Asymptotic rigidity of codimension-1 isometric immersions via quantitative estimates

1. Problem Statement

The paper addresses the asymptotic rigidity of isometric immersions between Riemannian manifolds where the codomain has a higher dimension than the domain (specifically, codimension 1).

• Context: Classical rigidity results (like Liouville's theorem) state that maps with derivatives in $SO(d)$ are rigid (isometries). In elasticity, this is generalized to "quantitative rigidity": if the "elastic energy" (deviation from isometry) is small, the map is close to a true isometry.
• The Challenge: Standard quantitative rigidity estimates (e.g., Friesecke–James–Müller) apply to equidimensional maps ($\mathbb{R}^d \to \mathbb{R}^d$). They fail for codimension-1 immersions ($M^d \to N^{d+1}$) because the derivative $du$ only captures local stretching, not the bending of the manifold in the ambient space.
• Goal: To prove that if a sequence of immersions has vanishing stretching energy and bounded (or vanishing) bending energy, the sequence converges to a smooth isometric immersion. This recovers and offers an alternative proof to results by Alpern, Kupferman, and Maor [1, 2], who used a more complex "Young measure" approach.

2. Methodology

The author employs a local-to-global strategy based on reducing the Riemannian problem to the Euclidean setting, utilizing the Friesecke–James–Müller (FJM) rigidity estimate.

A. Energy Functionals

The paper defines an elastic energy functional $E(u)$ for an immersion $u: M \to N$:

1. Stretching Energy ($E_s$): Measures the deviation of the differential $du$ from being an isometry.
   $$E_s(u) = \int_M \text{dist}^p(du, O(T_xM, T_{u(x)}N)) \, d\text{vol}_g$$
2. Bending Energy ($E_b$): Measures the curvature of the immersion via the shape operator $S_u$.
   $$E_b(u) = \int_M |du \circ S_u|^p \, d\text{vol}_g$$
   Note: A reference shape operator $S$ can be included to measure deviation from a specific target shape.

B. Core Strategy

1. Reduction to Euclidean Space: Instead of working intrinsically on the manifolds, the target manifold $N$ is isometrically embedded into a high-dimensional Euclidean space $\mathbb{R}^D$ (via Nash's embedding). The immersion is viewed as a map $\bar{u}: M \to \mathbb{R}^D$.
2. Local Rigidity Estimate (Theorem 3.1):
   • The author proves a quantitative rigidity estimate for maps defined on a small cube $Q \subset \mathbb{R}^d$ into $\mathbb{R}^D$.
   • Key Insight: If the bending energy is small, the unit normal vector field $\nu_u$ varies slowly. Consequently, the image of the immersion is locally close to a $d$-dimensional affine subspace.
   • Projection: The map is projected onto this local tangent subspace, effectively turning the codimension-1 problem into an equidimensional problem ($\mathbb{R}^d \to \mathbb{R}^d$).
   • Application of FJM: The standard FJM estimate is applied to this projected map. The error introduced by the projection is controlled by the variation of the normal vector (which is bounded by the bending energy).
3. Global Compactness:
   • Using a partition of unity and the local estimates, the author constructs a sequence of piecewise constant approximations.
   • The Fréchet-Kolmogorov theorem is used to establish compactness, proving that the sequence of derivatives converges in $L^p$.

3. Key Contributions

• Alternative Proof: Provides an elementary proof of the asymptotic rigidity theorem for codimension-1 immersions, avoiding the sophisticated machinery of Young measures used in previous works [1, 2].
• Quantitative Rigidity Estimate: Establishes a new local quantitative estimate (Theorem 3.1) that bounds the $L^p$ distance between the differential of an immersion and a rotation by the sum of stretching energy, bending energy, and the oscillation of the metric.
• Handling Bending: Explicitly incorporates the shape operator and the normal vector field into the rigidity framework, showing how bending energy controls the deviation from equidimensionality.
• Independence of Target Geometry: The method relies on embedding $N$ into $\mathbb{R}^D$, making the proof robust regardless of the specific curvature of $N$ (unlike earlier results that required constant sectional curvature).

4. Main Results

Theorem 3.1 (Local Quantitative Rigidity)

For a Sobolev immersion $u$ on a domain $Q$, there exists a rotation $R$ such that:
$$\int_Q |D\bar{u} - R|^p \leq C \left( |Q|(\text{osc}_Q g)^p + E_s(u) + \text{diam}(Q)E(u) \right)$$
This confirms that small total energy implies the map is locally close to a rigid rotation.

Theorem 4.1 (Asymptotic Rigidity)

Let $(u_k)$ be a sequence of immersions in $\text{Immp}(M; N)$ such that:

1. Stretching energy $E_s(u_k) \to 0$.
2. Bending energy $E_b(u_k)$ is bounded (or converges to 0 if a reference shape is specified).

Conclusion:

• There exists a subsequence converging in $W^{1,p}$ to a smooth isometric immersion $u$.
• The unit normals $\nu_{u_k}$ converge to $\nu_u$ in $W^{1,p}$.
• If the bending energy vanishes relative to a reference shape $S$, then the limit shape operator $S_u$ equals $S$ almost everywhere.

5. Significance

• Simplification: The paper demonstrates that deep results in non-Euclidean elasticity can be derived using elementary Euclidean techniques (specifically the FJM estimate) combined with geometric projections, rather than requiring advanced measure-theoretic tools.
• Foundation for Dimension Reduction: The results are crucial for the rigorous derivation of lower-dimensional models (like plate and shell theories) in non-Euclidean elasticity, where the reference metric is curved and the target space is higher-dimensional.
• Independence: The rigidity estimates derived are of independent interest for the analysis of Sobolev maps between manifolds and could be applied to other problems involving geometric variational problems with bending penalties.

In summary, Ba¸stu˘g successfully bridges the gap between equidimensional rigidity and codimension-1 rigidity by showing that bending energy controls the "dimensionality" of the image, allowing the powerful Euclidean rigidity estimates to be applied locally.