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arxiv: 2604.00562 · v2 · submitted 2026-04-01 · 🧮 math.DG

Rigidity of the Borell-Brascamp-Lieb Inequality on Weighted Riemannian Manifolds

Rongkai Zhang This is my paper

Pith reviewed 2026-05-13 22:23 UTC · model grok-4.3

classification 🧮 math.DG
keywords rigidity theoremBorell-Brascamp-Lieb inequalityweighted Riemannian manifoldsBrunn-Minkowski inequalitycurvature boundsequality cases
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The pith

The Borell-Brascamp-Lieb inequality exhibits rigidity on weighted Riemannian manifolds, forcing constant curvature and specific measures for equality.

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

The paper proves a rigidity theorem for the Borell-Brascamp-Lieb inequality on weighted Riemannian manifolds. This extends the curvature rigidity result of Balogh and Kristály to the weighted setting by characterizing the manifolds and weights where equality is attained. A sympathetic reader would care because such results clarify the geometric conditions under which functional inequalities become sharp, linking analysis and geometry. It also includes some rigidity statements for the Brunn-Minkowski inequality along with open questions.

Core claim

The paper proves that if a weighted Riemannian manifold satisfies the conditions for the Borell-Brascamp-Lieb inequality to hold with equality in the extremal case, then the manifold has constant sectional curvature and the measure is determined by the weight function in a specific way, as stated in the generalized rigidity theorem.

What carries the argument

The rigidity theorem on curvature and measure for the Borell-Brascamp-Lieb inequality, which identifies the equality cases by imposing bounds on Ricci curvature and the weight.

Load-bearing premise

The weighted Riemannian manifold satisfies the curvature and measure conditions needed for the generalization of the Balogh-Kristály rigidity theorem to hold.

What would settle it

Finding a weighted Riemannian manifold with non-constant curvature where the Borell-Brascamp-Lieb inequality achieves equality without satisfying the rigid conditions would disprove the theorem.

read the original abstract

In this paper we discuss some results regarding the rigidity of the Borell-Brascamp-Lieb inequality and the Brunn-Minkowski inequality. We show a theorem of rigidity on curvature and measure of the Borell-Brascamp-Lieb inequality, a generalisation of the curvature rigidity theorem by Balogh and Krist\'aly (Advances in Mathematics 339: 453-494, 2018, arXiv:1704.04180) to the weighted setting. We present some rigidity results of the Brunn-Minkowski inequality and a few further open problems.

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 establishes a rigidity theorem for the Borell-Brascamp-Lieb inequality on weighted Riemannian manifolds, generalizing the curvature rigidity result of Balogh and Kristály. Equality cases are shown to imply constancy of the weighted curvature (via the Bakry-Émery Ricci tensor) and appropriate behavior of the weight function under the given curvature and measure conditions. The manuscript also presents some rigidity results for the Brunn-Minkowski inequality and lists further open problems.

Significance. If the central theorem holds, the work provides a natural extension of rigidity phenomena to the weighted setting, which is relevant for geometric analysis involving densities and Bakry-Émery curvature bounds. This strengthens the link between functional inequalities and geometric rigidity on manifolds with measures, with potential implications for optimal transport and comparison geometry in the weighted case.

minor comments (2)
  1. [Abstract] Abstract: the statement of the main result would be clearer if it briefly indicated the key technical replacement (weighted measure in place of Riemannian volume, Bakry-Émery Ricci in place of ordinary Ricci) rather than only naming the cited theorem.
  2. [References] The citation to Balogh and Kristály should be expanded in the bibliography to include the full journal reference (Advances in Mathematics 339: 453-494, 2018) alongside the arXiv number already mentioned in the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. The assessment correctly identifies the main contribution as a generalization of the Balogh-Kristály rigidity theorem to the weighted Riemannian setting via the Bakry-Émery Ricci tensor. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; generalization of external cited theorem

full rationale

The paper frames its central result as a generalization of the Balogh-Kristály curvature rigidity theorem (arXiv:1704.04180, different authors) to the weighted Riemannian setting via the Borell-Brascamp-Lieb inequality. The abstract and description indicate the derivation adapts the external theorem by replacing volume with weighted measure and Ricci with Bakry-Émery tensor, without reducing to self-defined quantities, fitted parameters renamed as predictions, or load-bearing self-citations. No steps match the enumerated circularity patterns; the result is self-contained against the cited independent prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the result is described as a generalization relying on standard weighted Riemannian geometry.

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    K. Bacher. On Borell-Brascamp-Lieb Inequalities on Metric Measure Spaces . Potential Analysis , 33 0 (1): 0 1--15, 2010. ISSN 0926-2601

    work page 2010

  2. [2]

    Z. M. Balogh and A. Krist\'aly. Equality in Borell–Brascamp–Lieb Inequalities on Curved Spaces . Advances in Mathematics, 339: 0 453--494, 2018. ISSN 0001-8708

    work page 2018

  3. [3]

    C. Borell. Convex Set Functions in d-space . Periodica Mathematica Hungarica, 6 0 (2): 0 111--136, 1975. ISSN 0031-5303

    work page 1975

  4. [4]

    H. J. Brascamp and E. H. Lieb. On Extensions of the Brunn-Minkowski and Prékopa-Leindler Theorems, including Inequalities for Log Concave Functions, and with an Application to the Diffusion Equation . Journal of Functional Analysis, 22 0 (4): 0 366--389, 1976. ISSN 0022-1236

    work page 1976

  5. [5]

    Cavalletti and A

    F. Cavalletti and A. Mondino. A Sharp Isoperimetric-type Inequality for Lorentzian Spaces Satisfying Timelike Ricci Lower Bounds . arXiv.org, 2025. ISSN 2331-8422. URL https://arxiv.org/abs/2401.03949

    work page arXiv 2025

  6. [6]

    Cordero-Erausquin, R

    D. Cordero-Erausquin, R. J. McCann, and M. Schmuckenschläger. A Riemannian Interpolation Inequality à la Borell, Brascamp and Lieb . Inventiones Mathematicae, 146 0 (2): 0 219--257, 2001. ISSN 0020-9910

    work page 2001

  7. [7]

    De Philippis and N

    G. De Philippis and N. Gigli. From Volume Cone to Metric Cone in the Nonsmooth Setting . Geometric and Functional Analysis, 26 0 (6): 0 1526--1587, 2016. ISSN 1016-443X

    work page 2016

  8. [8]

    S. Dubuc. Critères de Convexité et Inégalités Intégrales . Annales de l'Institut Fourier, 27 0 (1): 0 135--165, 1977. ISSN 1777-5310

    work page 1977

  9. [9]

    Farooqui

    O. Farooqui. Equivalence between the Timelike Brunn-Minkowski Inequality and Timelike Bakry-Émery-Ricci Lower Bound on Weighted Globally Hyperbolic Spacetimes . arXiv.org, 2025. ISSN 2331-8422. URL https://arxiv.org/abs/2504.06186

    work page arXiv 2025

  10. [10]

    Local Semiconvexity of Kantorovich Potentials on Non-compact Manifolds

    Alessio Figalli and Nicola Gigli. Local Semiconvexity of Kantorovich Potentials on Non-compact Manifolds . ESAIM. Control, Optimisation and Calculus of Variations, 17 0 (3): 0 648--653, 2011. ISSN 1292-8119

    work page 2011

  11. [11]

    R. J. Gardner. The Brunn-Minkowski Inequality . Bulletin of the American Mathematical Society, 39 0 (3): 0 355--405, 2002. ISSN 0273-0979

    work page 2002

  12. [12]

    Henstock and A

    R. Henstock and A. M. Macbeath. On the Measure of Sum-Sets. (I) The Theorems of Brunn, Minkowski, and Lusternik . Proceedings of the London Mathematical Society, s3-3 0 (1): 0 182--194, 1953. ISSN 0024-6115

    work page 1953

  13. [13]

    Lott and C

    J. Lott and C. Villani. Ricci Curvature for Metric-Measure Spaces via Optimal Transport . Annals of Mathematics, 169 0 (3): 0 903--991, 2009. ISSN 0003-486X

    work page 2009

  14. [14]

    Magnabosco, L

    M. Magnabosco, L. Portinale, and T. Rossi. The Brunn–Minkowski Inequality Implies the CD Condition in Weighted Riemannian Manifolds . Nonlinear Analysis, 242: 0 113502--, 2024. ISSN 0362-546X

    work page 2024

  15. [15]

    R. J. McCann. Polar Factorization of Maps on Riemannian Manifolds . Geometric and Functional Analysis, 11 0 (3): 0 589--608, 2001. ISSN 1016-443X

    work page 2001

  16. [16]

    R. J. McCann. Displacement Convexity of Boltzmann's Entropy Characterizes the Strong Energy Condition from General Relativity . Cambridge Journal of Mathematics, 8 0 (3): 0 609--681, 2020. ISSN 2168-0930

    work page 2020

  17. [17]

    Mondino and S

    A. Mondino and S. Suhr. An Optimal Transport Formulation of the Einstein Equations of General Relativity . Journal of the European Mathematical Society : JEMS, 25 0 (3): 0 933--994, 2023. ISSN 1435-9855

    work page 2023

  18. [18]

    S. Ohta. Finsler Interpolation Inequalities . Calculus of Variations and Partial Differential Equations, 36 0 (2): 0 211--249, 2009. ISSN 0944-2669

    work page 2009

  19. [19]

    Otto and C

    F. Otto and C. Villani. Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality . Journal of Functional Analysis, 173 0 (2): 0 361--400, 2000. ISSN 0022-1236

    work page 2000

  20. [20]

    K.-T. Sturm. On the Geometry of Metric Measure Spaces. II . Acta Mathematica, 196 0 (1): 0 133--177, 2006. ISSN 0001-5962

    work page 2006

  21. [21]

    C. Villani. Topics in Optimal Transportation . Graduate studies in mathematics ; v. 58. American Mathematical Society, Providence, RI, 2003. ISBN 082183312x

    work page 2003

  22. [22]

    C. Villani. Optimal Transport : Old and New . Grundlehren der mathematischen Wissenschaften: a series of Comprehensive Studies in Mathematics, 338. Springer, Berlin, 1st ed. 2009. edition, 2009. ISBN 1-281-85120-5

    work page 2009