Rigidity of codimension-1 isometric immersions in complete manifolds

1. Problem Statement

The paper addresses the asymptotic rigidity of isometric immersions in the context of codimension-1 maps between Riemannian manifolds.

• Context: Classical rigidity results (e.g., Liouville's theorem, Reshetnyak's theorem, Friesecke-James-Müller estimates) state that if the gradient of a map is close to the set of rotations (isometries), the map itself is close to a rigid motion. These results typically apply to maps between spaces of the same dimension ($d \to d$).
• The Gap: In codimension-1 (maps from a $d$-dimensional manifold $M$ to a $(d+1)$-dimensional manifold $N$), standard "stretching" energy (measuring deviation from isometry) is insufficient to guarantee rigidity. A curve in $\mathbb{R}^2$ can have a unit derivative (isometric) but still bend arbitrarily. The lack of rigidity arises because the energy does not penalize bending.
• The Challenge: Previous work by Alpern, Kupferman, and Maor [2] established asymptotic rigidity for codimension-1 maps, but only when the target manifold $N$ is compact. This paper aims to extend these results to complete, non-compact target manifolds $N$.
• The Difficulty: In non-compact settings, sequences of maps may "escape to infinity" or lack compactness, making standard convergence arguments fail. Furthermore, Sobolev maps into non-compact manifolds are generally discontinuous, complicating the use of local charts.

2. Mathematical Framework and Definitions

Energy Functional

The author defines a total energy $E(u)$ for an immersion $u: M \to N$ consisting of two parts:

1. Stretching Energy ($E_s$): Measures the deviation of the differential $du$ from the set of linear isometries.
   $$E_s(u) = \int_M \text{dist}^p(du, O(T_xM, T_{u(x)}N)) \, d\text{vol}_g$$
2. Bending Energy ($E_b$): Measures the variation of the unit normal vector field $\nu_u$. It involves the shape operator $S_u$ induced by the immersion.
   $$E_b(u) = \int_M |du \circ (S_u - S)|^p \, d\text{vol}_g$$
   Here, $S$ is a reference shape operator. If $S=0$, this measures the covariant derivative of the normal vector.

Function Spaces

The paper utilizes colocally weakly differentiable (c.w.d.) maps (Convent and Van Schaftingen [6]) to define Sobolev spaces $W^{1,p}(M; N)$ without embedding $N$ into Euclidean space. This intrinsic approach is crucial for handling complete manifolds.

3. Methodology

The proof strategy avoids the use of Young measures (employed in previous works like [2]) and instead relies on local quantitative rigidity estimates derived via a reduction to the Euclidean setting.

Key Technical Steps:

1. Local Quantitative Rigidity (Theorem 3.7):

   • The author proves a local rigidity estimate for Sobolev immersions defined on a cube $Q \subset \mathbb{R}^d$ mapping into a manifold $N$.
   • Strategy: Use $\epsilon$-isometric charts (charts where the metric is close to Euclidean and Christoffel symbols are small) to map the problem to $\mathbb{R}^{d+1}$.
   • Extension: Since Sobolev maps are not necessarily continuous, the image of a small region might not fit in a single chart. The author uses extended charts (globally defined maps that are diffeomorphisms only on a specific region) to handle this.
   • Projection: The map is projected onto a $d$-dimensional hyperplane. The deviation of the normal vector (controlled by bending energy) ensures the projection error is small.
   • Result: The distance of the map to a rigid motion is bounded by the stretching energy, bending energy, and the oscillation of the metric.

2. Handling Non-Compactness:

   • To prevent sequences from escaping to infinity, the paper assumes a uniform bound on the $L^p$ distance of the sequence to a fixed point $q_0 \in N$ (Condition 4.2).
   • The proof utilizes a partitioning argument: The domain $M$ is divided into small cubes. On "good" cubes where the image stays within a compact region of $N$, the local rigidity estimate applies. On "bad" cubes (where the map might be far away), the measure is shown to be negligible due to the energy bounds and Poincaré inequalities.

3. Convergence:

   • Using the local estimates and the compactness of $M$, the author constructs a Cauchy subsequence in the homogeneous Sobolev space $\dot{W}^{1,p}$.
   • The limit is shown to be an isometric immersion with the correct normal vector field.

4. Key Results

Main Theorem (Theorem 4.1)

Let $(M, g)$ be a compact, oriented, connected $d$-dimensional Riemannian manifold, and $(N, h)$ be a complete, oriented, connected $(d+1)$-dimensional Riemannian manifold.
If a sequence of immersions $(u_k) \subset \text{Imm}^p(M; N)$ satisfies:

1. $E_s(u_k) \to 0$ (vanishing stretching energy),
2. $\limsup E_b(u_k) < \infty$ (bounded bending energy),
3. $\limsup \int_M d_h^p(u_k, q_0) \, d\text{vol}_g < \infty$ (bounded distance to a reference point),

Then, up to a subsequence, $u_k$ converges in $\dot{W}^{1,p}$ to an isometric immersion $u$.
Furthermore, if the bending energy relative to a reference shape operator $S$ vanishes ($E_b^S(u_k) \to 0$) and $|S| \in L^{p'}$, then the limit immersion satisfies $S_u = S$ almost everywhere.

Technical Contributions

• Theorem 3.7: A new local quantitative rigidity estimate for Sobolev immersions into Riemannian manifolds, generalizing the Friesecke-James-Müller (FJM) estimate to the codimension-1 case with bending terms.
• Intrinsic Approach: The proof is entirely intrinsic, avoiding isometric embeddings of $N$ into Euclidean space, which makes the results independent of the specific embedding and applicable to general complete manifolds.
• Avoidance of Young Measures: The method provides a more direct analytical framework compared to the Young measure approach used in [2].

5. Significance and Impact

• Extension of Rigidity Theory: This work bridges a critical gap in geometric analysis by extending asymptotic rigidity results from compact targets to complete non-compact targets. This is essential for applications in elasticity where the ambient space is unbounded.
• Codimension-1 Flexibility: It rigorously demonstrates that adding a bending energy term restores rigidity in codimension-1, quantifying exactly how much bending control is needed to suppress oscillations.
• Methodological Flexibility: The use of extended charts and local quantitative estimates offers a flexible toolkit that can likely be adapted to higher codimensions or other geometric variational problems.
• Applications: The results are relevant for the rigorous derivation of lower-dimensional models (plates, shells, rods) in non-Euclidean elasticity, particularly when the material is embedded in a curved or unbounded space.

In summary, Ba¸stu˘g provides a robust analytical framework proving that sequences of codimension-1 immersions with vanishing stretching and controlled bending energies converge to isometric immersions, even when the target manifold is complete and non-compact.