Pith. sign in

REVIEW 6 major objections 6 minor 9 references

Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds

T0 review · 6 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Cylinder manifolds inherit Euclidean-style rearrangement theory

desk verdict A well-intentioned draft whose central rearrangement on product manifolds is not well-posed and whose Euclidean proofs contain several false steps. read the letter →

arxiv 2411.15412 v1 pith:ZYGP4DOE submitted 2024-11-23 math.DG

Richard Stone This is my paper · ORCID
classification math.DG MSC 53C2053C2149Q20
keywords symmetricdecreasingrearrangementRiemannianmanifoldsco-areaformulaisoperimetricinequalityPólya-Szegőlayer-cakedecompositiontubevolumeHardy-Littlewood
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that the symmetric decreasing rearrangement of a function, a standard tool in Euclidean analysis, can be defined and used on Riemannian manifolds that are cylinders of the form $M = (0,\infty) \times \Sigma$, where $\Sigma$ is a smooth oriented manifold. The central result is a level-set identity: the super-level sets of the rearranged function are exactly the rearrangements of the original super-level sets, with rearrangement to a tube $(0,r^*) \times \Sigma$ determined by equal volume. The paper also aims to prove a smooth co-area formula on such manifolds, which would let perimeter, isoperimetric, and Pólya–Szegő inequalities be re-formulated in this setting. A sympathetic reader would care because these inequalities are powerful tools for PDEs and geometry, and a clean manifold version would extend them beyond flat space.

What carries the argument

The central machinery is the tube-volume function $r \mapsto \int_{(0,r)\times\Sigma} \omega$, used to define the rearrangement $A^*$ by the equation $\int_{(0,r^*)\times\Sigma} \omega = \operatorname{Vol}(A)$, together with the layer-cake decomposition $f^*(x) = \int_0^\infty \chi_{\{f>t\}^*}(x)\,dt$. The paper proves the level-set identity by splitting the layer-cake integral at $t$ and comparing super-level sets, a direct transposition of the Euclidean argument. For the co-area formula, the load-bearing object is the Jacobian $J_\Phi$ of a smooth map between Riemannian manifolds, defined by the Gram determinant, with Sard's theorem used to discard critical points. These two threads—tube-volume rearrangement and the Jacobian-based co-area identity—are what would carry the Euclidean rearrangement theory into the manifold setting.

What would settle it

Take $M = (0,\infty) \times \mathbb{R}^{n-1}$ with the product metric and choose any nonempty bounded open set $A$; then $\int_{(0,r)\times\mathbb{R}^{n-1}} dV = \infty$ for every $r > 0$, so no finite $r^*$ satisfies the defining volume equality, and Definition 1.3 has no output.

Watch

Extended reading notes

Core claim

On a smooth oriented manifold $M^n = (0,\infty) \times \Sigma^{n-1}$ with volume form $\omega$, the paper defines the symmetric rearrangement of an open set $A$ to be the tube $A^* = (0,r^*) \times \Sigma$ satisfying $\int_{A^*} \omega = \int_A \omega$. The function rearrangement is built by layer-cake integration of rearranged super-level sets. The paper's main theorem states that $\{x \in M : f^*(x) > t\} = \{x \in M : f(x) > t\}^*$ for every $t > 0$, and the paper claims this mimics the Euclidean result. The paper further claims a generalised co-area formula for smooth maps $\Phi: M^m \to N^n$, which yields the usual formula $\int_M f |\operatorname{grad} \Phi| \, dV_g = \int_{\mathbb{R}} dy \int_{\Phi^{-1}(y)} f \, dH^{m-1}$. On the basis of these, it recasts the isoperimetric inequality $\operatorname{Per}(A) \ge \operatorname{Per}(A^*)$ and the Pólya–Szegő inequality $\|\operatorname{grad} f\|_p \ge \|\operatorname{grad} f^*\|_p$ as analogs on $M$.

Load-bearing premise

The whole construction rests on the assumption that the volume of the tube $(0,r)\times\Sigma$ is finite for each $r$ and grows from 0 to infinity as $r$ runs from 0 to $\infty$, which fails when $\Sigma$ is noncompact, as in $\Sigma = \mathbb{R}^{n-1}$ where every tube has infinite volume.

Editorial extensions

If this is right

  • If Theorem 4.1 is correct, the Hardy–Littlewood inequality and the non-expansivity of rearrangement under convex integrands transfer verbatim to these product manifolds, yielding new proofs of the isoperimetric and Pólya–Szegő inequalities there.
  • A valid smooth co-area formula on $M$ justifies the perimeter definition $\operatorname{Per}(A) = \int_{\partial A} dH^{n-1}$ and allows the standard layer-cake proof of Pólya–Szegő to be rerun.
  • The rearrangement machinery would give a route to Faber–Krahn-type eigenvalue bounds and Talenti-type comparison results for the Laplacian on these manifolds.
  • The paper's definitions of the isoperimetric and Pólya–Szegő inequalities on $M$ would become usable tools for proving geometric bounds on such manifolds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The compactness of $\Sigma$ is likely a necessary condition for the tube-volume rearrangement: when $\Sigma$ is noncompact (e.g., $\mathbb{R}^{n-1}$ with the product metric), the volume of every tube $(0,r)\times\Sigma$ is infinite, so the defining equation for $r^*$ has no finite solution.
  • The co-area formula of Section 5 is probably independent of the rearrangement definition, since it is derived from the Jacobian and Sard's theorem; this part of the paper could survive even if the rearrangement construction fails.
  • On a compact cross-section such as $S^{n-1}$ with a warped product metric whose tube volumes grow appropriately, the paper's level-set identity should hold; this is a concrete testable prediction of the arguments.
  • The failure for noncompact cross-sections points toward a broader lesson: rearrangement by volume equality only works when the reference tubes provide a finite exhaustion of the manifold, and for noncompact manifolds one would need a reference measure with finite-total-volume truncations.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

6 major / 6 minor

Summary. The paper develops symmetric decreasing rearrangement theory in Euclidean space and then attempts to extend it to Riemannian manifolds of the form M^n=(0,∞)×Σ^{n-1}. The Euclidean part proves (or claims to prove) the Hardy-Littlewood inequality, non-expansivity of rearrangement for convex integrands, Pólya-Szegő and isoperimetric inequalities, with applications to Poisson's equation and Faber-Krahn. The manifold part defines a symmetric rearrangement of sets and functions using tube volumes (0,r)×Σ, states a level-set characterization for the rearrangement of functions, and proves a smooth co-area formula and perimeter definitions on these manifolds. The central advertised result is that the Euclidean rearrangement theory, including the level-set identity {f*>t}={f>t}*, carries over to M^n; the paper also presents three Euclidean proofs of the isoperimetric inequality, one of which proceeds via Pólya-Szegő.

Significance. If the main results were correct, the paper would supply a natural manifold analogue of symmetric decreasing rearrangement for warped-product-like spaces and a self-contained proof of a smooth co-area formula, with potential applications to isoperimetric and Pólya-Szegő inequalities on such manifolds. The paper also collects several standard Euclidean rearrangement arguments and gives some credit to earlier sources. However, the central manifold definition is not well-posed for natural examples such as (0,∞)×R^{n-1} or finite-volume warped products, and several load-bearing proofs in the Euclidean section rest on false identities. The co-area theorem proved is a standard result whose proof here is incomplete. As it stands, the manuscript does not establish its advertised results.

major comments (6)
  1. [Definition 1.3 and Appendix A] The symmetric rearrangement A* is not well-defined for the stated generality. The existence proof in Appendix A asserts that for every finite-volume A there exist r1<r2 with f(r1)<∫_A ω<f(r2), where f(r)=∫_{(0,r)×Σ}ω, but this is exactly what needs proof and is false under the only stated hypothesis (6), which is lim_{r→0} f(r)=0. For M=(0,∞)×R^{n-1} with the product metric, f(r)=∞ for every r>0, so no finite-volume A admits an equal-volume tube. For a finite-volume warped product such as g=dr^2+(1+r)^{-2}g_S on (0,∞)×S^{n-1}, f is bounded and tubes cannot match volumes exceeding the total volume. Consequently A* is undefined for many natural A, and Theorem 4.1, Corollary 4.1, and all subsequent manifold statements quantify over objects that do not exist. The paper needs an explicit hypothesis that f is finite, strictly increasing, and unbounded (or must restrict to sets for which such r* exists); this is a failure of Definition 1.3, not a mere regularity nuisance.
  2. [Theorem 5.2] The proof of the generalised co-area formula is incomplete. Lemma 5.3 is applied to conclude ∫_M f JΦ dH^m = ∫_{R(Φ)} dH^n(y) ∫_{Φ^{-1}[y]} f dH^{m-n}, but Lemma 5.3 holds only at points where dim ker Φ_* = m-n, i.e. at regular points. Sard's theorem, as cited, only states that the set of critical values has measure zero in N; it does not imply that the set of critical points has measure zero in M, which is needed to discard the complementary part of the domain integral. The proof also does not justify measurability of the inner integral or the validity of passing from an identity of differential forms at regular points to an integral over all of M. The announced smooth co-area formula therefore is not proved as written.
  3. [Section 3.1, Eq. (45)] The identity |f(x)-g(x)| = sup_{t∈N} ([f(x)-t]_+ 1_{g≤t} + [g(x)-t]_+ 1_{f≤t}) is false. For example, take f=2.5 and g=1.5; for integer t the expression equals 0 at t=1 and 0.5 at t=2, so the supremum is 0.5, while |f-g|=1. Since this identity is the basis for the proof of the 'Rearrangement Decreases L^p Distance' theorem (Theorem 3.2, second numbering), that proof is invalid. The statement may be true, but the presented argument does not establish it.
  4. [Theorem 3.3] The proof of non-expansivity for convex J uses the representation J_+(f(x)-g(x)) = ∫_{g(x)}^{f(x)} J'_+(s)(f(x)-s) ds. This is incorrect: for J_+(t)=t^2, the right side is not (f-g)^2, and the dimensions do not match. A correct representation would involve J'_+(f(x)-s) as a function of the integration variable, and the subsequent step with the indicator X_{g≤s} does not follow from the displayed formula. The case f<g is also handled by the same formula without explanation. Thus the proof of Theorem 3.3 is invalid as written.
  5. [Lemma 3.7, Eqs. (84)-(86)] The Brunn-Minkowski argument for the sharp isoperimetric inequality contains a scaling error. The volume of the ε-ball is ω_n ε^n, so its 1/n-th power is ε ω_n^{1/n}, not (εω_n)^{1/n} as written. With the expression used, the lower bound in Eq. (86) behaves like ε^{1/n-1} as ε→0 and diverges to +∞; the limit evaluated via L'Hôpital is not the limit of the displayed expression. The proof of Lemma 3.7 is therefore invalid as written, although it could be repaired by replacing (εω_n)^{1/n} with ε ω_n^{1/n}.
  6. [Lemma 3.5 and Section 3.6 proof (1)] There is a circular dependence in the paper's derivation of the two main inequalities. Lemma 3.5 proves the Pólya-Szegő inequality by invoking the isoperimetric inequality: the middle inequality in Eq. (79) compares Per({f>t}) with Per({f*>t}) and uses Lemma 3.4. In Section 3.6, proof (1) proves the isoperimetric inequality by assuming the Pólya-Szegő inequality. Since Lemma 3.5 is the paper's only proof of Pólya-Szegő, the chain Lemma 3.5 → proof (1) is circular. The alternative proofs (2) and (3) avoid this particular circle, but the paper does not acknowledge the logical dependence or present Pólya-Szegő as conditional on the isoperimetric inequality.
minor comments (6)
  1. [Theorem 3.1] The proof writes |lim inf_{s→t} {f>s}| = |{f≥t}|, which is not justified as s→t from above; the correct limiting identity involves s↑t, and the inclusion used in the proof is weaker than the equality claimed. The argument should be rephrased using monotone convergence of the sets {f>s} as s varies.
  2. [Corollary 3.6] The proof uses the identity (Δf)^* = Δf^* in Eq. (104), which is false for general non-radial f. This invalidates the alternative proof of the p=2 Pólya-Szegő inequality. This is not the main proof of Pólya-Szegő, but the statement as written is incorrect.
  3. [Throughout] There are two theorems numbered Theorem 3.2 (Hardy-Littlewood and Rearrangement Decreases L^p Distance); renumbering would avoid confusion.
  4. [Eq. (104)] The factor written as '1/(4πt)' should be the heat-kernel normalization, and 'e^{x^2}' should read 'e^{-|x|^2/4t}'. Also, the expression is evaluated at t=0 after differentiating, but the displayed formula contains t in the denominator without explaining the limiting procedure.
  5. [Appendix A] In the uniqueness proof, the step 'ω>0, so (r~*,r*) has measure 0' assumes that ∫_{Σ} α_r is finite and positive for almost every r; for noncompact Σ this may fail. This is subsumed by Major Comment 1, but should be clarified if the volume-growth hypotheses are added.
  6. [Section 4.1] The tensor review and the orientability calculation in Section 4.1 and Appendix B are not used in the later manifold arguments; the paper could be shortened by moving or removing this material.

Circularity Check

1 steps flagged · score 4.0 of 10

Local circular loop in the '3 different proofs' section: isoperimetric proof (1) assumes Pólya-Szegő, while Pólya-Szegő was proven using the isoperimetric inequality; independent proofs (2) and (3) keep the main theorems non-circular.

  1. other [Section 3.6 proof (1), p. 16, in conjunction with Lemma 3.5, p. 11]
    "Proof. Assuming the Pólya-Szegő Inequality, Per(A) ≈ ∫ |∇φδ|dx = ||∇φδ||1 ≥ ||∇φ∗δ||1 = ∫ |∇φ∗δ|dx ≈ Per(A∗). Taking limits gives the exact result."

    This 'proof' of the isoperimetric inequality uses Pólya-Szegő (Lemma 3.5) as its hypothesis. Lemma 3.5's own proof is not independent: it invokes the isoperimetric inequality at the step 'and use the isoperimetric inequality to get' before equation (79). Hence in this branch ISO is derived from PS and PS is derived from ISO, a mutual-dependence loop rather than a proof from first principles. The circularity is partial: proof (2) from Brunn-Minkowski and proof (3) from Riesz's inequality establish the isoperimetric inequality without this loop, and the manifold section does not depend on proof (1).

full rationale

The principal circularity is internal to Section 3.6's claim of '3 Different Proofs of the Isoperimetric Inequality'. Proof (1) is explicitly conditional on Pólya-Szegő, and Pólya-Szegő (Lemma 3.5) was derived using the isoperimetric inequality, so the two statements are mutually dependent in that branch. I count this as a partial circularity but not a fatal one: proofs (2) and (3) provide independent derivations of the isoperimetric inequality, and the manifold-level Theorems 4.1 and 5.2 do not rely on the loop. There are no load-bearing self-citations; the cited external texts (Lieb-Loss, Chavel, Burchard) supply independent support. I also note, without counting it toward the circularity score, a separate correctness gap: Definition 1.3 and Appendix A assume an r* exists by intermediate value, but hypothesis (6) only gives decay to 0 as r→0; finiteness and unboundedness of the tube-volume function are unproved and fail for noncompact Σ, so the manifold rearrangement can be undefined. That is a missing-support issue, not a circular reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central development rests on standard external theorems and on several unstated assumptions about the tube-volume function and the smoothness of level sets. It also relies on two false identities in proof steps. No free parameters or new entities are used.

assumptions (5)
  • domain assumption Tube volume r↦∫_{(0,r)×Σ}ω is finite, continuous, tends to 0 at 0, and is unbounded.
    Needed for Definition 1.3 and Appendix A existence and uniqueness of A*; not stated; false for (0,∞)×R^(n-1) with product volume.
  • domain assumption Level sets of f are smooth hypersurfaces and ∂{f>t}={f=t} for the perimeter and co-area arguments.
    Used in Definition 3.2, Lemma 3.3, and the Pólya-Szegő proof; the paper only notes ∇f≠0 in one lemma, which does not cover all cases.
  • ad hoc to paper The identities |f-g|=sup over integer t of [f-t]_+X_{g≤t}+[g-t]_+X_{f≤t}, and the J_+ representation in Theorem 3.3, are valid.
    These identities are asserted in the proofs of Theorem 3.2 and Theorem 3.3 and are false as written, so those proofs fail.
  • ad hoc to paper (Δf)^* = Δf^*.
    Used in Corollary 3.6 equation (104); false for symmetric decreasing rearrangement, invalidating the alternative proof of the p=2 Pólya-Szegő inequality.
  • standard math Standard external results: area and co-area formulas, Brunn-Minkowski, Riesz rearrangement inequality, Sard's theorem.
    Quoted from references [3], [4], [6], and [7]; accepted as background, but the paper's applications of them contain errors.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds." pith.science (2026). https://pith.science/paper/ZYGP4DOE

@misc{pith2026241115412,
  author       = {Pith},
  title        = {Pith review of: Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZYGP4DOE}},
  note         = {Machine review of arXiv:2411.15412}
}
abstract

This paper starts by introducing results from geometric measure theory to prove symmetric decreasing rearrangement inequalities on $\mathbb{R}^n$, which give multiple proofs of the isoperimetric and P\'{o}lya-Szeg\H{o} inequalities. Then we consider smooth oriented Riemannian manifolds of the form $M^n = (0,\infty)\times \Sigma^{n-1}$, and test what results carry over from the $\mathbb{R}^n$ setting or what assumptions about $M^n$ need to be added. Of particular interest was proving the smooth co-area formula in the Riemannian manifolds setting and re-formulating particular geometric inequalities.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    Federico II

    Almut Burchard. A Short Course on Rearrangement Inequalities , Universita di Napoli “Federico II” , June 2009

    work page 2009

  2. [2]

    Sobolev and Isoperimetric Inequalities on Riemannian Mani folds, MIT, 2022

    Paige Dote. Sobolev and Isoperimetric Inequalities on Riemannian Mani folds, MIT, 2022

    work page 2022

  3. [3]

    Lectures on Geometric Measure Theory , Stanford University, 1984

    Simon, L. Lectures on Geometric Measure Theory , Stanford University, 1984

    work page 1984

  4. [4]

    Isoperimetric Inequalities: Differential Geometric and An alytic Perspectives , Cambridge University Press, 2001

    Isaac Chavel. Isoperimetric Inequalities: Differential Geometric and An alytic Perspectives , Cambridge University Press, 2001

    work page 2001

  5. [5]

    Sobolev Spaces, Department of Mathematics, Aalto University, 2024

    Juha Kinnunen. Sobolev Spaces, Department of Mathematics, Aalto University, 2024

    work page 2024

  6. [6]

    Lieb, Michael Loss

    Elliott H. Lieb, Michael Loss. Analysis, American Mathematical Society; Second edition, 2001

    work page 2001

  7. [7]

    Riemannian Geometry: A Modern Introduction , Cambridge University Press, 1994

    Isaac Chavel. Riemannian Geometry: A Modern Introduction , Cambridge University Press, 1994

    work page 1994

  8. [8]

    John M. Lee. Riemannian Manifolds: An Introduction to Curvature , Springer-Verlag New York, Inc, 1997

    work page 1997

Show all 9 references
  1. [9]

    John M. Lee. Introduction to Riemannian Manifolds , Springer International Publishing AG, 2018

    2018

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.