Asymptotic rigidity of codimension-1 isometric immersions via quantitative estimates

Abstract: We offer an alternative approach to the asymptotic rigidity of codimension-1 isometric immersions via quantitative rigidity estimates. We show that an immersion between compact manifolds $M$ and $N$ of dimensions $d$ and $d + 1$, respectively, with small stretching plus bending energy is close to an isometric immersion. In this way, we recover the results of Alpern, Kupferman, and Maor. In contrast to their intrinsic approach, we reduce the problem to the equidimensional Euclidean setting and apply the Friesecke-James-Müller rigidity estimate to obtain quantitative results. This yields an elementary proof based on Euclidean techniques. The rigidity estimates are of independent interest.

• The paper introduces a novel Euclidean reduction method to derive quantitative rigidity estimates for codimension-1 isometric immersions.
• It establishes an L^p-based local rigidity theorem, showing that small elastic energy ensures the deformation gradient is close to a rotation with measurable error bounds.
• The study proves asymptotic convergence to genuine isometric immersions and discusses prospects for extending the method to non-compact and higher codimension settings.

Asymptotic Rigidity of Codimension-1 Isometric Immersions via Quantitative Estimates

Context and Motivation

The study of rigidity theorems for isometric immersions occupies a central role in both geometric analysis and the mechanics of elastic bodies. Classical results such as Liouville's theorem and the Friesecke–James–Müller (FJM) quantitative rigidity estimate provide foundational tools for understanding deformations that preserve geometric and metric properties. While the FJM theorem and its extensions furnish powerful quantitative and asymptotic rigidity statements in the equidimensional Euclidean regime, analogous results for isometric immersions of codimension one between Riemannian manifolds require accounting for both local stretching and bending, as encoded by the first and second fundamental forms.

This work provides an alternative, Euclidean-based approach to the asymptotic rigidity of codimension-1 isometric immersions for compact manifolds, focusing on quantitative estimates. The approach diverges from previous intrinsic or Young measure-based proofs, instead leveraging localized FJM-type rigidity estimates after reduction to the Euclidean setting.

Quantitative Rigidity and the Energy Framework

The core idea is to analyze immersions $u : M \to N$, where $M$ and $N$ are compact oriented Riemannian manifolds of dimensions $d$ and $d+1$, respectively. The space $\Imm_p(M; N)$ consists of Sobolev immersions with $\operatorname{rank} du_x = d$ almost everywhere and whose induced unit normal field is in $W^{1,p}$. The relevant energy functional, which integrates both stretching and bending, is defined as

$$E(u) = \int_M \operatorname{dist}_{g, h}^p(du_x, \mathrm{O}((T_x M, g_x), (T_{u(x)}N, h_{u(x)})))\, d\mathrm{vol}_g(x) + \int_M |du_x \circ (S_u(x) - S(x))|_{g, h}^p \, d\mathrm{vol}_g(x)$$

where $S_u$ is the shape operator induced by $M$0 and $M$1 is a reference shape operator.

The first term penalizes metric distortion (stretching), and the second encapsulates deviations of the actual shape operator from the reference (bending). This is a natural generalization of the elastic energy functionals in non-Euclidean elasticity.

Main Contributions

Reduction to the Euclidean Setting

Unlike the intrinsic geometric analyses in previous literature, the paper reduces the isometric immersion problem to Euclidean local neighborhoods, employing Nash's embedding theorem to isometrically embed the target manifold in a Euclidean space. In this regime, the FJM estimate yields sharp $M$2-control of the deformation gradient in terms of its distance from the local isometry group. This transference is facilitated by controlling the variations in tangent and normal spaces of the immersion.

Local Rigidity Estimate

The cornerstone of the analysis is a quantitative local rigidity theorem: if $M$3 is a Sobolev immersion with small elastic energy on a sufficiently small domain, then (after projection) its derivative is $M$4-close to a rotation, up to controllable errors depending on the metric oscillation, energy, and the diameter of the domain. Formally, for $M$5, where $M$6 is a small cube with uniformly controlled metric, there exist $M$7 and $M$8 such that

$M$9

where $N$0 is the stretching energy and $N$1 combines bending and $N$2-norms of $N$3. This estimate crucially allows for patching together local approximations and establishing compactness for sequences with vanishing energies.

Asymptotic Rigidity Theorem

The main asymptotic rigidity theorem demonstrates that any sequence $N$4 of codimension-1 Sobolev immersions with vanishing stretching energy and uniformly bounded bending energy

$N$5

subsequentially converges in $N$6 to an isometric immersion $N$7. Furthermore, if the bending energy relative to a reference shape operator $N$8 vanishes asymptotically, the limiting shape coincides with the reference: $N$9 almost everywhere.

The convergence is achieved by local Euclidean approximation, combinatorial covering and partitioning, patching local results via compactness, and exploiting the uniform boundedness of the energy and metric coefficients.

Notably, the proof does not require the regularity theory or Young measure approach of earlier works but depends essentially on the local quantitative rigidity and careful control of metric and geometric error terms.

Key Technical Elements

• Metric and Projection Control: Lemmas provide control over oscillations in the local metric, equivalence of norms, and variations in tangent and normal spaces. These are vital for transferring Euclidean estimates to the Riemannian context.
• Second Fundamental Form Estimates: The shape operator $d$0 enters both the bending energy and the error analysis. A crucial estimate relates the $d$1-norm of $d$2 (the derivative of the lifted unit normal) to the energy, ensuring compactness of the normal fields.
• Piecewise Approximation and Compactness: Partitioning, local approximation, and the Fréchet-Kolmogorov theorem are used to pass from local to global rigidity. Weak and strong convergence arguments follow.

Implications and Future Directions

The alternative approach developed in this work establishes that asymptotic rigidity for codimension-1 isometric immersions follows directly from quantitative, local Euclidean rigidity estimates and partition approaches, avoiding the machinery of intrinsic geometry or Young measure concentration-compactness. This method yields explicit $d$3 control and a transparent argument, suggesting a template for similar analysis in related problems.

Potential directions include:

• Non-compact Targets: The author references work-in-progress extending the statements to complete, non-compact target manifolds, requiring more technical Sobolev and geometric analysis.
• Higher Codimension and Additional Constraints: While the present work focuses on codimension one, the extension to higher codimension immersions, where the geometry of normal bundles is more complex, is of interest.
• Applications to Non-Euclidean Growth/Elasticity: These results can inform the rigorous derivation of dimensionally reduced models in elasticity, especially for thin shells and non-Euclidean plates, where the interaction between stretching and bending dominates.

The rigidity estimates themselves may serve broader roles in quantitative geometry, inverse problems, and geometric variational analysis.

Conclusion

This paper provides a concise, elementary, and quantitatively precise proof of the asymptotic rigidity of codimension-1 isometric immersions between compact Riemannian manifolds, as sequences with vanishing (stretching + bending) energies converge to genuine isometric immersions. The reduction to Euclidean rigidity, via local projections and energy control, not only recovers but sharpens and clarifies previous intrinsic results. These methods open avenues for direct, quantitative analysis of rigidity problems in geometric analysis and the calculus of variations (2604.09387).