration: You do this over and over. Each time you add ripples, you fix the remaining errors, but you create tiny new errors. You keep doing this, making the ripples smaller and faster, until the door looks smooth to the naked eye, but mathematically, it has been folded perfectly.

The Innovation: The "Three-Frequency" Dance

The previous best method (by a team including Inauen in a 2023 paper) was like a dance where the dancers moved at different speeds.

• The Old Way: To fix the errors, the dancers had to speed up their steps very quickly. If they didn't, the "noise" (the error) would get too loud, and the math would break. This forced the fabric to be quite rough.
• Inauen's New Way: He realized that the "noise" wasn't just one big roar. It was a complex chord made of three different notes (frequencies):
  1. The speed of the new ripple being added.
  2. The speed of the ripples already there.
  3. The speed of the "glue" (the coefficients) holding them together.

Inauen discovered that if you arrange these three frequencies carefully, they can cancel each other out like noise-canceling headphones.

The Analogy:
Imagine you are trying to silence a loud engine.

• Old Method: You had to build a massive wall (make the fabric very rough) to block the noise.
• New Method: Inauen realized the engine has a specific rhythm. By playing a counter-rhythm (integration by parts) that perfectly matches the engine's vibration, the noise disappears. This allows him to use a much thinner wall (smoother fabric) and still get silence.

The "Subfamily" Secret

The paper introduces a clever organizational trick. Imagine you are organizing a library.

• Old Way: You tried to shelve every book in one giant, chaotic pile.
• New Way: Inauen realized the books (the mathematical errors) belong to specific "families."
  • Some errors are "North-South" errors.
  • Some are "East-West" errors.
  • Some are "Diagonal" errors.

He groups the errors into families. When he fixes the "North-South" family, he doesn't accidentally mess up the "East-West" family because he knows exactly how they interact. By handling them in specific groups (subfamilies), he can keep the "noise" low enough to allow for a smoother fabric.

Why Does This Matter?

This isn't just about folding paper. It's about understanding the fundamental limits of geometry and physics.

• Fluid Dynamics: This same math helps explain how fluids (like air or water) behave. It suggests that even very smooth fluids might have "hidden" turbulence that we can't see but that affects energy conservation.
• Material Science: It helps us understand the limits of how much we can bend or shape materials before they break or lose their shape.

The Bottom Line

Dominik Inauen has taken a mathematical "magic trick" (folding a shape without stretching it) and shown that the trick works with smoother materials than anyone thought possible. He did this by listening more carefully to the "music" of the errors and finding a way to cancel them out, rather than just shouting over them.

He proved that the "tipping point" between rigidity and flexibility is lower than we thought, bringing us one step closer to solving the ultimate puzzle: Exactly how smooth can a shape be before it becomes impossible to fold?

Technical Summary

Here is a detailed technical summary of the paper "Flexibility of Codimension One $C^{1,\theta}$ Isometric Immersions" by Dominik Inauen.

1. Problem Statement

The paper addresses the isometric immersion problem for Riemannian manifolds. Specifically, it investigates the existence of isometric immersions $u: M \to \mathbb{R}^{n+1}$ for a smooth $n$-dimensional Riemannian manifold $(M, g)$, where the immersion has limited regularity, specifically belonging to the Hölder space $C^{1,\theta}$.

• The Equation: The condition for an isometric immersion is given by the system of non-linear PDEs:
  $$Du^T Du = g$$
  where $Du$ is the Jacobian of the map.
• The Dichotomy:
  • Rigidity: For high regularity (e.g., $C^2$), isometric immersions are often rigid (unique up to rigid motions), as seen in the Weyl problem for $S^2$.
  • Flexibility: The classical Nash–Kuiper theorem (1950s) established that for $C^1$ regularity, any short immersion (where the induced metric is strictly less than $g$) can be approximated by an isometric immersion. This implies an abundance of solutions.
• The Open Question: What is the critical Hölder exponent $\theta_0$ that separates rigidity from flexibility?
  • It is known that rigidity holds for $\theta > 2/3$ (Borisov).
  • Flexibility is known for $\theta < 1/(1+n(n+1))$ (Borisov) and improved to $\theta < 1/(1+n(n-1))$ in recent work [9].
  • The paper focuses on the codimension one case ($m=n+1$) for $n \ge 3$, aiming to improve the known flexibility threshold.

2. Methodology

The paper employs the Convex Integration method, a technique pioneered by Nash and Kuiper and refined by Gromov, to construct solutions to underdetermined PDEs. The core of the work lies in a refined iterative integration by parts scheme.

A. The Nash Iteration Scheme

The solution $u$ is constructed as the limit of a sequence of smooth short immersions $\{u_q\}$.

1. Defect Decomposition: At each step $q$, the metric deficit $g - Du_q^T Du_q$ is decomposed into a sum of "primitive metrics" (rank-1 tensors):
   $$g - Du_q^T Du_q = \sum a_i^2 \eta_i \otimes \eta_i$$
2. Perturbation: A high-frequency oscillatory perturbation is added to $u_q$ to cancel these primitive metrics. The ansatz involves corrugations in directions $\eta_i$ with frequencies $\nu_i$.
3. Error Control: The goal is to ensure the new metric deficit decays exponentially while controlling the growth of the $C^2$ norm (which determines the Hölder regularity of the limit).

B. The Key Innovation: Refined Integration by Parts

Previous works (specifically [9]) used an iterative integration by parts procedure to cancel leading-order error terms. However, this procedure introduced "remainders" that could not be canceled if the frequency scaling was too slow, forcing a rapid growth of frequencies and limiting the Hölder exponent.

Inauen's improvement relies on a structural analysis of error terms involving three distinct frequency scales:

1. $\nu_i$: The frequency of the new perturbation.
2. $\nu_{i-1}$: The frequency at which the previous map (and its tangent/normal fields) oscillates.
3. $\nu_0$: The frequency at which the amplitude coefficients oscillate.

The "Subfamily" Strategy:
The author introduces a specific ordering of the primitive metric directions $\eta_{ij}$ based on the standard basis vectors $e_k$.

• Subfamilies: The directions are grouped into subfamilies $\{ \eta_{kl} \}_{l \ge k}$.
• Within a Subfamily: When the current direction $\eta_i$ and the previous direction $\eta_{i-1}$ belong to the same subfamily, the interaction of the frequencies allows the "large remainder" term (which usually forces a frequency jump) to be decomposed further. Specifically, the term $A(\nu_{i-1}x \cdot \eta_{i-1}, x) \otimes \eta_{i-1}$ can be integrated by parts such that the remainder lies in a lower-dimensional subspace $V_{k+1}$, which can be canceled by future perturbations in the sequence.
• Between Subfamilies: When crossing from one subfamily to the next, the standard frequency jump (multiplying by a large constant $K$) is still required.

Resulting Frequency Scaling:
Instead of the exponential growth $\nu_i \approx K^i \nu_0$ required in previous works, the refined scheme allows frequencies to remain nearly constant within a subfamily.

• For $n$ dimensions, there are $n$ subfamilies.
• The frequency only needs to jump $n-1$ times (once per subfamily transition).
• The final frequency scales as $\nu_{final} \approx K^{n-1} \nu_0$ rather than $K^{n^*} \nu_0$ (where $n^* = n(n+1)/2$).

3. Key Contributions

1. Improved Hölder Exponent: The paper proves that for any short immersion, there exists a $C^{1,\theta}$ isometric immersion for:
   $$\theta < \frac{1}{1 + 2(n-1)}$$
   This improves upon the previous best-known exponent $\theta < \frac{1}{1 + n(n-1)}$ for all $n \ge 3$.
   • Example: For $n=3$, the new bound is $\theta < 1/5 = 0.2$, whereas the previous was $\theta < 1/7 \approx 0.14$.
2. Structural Analysis of Error Terms: The paper provides a detailed algebraic decomposition (Lemma 2.3) showing how error terms involving mixed frequencies can be handled if the directions $\eta_i$ and $\eta_{i-1}$ share specific geometric properties (belonging to the same "subfamily").
3. Refined Iterative Integration by Parts: The construction of the corrector vector fields $w$ is optimized to exploit the three-frequency structure, reducing the "loss of derivatives" in the iteration.

4. Main Results

Theorem 1.1: Let $n \ge 2$ and $g \in C^2$ be a Riemannian metric on a bounded domain $\Omega \subset \mathbb{R}^n$. For any short immersion $u \in C^1$, any $\epsilon > 0$, and any Hölder exponent $\theta < \frac{1}{1 + 2(n-1)}$, there exists an isometric immersion $\tilde{u} \in C^{1,\theta}$ such that $\|u - \tilde{u}\|_0 < \epsilon$.

• Extension: The result extends to embeddings and compact manifolds via standard arguments and global frameworks.
• Monge-Ampère: The same mechanism applies to the very weak Monge-Ampère equation, yielding an analogous improvement in the Hölder exponent.

5. Significance

• Closing the Gap: The result moves the known flexibility threshold closer to the conjectured "Onsager threshold" of $\theta = 1/3$ (inspired by fluid dynamics and the Onsager conjecture). While it does not reach $1/3$, it significantly narrows the gap between the known flexibility bound and the rigidity bound ($\theta > 2/3$).
• Methodological Advancement: The paper demonstrates that the "loss of derivatives" in convex integration is not merely a technical artifact but can be mitigated by a deeper understanding of the algebraic structure of the error terms and the interaction of multiple frequency scales.
• Optimality Question: The work suggests that the exponent $1/(1+2(n-1))$might be closer to the true critical threshold for codimension one, challenging previous assumptions that the exponent was limited by the number of equations ($n(n+1)/2$).

In summary, Inauen's work represents a significant step forward in the regularity theory of isometric immersions, utilizing a sophisticated frequency analysis to push the boundary of flexibility in codimension one.