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arxiv: 2604.09387 · v1 · submitted 2026-04-10 · 🧮 math.AP · math.DG

Asymptotic rigidity of codimension-1 isometric immersions via quantitative estimates

Mert Ba\c{s}tu\u{g} This is my paper

Pith reviewed 2026-05-10 17:30 UTC · model grok-4.3

classification 🧮 math.AP math.DG
keywords asymptotic rigiditycodimension-1 immersionsisometric immersionsquantitative estimatesstretching energybending energycompact manifolds
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The pith

An immersion between compact manifolds of dimensions d and d+1 with small stretching plus bending energy must stay close to an isometric immersion.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that immersions with small total stretching plus bending energy between compact manifolds of consecutive dimensions must be close to actual isometric immersions. This provides a quantitative version of rigidity for such maps. The approach reduces the manifold problem to a Euclidean one and invokes a known rigidity estimate there. This matters because it offers an elementary proof of asymptotic rigidity results that were previously obtained by more intrinsic geometric methods.

Core claim

We show that an immersion between compact manifolds M and N of dimensions d and d+1 with small stretching plus bending energy is close to an isometric immersion, by reducing the problem to the equidimensional Euclidean setting and applying a quantitative rigidity estimate valid there. This yields an elementary proof based on Euclidean techniques and recovers prior asymptotic rigidity results while producing estimates of independent interest.

What carries the argument

The reduction of the codimension-1 immersion problem on compact manifolds to the equidimensional Euclidean setting, which allows direct application of a quantitative rigidity estimate that bounds distance to isometries in terms of the combined stretching and bending energy.

If this is right

  • Immersions become asymptotically rigid as the stretching plus bending energy tends to zero.
  • The distance to an isometric immersion is controlled quantitatively by the size of the energy.
  • Prior asymptotic rigidity results for codimension-1 immersions are recovered via this Euclidean reduction.
  • The resulting estimates apply independently to other energy-based problems on manifolds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique might extend to non-compact manifolds provided suitable decay conditions control the energy at infinity.
  • Analogous quantitative estimates could simplify rigidity proofs for immersions in codimension greater than one.
  • Numerical checks on explicit manifolds such as spheres could test how sharp the constants in the energy-to-distance bounds are.

Load-bearing premise

The reduction of the codimension-1 immersion problem on compact manifolds to the equidimensional Euclidean setting preserves the quantitative estimates without significant loss or additional errors.

What would settle it

A sequence of immersions on a compact manifold such as the sphere or torus where the combined stretching and bending energy tends to zero but the distance to any isometric immersion remains bounded away from zero.

read the original 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-M\"uller rigidity estimate to obtain quantitative results. This yields an elementary proof based on Euclidean techniques. The rigidity estimates are of independent interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper presents an alternative proof of asymptotic rigidity for codimension-1 isometric immersions of compact manifolds. It claims that an immersion f: M^d → N^{d+1} with small stretching-plus-bending energy is quantitatively close (in W^{2,2} or similar norms) to an isometric immersion, obtained by reducing the problem to the equidimensional Euclidean setting and invoking the Friesecke-James-Müller rigidity estimate. This recovers the results of Alpern-Kupferman-Maor via Euclidean techniques rather than an intrinsic approach.

Significance. If the reduction step preserves the quantitative constants (depending only on the fixed geometry of M and N), the work supplies an elementary, Euclidean-based proof of a known rigidity result together with quantitative estimates that are of independent interest. The approach avoids intrinsic curvature arguments and may facilitate extensions to other settings where Euclidean rigidity tools are available.

minor comments (2)
  1. §2: The precise definition of the stretching and bending energies (e.g., how the second fundamental form enters the bending term) should be stated explicitly before the reduction argument begins, to make the transfer of constants transparent.
  2. The statement of the main theorem could include the explicit dependence of the closeness constant on the geometry of M and N (diameter, curvature bounds, etc.) rather than leaving it implicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, for highlighting the elementary Euclidean approach and the independent interest of the quantitative estimates, and for recommending acceptance. We are pleased that the reduction to the equidimensional setting and application of the Friesecke-James-Müller estimate are viewed as providing a useful alternative to the intrinsic methods of Alpern-Kupferman-Maor.

Circularity Check

0 steps flagged

No significant circularity; derivation reduces to independent FJM estimate

full rationale

The paper derives its main result by reducing the codimension-1 compact manifold immersion problem to the equidimensional Euclidean setting and invoking the established Friesecke-James-Müller quantitative rigidity estimate. This reduction transfers the estimates with constants controlled solely by the fixed manifold geometry, introducing no uncontrolled errors in the small-energy regime. The FJM estimate originates from independent prior literature with no author overlap, serving as external input rather than a self-citation chain. No self-definitional steps, fitted inputs renamed as predictions, or ansatz smuggling appear in the provided derivation outline. The approach is self-contained against external benchmarks, with the central claim retaining independent content beyond the reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions in differential geometry and analysis, including compactness of the manifolds and the validity of reducing the problem to Euclidean space while preserving quantitative control.

axioms (2)
  • domain assumption The manifolds M and N are compact.
    Explicitly stated in the abstract as the setting for the immersions and the result.
  • domain assumption The Friesecke-James-Müller rigidity estimate applies directly after reduction to the equidimensional Euclidean setting.
    This is the key step invoked to obtain the quantitative closeness to isometry.

pith-pipeline@v0.9.0 · 5407 in / 1439 out tokens · 116315 ms · 2026-05-10T17:30:42.013797+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Rigidity of codimension-1 isometric immersions in complete manifolds

    math.AP 2026-04 unverdicted novelty 6.0

    Vanishing elastic energy implies subsequence convergence to an isometric immersion for codimension-1 maps into complete manifolds.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

  1. [1]

    Asymptotic rigidity for shells in non-Euclidean elasticity

    Itai Alpern, Raz Kupferman, and Cy Maor. Asymptotic rigidity for shells in non-Euclidean elasticity. J. Funct. Anal., 283(6):Paper No. 109575, 38, 2022

    work page 2022

  2. [2]

    Stability of isometric immersions of hypersurfaces.Forum Math

    Itai Alpern, Raz Kupferman, and Cy Maor. Stability of isometric immersions of hypersurfaces.Forum Math. Sigma, 12:Paper No. e43, 27, 2024

    work page 2024

  3. [3]

    Rigidity of codimension-1 isometric immersions in complete manifolds

    Mert Ba¸ stu˘ g. Rigidity of codimension-1 isometric immersions in complete manifolds. Forthcoming

    work page

  4. [4]

    Optimal rigidity estimates for maps of a compact Riemannian manifold to itself.SIAM J

    Sergio Conti, Georg Dolzmann, and Stefan M¨ uller. Optimal rigidity estimates for maps of a compact Riemannian manifold to itself.SIAM J. Math. Anal., 56(6):8070–8095, 2024

    work page 2024

  5. [5]

    Rigidity and gamma convergence for solid-solid phase transitions with SO(2) invariance.Comm

    Sergio Conti and Ben Schweizer. Rigidity and gamma convergence for solid-solid phase transitions with SO(2) invariance.Comm. Pure Appl. Math., 59(6):830–868, 2006

    work page 2006

  6. [6]

    Intrinsic co-local weak derivatives and Sobolev spaces between manifolds.Ann

    Alexandra Convent and Jean Van Schaftingen. Intrinsic co-local weak derivatives and Sobolev spaces between manifolds.Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 16(1):97–128, 2016

    work page 2016

  7. [7]

    James, Maria Giovanna Mora, and Stefan M¨ uller

    Gero Friesecke, Richard D. James, Maria Giovanna Mora, and Stefan M¨ uller. Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence.C. R. Math. Acad. Sci. Paris, 336(8):697–702, 2003. 14

    work page 2003

  8. [8]

    James, and Stefan M¨ uller

    Gero Friesecke, Richard D. James, and Stefan M¨ uller. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity.Comm. Pure Appl. Math., 55(11):1461–1506, 2002

    work page 2002

  9. [9]

    James, and Stefan M¨ uller

    Gero Friesecke, Richard D. James, and Stefan M¨ uller. A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence.Arch. Ration. Mech. Anal., 180(2):183–236, 2006

    work page 2006

  10. [10]

    Reshetnyak rigidity for Riemannian manifolds.Arch

    Raz Kupferman, Cy Maor, and Asaf Shachar. Reshetnyak rigidity for Riemannian manifolds.Arch. Ration. Mech. Anal., 231(1):367–408, 2019

    work page 2019

  11. [11]

    Scaling laws for non-Euclidean plates and theW 2,2 isometric immersions of Riemannian metrics.ESAIM Control Optim

    Marta Lewicka and Mohammad Reza Pakzad. Scaling laws for non-Euclidean plates and theW 2,2 isometric immersions of Riemannian metrics.ESAIM Control Optim. Calc. Var., 17(4):1158–1173, 2011

    work page 2011

  12. [12]

    Mathematical problems in thin elastic sheets: scaling limits, packing, crumpling and singularities

    Stefan M¨ uller. Mathematical problems in thin elastic sheets: scaling limits, packing, crumpling and singularities. InVector-valued partial differential equations and applications, volume 2179 ofLecture Notes in Math., pages 125–193. Springer, Cham, 2017. 15

    work page 2017