pith. sign in

arxiv: 2604.11130 · v1 · submitted 2026-04-13 · 🧮 math.AP · math.DG

Rigidity of codimension-1 isometric immersions in complete manifolds

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

Pith reviewed 2026-05-10 16:01 UTC · model grok-4.3

classification 🧮 math.AP math.DG
keywords isometric immersionscodimension oneelastic energyrigiditycomplete manifoldsasymptotic convergencequantitative estimates
0
0 comments X

The pith

Sequences of immersions with vanishing elastic energy into complete manifolds converge to isometric immersions.

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

The paper establishes that a sequence of codimension-one immersions from a compact manifold into a complete higher-dimensional manifold, with elastic energy measuring stretch and bend going to zero, admits a subsequence converging to an exact isometric immersion. This extends prior results that required the target manifold to be Euclidean or compact, by overcoming the analytical issues from non-compactness. The argument proceeds through local quantitative rigidity estimates that reduce the problem locally to the Euclidean case. A reader would care because the result broadens the settings where one can conclude that near-isometries are close to true isometries without additional compactness assumptions on the ambient space.

Core claim

We establish an asymptotic rigidity result for isometric immersions of codimension-1. We consider a sequence of immersions from a compact d-dimensional manifold into a complete (d+1)-dimensional manifold whose elastic energies vanish asymptotically and show that such a sequence admits a subsequence converging to an isometric immersion. The proof relies on local quantitative rigidity estimates obtained via reduction to the Euclidean setting.

What carries the argument

Local quantitative rigidity estimates obtained via reduction to the Euclidean setting, which control the deviation from isometry on small charts even when the target manifold lacks compactness.

If this is right

  • A subsequence of the given immersions converges in a suitable topology to a limit that is an isometric immersion.
  • The result applies without requiring compactness of the target manifold.
  • The proof supplies a framework based on local Euclidean reductions that avoids Young measures.
  • The same local estimates may support similar rigidity statements for other energies or nearby geometric settings.

Where Pith is reading between the lines

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

  • The local-reduction technique could be tested on immersions into hyperbolic space or other model non-compact manifolds to check convergence rates explicitly.
  • One could ask whether analogous statements hold for higher codimension or for sequences that are only approximately codimension one.
  • The avoidance of Young measures suggests the method might adapt to numerical approximation schemes for thin elastic structures in unbounded domains.

Load-bearing premise

The local quantitative rigidity estimates obtained via reduction to the Euclidean setting continue to hold when the target is a general complete manifold rather than Euclidean space or a compact manifold.

What would settle it

An explicit sequence of immersions from a compact surface into hyperbolic three-space whose elastic energy tends to zero but for which no subsequence converges to an isometric immersion would disprove the claim.

read the original abstract

We establish an asymptotic rigidity result for isometric immersions of codimension-1. Specifically, we consider a sequence of immersions from a compact $d$-dimensional manifold into a complete $(d+1)$-dimensional manifold whose elastic energies vanish asymptotically, where the elastic energy quantifies both stretching and bending. We show that such a sequence admits a subsequence converging to an isometric immersion. This extends a result of Alpern, Kupferman, and Maor to the case of complete target manifolds, where the lack of compactness introduces additional analytical difficulties. The proof is based on an approach using local quantitative rigidity estimates, obtained via a reduction to the Euclidean setting. This method avoids the use of Young measures and provides a flexible framework that may be 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

2 major / 2 minor

Summary. The paper establishes an asymptotic rigidity result for sequences of codimension-1 immersions from a compact d-dimensional manifold into a complete (d+1)-dimensional manifold. When the elastic energy (measuring both stretching and bending) vanishes asymptotically, a subsequence converges to an isometric immersion. This extends the result of Alpern, Kupferman, and Maor from compact or Euclidean targets to general complete manifolds by using local quantitative rigidity estimates obtained via reduction to the Euclidean setting, avoiding Young measures.

Significance. If the central claim holds, the result is significant for geometric analysis and nonlinear elasticity, as it handles the lack of compactness in complete targets while providing a flexible local-reduction framework that may apply more broadly. The avoidance of Young measures and the explicit extension of prior work are strengths; the method's potential independent interest is noted in the abstract.

major comments (2)
  1. [Proof of the main result (reduction and compactness extraction)] The load-bearing step is the claim that local quantitative rigidity estimates (obtained by reduction to the Euclidean case) remain valid with constants that permit a global compactness argument on the complete target. The skeptic's concern is valid here: in a general complete (d+1)-manifold, sectional curvatures and injectivity radii need not be uniformly bounded, so the distortion constants in the C^1 or W^{2,2} closeness to isometries become point-dependent. Without an a-priori bound on the images of the sequence (derived from vanishing energy) or a uniform-control argument, the diagonal extraction may fail to produce a globally isometric limit. This must be addressed explicitly in the proof of the main theorem.
  2. [Section on local quantitative rigidity estimates] The abstract states that the reduction technique 'avoids the use of Young measures and provides a flexible framework.' However, the manuscript must verify that the Euclidean rigidity constants (from the cited quantitative estimates) transfer without additional curvature-dependent error terms that accumulate when the basepoint varies over the complete manifold. If the local charts or normal coordinates introduce pointwise dependence, the vanishing-energy hypothesis alone may not suffice for uniform control.
minor comments (2)
  1. [Introduction] The notation for the elastic energy functional should be introduced with an explicit formula (including the precise stretching and bending terms) already in the introduction, rather than deferred to the preliminaries, to make the vanishing-energy hypothesis immediately readable.
  2. [Introduction] A brief comparison table or paragraph contrasting the new result with Alpern-Kupferman-Maor (compact case) and with existing results on non-compact targets would clarify the precise novelty and the additional analytical difficulties mentioned.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the two major comments point by point below, providing clarifications on the uniformity arguments and committing to revisions that make the relevant estimates explicit.

read point-by-point responses

  1. Referee: [Proof of the main result (reduction and compactness extraction)] The load-bearing step is the claim that local quantitative rigidity estimates (obtained by reduction to the Euclidean case) remain valid with constants that permit a global compactness argument on the complete target. The skeptic's concern is valid here: in a general complete (d+1)-manifold, sectional curvatures and injectivity radii need not be uniformly bounded, so the distortion constants in the C^1 or W^{2,2} closeness to isometries become point-dependent. Without an a-priori bound on the images of the sequence (derived from vanishing energy) or a uniform-control argument, the diagonal extraction may fail to produce a globally isometric limit. This must be addressed explicitly in the proof of the main theorem.

    Authors: We agree that an explicit uniformity argument is required and will add it to the revised proof. The vanishing of the elastic energy (which controls both the L^2 stretching term and the W^{2,2} bending term) yields, via the quantitative rigidity estimates, a uniform C^1 closeness of the immersions to isometries. Because the domain is compact, this closeness prevents the images from escaping to infinity in the complete target manifold; consequently the images lie in a compact subset. In a tubular neighborhood of this compact set the sectional curvatures are bounded above and the injectivity radii are bounded below by positive constants (by continuity of these quantities). These uniform geometric bounds allow the local Euclidean constants to be chosen independently of basepoint, so that the diagonal extraction produces a globally isometric limit. A new paragraph detailing this a-priori bound and its consequences for the compactness argument will be inserted in the proof of the main theorem. revision: yes

  2. Referee: [Section on local quantitative rigidity estimates] The abstract states that the reduction technique 'avoids the use of Young measures and provides a flexible framework.' However, the manuscript must verify that the Euclidean rigidity constants (from the cited quantitative estimates) transfer without additional curvature-dependent error terms that accumulate when the basepoint varies over the complete manifold. If the local charts or normal coordinates introduce pointwise dependence, the vanishing-energy hypothesis alone may not suffice for uniform control.

    Authors: We concur that the transfer of constants must be verified explicitly and will expand the relevant section. The reduction proceeds in normal coordinates centered at points of the image; the metric approximation error is O(r^2) with r the chart radius, controlled by the sectional curvature. The uniform image bound obtained from vanishing energy (as detailed in the response to the first comment) supplies a uniform positive lower bound on injectivity radius and upper bound on curvature throughout a fixed tubular neighborhood. Consequently the Euclidean rigidity constants carry over with a uniform multiplicative factor independent of basepoint. Because the domain is compact, only finitely many such charts are needed, so no accumulation of errors occurs. The revised section will include a precise statement of these error controls together with the resulting uniform constants. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external prior estimates via standard reduction

full rationale

The central claim extends the Alpern-Kupferman-Maor result to complete targets by invoking local quantitative rigidity estimates obtained through reduction to the Euclidean setting. This reduction and the estimates are drawn from prior independent work rather than defined in terms of the target convergence result or fitted to it. No self-citation chains, self-definitional steps, or renamings of known patterns appear as load-bearing elements in the derivation. The argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a theorem in differential geometry and analysis. It relies on standard background results about Riemannian manifolds and isometric immersions rather than introducing new fitted quantities or entities.

axioms (2)
  • standard math Standard properties of Riemannian metrics and isometric immersions on smooth manifolds hold, including the existence of local charts reducing to Euclidean space.
    Invoked implicitly when reducing local quantitative rigidity estimates to the Euclidean setting.
  • domain assumption The elastic energy functional is well-defined and lower semicontinuous for immersions into complete manifolds.
    Required for the asymptotic vanishing to imply convergence; appears in the setup of the sequence.

pith-pipeline@v0.9.0 · 5420 in / 1357 out tokens · 50279 ms · 2026-05-10T16:01:13.893041+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

  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]

    Asymptotic rigidity of codimension-1 isometric immersions via quantitative estimates,

    Mert Ba¸ stu˘ g. Asymptotic rigidity of codimension-1 isometric immersions via quantitative estimates,

    work page

  4. [4]

    https://arxiv.org/abs/2604.09387

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    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

  6. [6]

    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

  7. [7]

    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

  8. [8]

    Mathematics: Theory & Applications

    Manfredo Perdig˜ ao do Carmo.Riemannian geometry. Mathematics: Theory & Applications. Birkh¨ auser Boston, Inc., Boston, MA, portuguese edition, 1992

    work page 1992

  9. [9]

    Evans and Ronald F

    Lawrence C. Evans and Ronald F. Gariepy.Measure theory and fine properties of functions. Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015

    work page 2015

  10. [10]

    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

    work page 2003

  11. [11]

    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

  12. [12]

    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

  13. [13]

    Universitext

    Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine.Riemannian geometry. Universitext. Springer-Verlag, Berlin, third edition, 2004

    work page 2004

  14. [14]

    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

  15. [15]

    Lee.Manifolds and differential geometry, volume 107 ofGraduate Studies in Mathematics

    Jeffrey M. Lee.Manifolds and differential geometry, volume 107 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2009

    work page 2009

  16. [16]

    Lee.Introduction to Riemannian manifolds, volume 176 ofGraduate Texts in Mathematics

    John M. Lee.Introduction to Riemannian manifolds, volume 176 ofGraduate Texts in Mathematics. Springer, Cham, second edition, 2018

    work page 2018

  17. [17]

    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

  18. [18]

    Michor.Topics in differential geometry, volume 93 ofGraduate Studies in Mathematics

    Peter W. Michor.Topics in differential geometry, volume 93 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008

    work page 2008

  19. [19]

    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. 27

    work page 2017

  20. [20]

    Yu. G. Reshetnyak. Liouville’s conformal mapping theorem under minimal regularity hypotheses. Sibirsk. Mat. ˇZ., 8:835–840, 1967

    work page 1967

  21. [21]

    Yu. G. Reshetnyak. Stability of conformal mappings in multi-dimensional spaces.Sibirsk. Mat. ˇZ., 8:91–114, 1967

    work page 1967

  22. [22]

    On the differential geometry of tangent bundles of Riemannian manifolds

    Shigeo Sasaki. On the differential geometry of tangent bundles of Riemannian manifolds. II.Tohoku Math. J. (2), 14:146–155, 1962. 28

    work page 1962