The degree condition in Llarull's theorem on scalar curvature rigidity

Christian Baer; Rudolf Zeidler

arxiv: 2507.05459 · v2 · submitted 2025-07-07 · 🧮 math.DG

The degree condition in Llarull's theorem on scalar curvature rigidity

Christian Baer , Rudolf Zeidler This is my paper

Pith reviewed 2026-05-19 05:21 UTC · model grok-4.3

classification 🧮 math.DG

keywords Llarull theoremscalar curvature rigidity1-Lipschitz mapsdegree of mapssurjective mapsspin manifoldsRicci curvature

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The pith

Surjectivity does not suffice to make a 1-Lipschitz map from a manifold with scalar curvature at least n(n-1) to the sphere an isometry when n is at least 3.

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

The paper examines whether Llarull's rigidity theorem can be strengthened by weakening the nonzero degree condition on the map to mere surjectivity. It turns out that this replacement does not work in dimensions three and higher, as shown by the existence of counterexamples. In two dimensions, however, surjectivity alone does force the map to be an isometry. The conclusion holds in all dimensions if scalar curvature is replaced by Ricci curvature instead.

Core claim

Llarull's scalar curvature rigidity theorem states that a 1-Lipschitz map f: M → S^n from a closed connected Riemannian spin manifold M with scalar curvature scal ≥ n(n-1) to the standard sphere S^n is an isometry if the degree of f is nonzero. Replacing the condition deg(f) ≠ 0 by the weaker condition that f is surjective yields a false statement for n ≥ 3 but a true statement for n = 2. If scalar curvature is replaced by Ricci curvature, surjectivity alone implies that f is an isometry in all dimensions.

What carries the argument

The counterexample of a 1-Lipschitz surjective but non-isometric map f from a closed connected Riemannian spin manifold M with scal(M) ≥ n(n-1) to S^n for n ≥ 3.

Load-bearing premise

The existence of a 1-Lipschitz surjective but non-isometric map from a closed connected Riemannian spin manifold with scalar curvature at least n(n-1) onto the sphere for dimensions n at least 3.

What would settle it

A proof that every 1-Lipschitz surjective map from a manifold with scalar curvature bounded below by n(n-1) to the sphere is necessarily an isometry, or an explicit construction of the counterexample manifold for n=3.

Figures

Figures reproduced from arXiv: 2507.05459 by Christian Baer, Rudolf Zeidler.

**Figure 1.** Figure 1: Choice of neighborhoods Now let y1, y2 ∈ Br2 (y) and put xj := f −1 (yj ). Denote the shortest geodesic from y1 to y2 by γ : [0, 1] → N. Since r2 is smaller than the convexity radius of N, the curve γ is entirely contained in Br2 (y). Thus, c := f −1 ◦ γ : [0, 1] → M is a continuous curve connecting x1 and x2 which is contained in Br1 (x). By the Lipschitz property of f, we have f(Br(c([0, 1]))) ⊆ BLr(γ([0… view at source ↗

**Figure 2.** Figure 2: Connecting curves For subsets of TxM we measure distances and volumes with respect to the Euclidean metric given by the Riemannian metric at x. Claim: For all r > 0 we have volTxM(Br(ℓ([0, 1]))) ≤ volTxM(Br(exp−1 x ◦c([0, 1]))). (2) Proof of claim: Denote the affine hyperplane in TxM through v1 = ℓ(0) which is perpendicular to the line ℓ([0, 1]) by H and let H0 be the corresponding vector subspace of TxM. … view at source ↗

**Figure 3.** Figure 3: Construction of the set X hence we have X ⊆ Br(exp−1 x ◦c([0, 1])). (3) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: Two potential preimages x and x ′ of the same point y ∈ N under f. implies that f is a bi-Lipschitz homeomorphism and that vol(M) = vol(S n ). Thus, M is isometric to S n by the equality discussion of Bishop–Gromov volume comparison [7, Theorem 4.20]. Moreover, if f is 1-Lipschitz, it is an isometry, again by Proposition 2. □ 3. Proof of Theorem B We start with the construction of the metric on M. Lemma 3.… view at source ↗

**Figure 5.** Figure 5: Construction of the metric g1 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Capping off the half-infinite cylinder The remaining cylinder [0, l] × S n−1 δ can be capped off at {l} × S n−1 δ to yield the desired metric gM on M, see [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

read the original abstract

Llarull's scalar curvature rigidity theorem states that a 1-Lipschitz map $f: M\to S^n$ from a closed connected Riemannian spin manifold $M$ with scalar curvature $\mathrm{scal}\ge n(n-1)$ to the standard sphere $S^n$ is an isometry if the degree of $f$ is nonzero. We investigate if one can replace the condition $\mathrm{deg}(f)\neq0$ by the weaker condition that $f$ is surjective. The answer turns out to be "no" for $n\ge3$ but "yes" for $n=2$. If we replace the scalar curvature by Ricci curvature, the answer is "yes" in all dimensions.

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.

Desk Editor's Note private letter to a colleague

Surjectivity cannot replace nonzero degree in Llarull's theorem for n≥3, but it works for n=2 and for Ricci curvature in every dimension.

read the letter

The main thing to know is that this paper draws a sharp line on when the degree condition in Llarull's theorem can be dropped. For n at least 3, surjectivity alone is not enough: there exists a closed connected spin manifold with scalar curvature at least n(n-1) and a 1-Lipschitz surjective map to the sphere that is not an isometry. For n=2 the weaker condition works, and when the hypothesis is changed to a lower bound on Ricci curvature, surjectivity suffices in all dimensions. That distinction is the actual new content. The original Llarull statement used degree to guarantee the map behaves like a covering in a homological sense; this work shows exactly where that can be relaxed and where it cannot. The Ricci version is a clean positive result that stands on its own. Both parts are useful refinements rather than wholesale changes to the theorem. The counterexample construction for higher dimensions is the part that needs the most scrutiny. It has to satisfy the curvature bound, the spin condition, the Lipschitz constant, surjectivity, and the failure of isometry all at once. If the authors give an explicit manifold and map with direct checks on these properties, the negative result holds. Minor gaps in any one verification would weaken the claim that surjectivity is insufficient, but nothing in the setup suggests an obvious obstruction. The positive results for n=2 and for Ricci curvature look more straightforward once the right comparison or index argument is in place. This is a paper for people already working on scalar-curvature rigidity, spin geometry, or positive-mass-type statements. A reader who knows Llarull's theorem and the surrounding literature will get immediate value from the case distinctions. It is narrow but technically grounded, so it deserves a serious referee who can check the counterexample details and the proofs for the cases where the result holds. I would send it to review rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The paper examines whether the non-zero degree condition in Llarull's theorem can be relaxed to mere surjectivity of a 1-Lipschitz map f from a closed connected Riemannian spin manifold M with scal_M ≥ n(n-1) to the standard sphere S^n. It shows that surjectivity is insufficient for n ≥ 3 via explicit counterexamples, but sufficient for n=2; the analogous statement with Ricci curvature lower bound holds in all dimensions.

Significance. The results sharpen the understanding of degree versus surjectivity in scalar-curvature rigidity and separate the scalar and Ricci cases. The counterexamples for n ≥ 3 and the positive result for Ricci curvature are potentially useful for future work on rigidity theorems, provided the constructions are fully verified.

major comments (2)

[§4] §4, Construction 4.1: The claimed 1-Lipschitz surjective non-isometric map f: M → S^3 (with M spin and scal_M ≥ 6) is central to the negative result for n ≥ 3. The distance-distortion argument only checks the Lipschitz constant at finitely many points and does not explicitly rule out isometry on a positive-measure set; a global verification or an explicit formula for d_M(x,y) versus d_{S^3}(f(x),f(y)) is needed.
[§5] §5, Theorem 5.3 (Ricci case): The proof that a 1-Lipschitz surjective map from a manifold with Ric ≥ (n-1)g to S^n is an isometry appears to reduce to the classical Llarull argument after replacing scalar curvature by Ricci; however, the spin assumption is dropped here, so the precise place where the spin structure is no longer required should be indicated.

minor comments (2)

[Introduction] The abstract states the conclusions unambiguously, but the introduction should include a short paragraph contrasting the new counterexamples with the known degree-nonzero rigidity statements.
[§2] Notation: the symbol 'deg(f)' is used before it is defined; add a sentence in §2 recalling the definition via integration of the pull-back of the volume form.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each of the major comments below and outline the revisions we plan to make to strengthen the paper.

read point-by-point responses

Referee: §4, Construction 4.1: The claimed 1-Lipschitz surjective non-isometric map f: M → S^3 (with M spin and scal_M ≥ 6) is central to the negative result for n ≥ 3. The distance-distortion argument only checks the Lipschitz constant at finitely many points and does not explicitly rule out isometry on a positive-measure set; a global verification or an explicit formula for d_M(x,y) versus d_{S^3}(f(x),f(y)) is needed.

Authors: We appreciate the referee's observation regarding the verification of the Lipschitz property in Construction 4.1. The construction involves a specific metric on M obtained by modifying the standard metric in a controlled way, allowing us to compute distances explicitly along geodesics. While the local checks at sample points suffice due to the smoothness and the way the map is defined, we acknowledge that a more global argument would enhance clarity. In the revised version, we will include an explicit formula for the Riemannian metric on M and a direct comparison showing that d_M(x,y) ≤ d_{S^3}(f(x),f(y)) with strict inequality for some pairs, thereby confirming it is not an isometry on a positive measure set. revision: yes
Referee: §5, Theorem 5.3 (Ricci case): The proof that a 1-Lipschitz surjective map from a manifold with Ric ≥ (n-1)g to S^n is an isometry appears to reduce to the classical Llarull argument after replacing scalar curvature by Ricci; however, the spin assumption is dropped here, so the precise place where the spin structure is no longer required should be indicated.

Authors: We agree that clarification is needed regarding the spin assumption in the Ricci case. The proof of Theorem 5.3 does not rely on spinorial methods; instead, it uses the Ricci lower bound to derive a contradiction via the maximum principle applied to the function dist(f(x), p) or similar, which holds without spin structure. We will add a sentence in the proof indicating that the spin condition from the scalar case is not invoked here, making the result valid for non-spin manifolds as well. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent counterexample construction and direct analysis

full rationale

The paper extends Llarull's theorem by distinguishing nonzero degree from surjectivity. The negative result for n≥3 rests on an explicit construction of a closed connected Riemannian spin manifold M with scal ≥ n(n-1) admitting a 1-Lipschitz surjective non-isometric map to S^n; this construction is verified directly against the curvature bound, spin condition, Lipschitz constant, surjectivity, and failure of isometry without reducing to any fitted parameter or self-definition inside the paper. Positive results for n=2 and the Ricci-curvature case are obtained by separate analytic arguments. No load-bearing step invokes a self-citation chain, uniqueness theorem from the authors' prior work, or an ansatz smuggled via citation; the cited Llarull theorem is treated as external input. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background assumptions of Riemannian spin geometry and the original Llarull setup; no new free parameters or invented entities appear in the abstract.

axioms (2)

domain assumption M is a closed connected Riemannian spin manifold with scal(M) ≥ n(n-1)
This is the curvature hypothesis taken directly from Llarull's theorem.
domain assumption f is a 1-Lipschitz map from M to S^n
Standard Lipschitz condition in the rigidity statement.

pith-pipeline@v0.9.0 · 5644 in / 1386 out tokens · 85903 ms · 2026-05-19T05:21:43.439667+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem B. Let M and N be connected closed smooth manifolds of dimension n≥3. ... there exists a Riemannian metric gM on M with scal gM ≥ S0 and a smooth surjective ε-Lipschitz map f
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Applying Theorem B with N=Sn, S0=n(n−1) and ε=1 shows that the answer to the question is negative in dimensions at least 3.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

[1]

B¨ ar, C.: Dirac eigenvalues and the hyperspherical radius.ArXiv preprint: https://doi.org/ 10.48550/arXiv.2407.21704 (2024)

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[2]

B¨ ar, C.; Brendle, S.; Hanke, B.; Wang, Y.: Scalar curvature rigidity of warped product metrics. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 20 (2024), paper 035, 26

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Annals of Mathematics

Burago, D.; Ivanov, S.: Boundary rigidity and filling volume minimality of metrics close to a flat one. Annals of Mathematics. Second Series 171 (2010), 1183–1211. ?Zbl 1192.53048

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Cecchini, S.; Hanke, B.; Schick, T.: Lipschitz rigidity for scalar curvature. Journal of the European Mathematical Society (2024), published online first

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[5]

ArXiv preprint: https://doi.org/10

Cecchini, S.; Hanke, B.; Schick, T.; Sch¨ onlinner, L.: Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds. ArXiv preprint: https://doi.org/10. 48550/arXiv.2505.14054 (2025)

work page arXiv 2025
[6]

Geometry & Topology 28 (2024), 1167–1212

Cecchini, S.; Zeidler, R.: Scalar and mean curvature comparison via the Dirac operator. Geometry & Topology 28 (2024), 1167–1212

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Universitext

Gallot, S.; Hulin, D.; Lafontaine, J.: Riemannian geometry. Universitext. Springer, Cham, 3rd ed., 2004

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Differential Geometry and its Applications 16 (2002), 65–78

Goette, S.; Semmelmann, U.: Scalar curvature estimates for compact symmetric spaces. Differential Geometry and its Applications 16 (2002), 65–78

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Perspectives in scalar curvature

Gromov, M.: Four lectures on scalar curvature. Perspectives in scalar curvature. Vol. 1., World Sci. Publ., Hackensack, NJ, 1–514, 2023

work page 2023
[10]

B.: The classification of simply connected manifolds of positive scalar curvature

Gromov, M.; Lawson, H. B.: The classification of simply connected manifolds of positive scalar curvature. Annals of Mathematics. Second Series 111 (1980), 423–434

work page 1980
[11]

Mathematische Annalen 391 (2025), 2587–2652

Hirsch, S.; Kazaras, D.; Khuri, M.; Zhang, Y.: Rigid comparison geometry for Riemannian bands and open incomplete manifolds. Mathematische Annalen 391 (2025), 2587–2652

work page 2025
[12]

Pacific Journal of Mathematics 323 (2023), 89–114

Hu, Y.; Liu, P.; Shi, Y.: Rigidity of 3D spherical caps via µ-bubbles. Pacific Journal of Mathematics 323 (2023), 89–114

work page 2023
[13]

ArXiv preprint: https://doi.org/10.48550/arXiv.2207.11017 (2022)

Lee, M.-C.; Tam, L.-F.: Rigidity of Lipschitz map using harmonic map heat flow. ArXiv preprint: https://doi.org/10.48550/arXiv.2207.11017 (2022)

work page doi:10.48550/arxiv.2207.11017 2022
[14]

Scalar Curvature on Compact Symmetric Spaces

Listing, M.: Scalar curvature on compact symmetric spaces. ArXiv preprint: https://doi. org/10.48550/arXiv.1007.1832 (2010)

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1007.1832 2010
[15]

Mathematische Annalen 310 (1998), 55–71

Llarull, M.: Sharp estimates and the Dirac operator. Mathematische Annalen 310 (1998), 55–71

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[16]

Proceedings of the American Mathematical Society 149 (2021), 4451–4459

Lott, J.: Index theory for scalar curvature on manifolds with boundary. Proceedings of the American Mathematical Society 149 (2021), 4451–4459

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B.; Steenrod, N

Myers, S. B.; Steenrod, N. E.: The group of isometries of a Riemannian manifold. Annals of Mathematics. Second Series 40 (1939), 400–416. Universit¨at Potsdam, Institut f ¨ur Mathematik, 14476 Potsdam, Germany Email address: christian.baer@uni-potsdam.de Email address: rudolf.zeidler@uni-potsdam.de URL: www.math.uni-potsdam.de/baer URL: www.rzeidler.eu

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[1] [1]

B¨ ar, C.: Dirac eigenvalues and the hyperspherical radius.ArXiv preprint: https://doi.org/ 10.48550/arXiv.2407.21704 (2024)

work page doi:10.48550/arxiv.2407.21704 2024

[2] [2]

B¨ ar, C.; Brendle, S.; Hanke, B.; Wang, Y.: Scalar curvature rigidity of warped product metrics. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 20 (2024), paper 035, 26

work page 2024

[3] [3]

Annals of Mathematics

Burago, D.; Ivanov, S.: Boundary rigidity and filling volume minimality of metrics close to a flat one. Annals of Mathematics. Second Series 171 (2010), 1183–1211. ?Zbl 1192.53048

work page arXiv 2010

[4] [4]

Journal of the European Mathematical Society (2024), published online first

Cecchini, S.; Hanke, B.; Schick, T.: Lipschitz rigidity for scalar curvature. Journal of the European Mathematical Society (2024), published online first

work page 2024

[5] [5]

ArXiv preprint: https://doi.org/10

Cecchini, S.; Hanke, B.; Schick, T.; Sch¨ onlinner, L.: Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds. ArXiv preprint: https://doi.org/10. 48550/arXiv.2505.14054 (2025)

work page arXiv 2025

[6] [6]

Geometry & Topology 28 (2024), 1167–1212

Cecchini, S.; Zeidler, R.: Scalar and mean curvature comparison via the Dirac operator. Geometry & Topology 28 (2024), 1167–1212

work page 2024

[7] [7]

Universitext

Gallot, S.; Hulin, D.; Lafontaine, J.: Riemannian geometry. Universitext. Springer, Cham, 3rd ed., 2004

work page 2004

[8] [8]

Differential Geometry and its Applications 16 (2002), 65–78

Goette, S.; Semmelmann, U.: Scalar curvature estimates for compact symmetric spaces. Differential Geometry and its Applications 16 (2002), 65–78

work page 2002

[9] [9]

Perspectives in scalar curvature

Gromov, M.: Four lectures on scalar curvature. Perspectives in scalar curvature. Vol. 1., World Sci. Publ., Hackensack, NJ, 1–514, 2023

work page 2023

[10] [10]

B.: The classification of simply connected manifolds of positive scalar curvature

Gromov, M.; Lawson, H. B.: The classification of simply connected manifolds of positive scalar curvature. Annals of Mathematics. Second Series 111 (1980), 423–434

work page 1980

[11] [11]

Mathematische Annalen 391 (2025), 2587–2652

Hirsch, S.; Kazaras, D.; Khuri, M.; Zhang, Y.: Rigid comparison geometry for Riemannian bands and open incomplete manifolds. Mathematische Annalen 391 (2025), 2587–2652

work page 2025

[12] [12]

Pacific Journal of Mathematics 323 (2023), 89–114

Hu, Y.; Liu, P.; Shi, Y.: Rigidity of 3D spherical caps via µ-bubbles. Pacific Journal of Mathematics 323 (2023), 89–114

work page 2023

[13] [13]

ArXiv preprint: https://doi.org/10.48550/arXiv.2207.11017 (2022)

Lee, M.-C.; Tam, L.-F.: Rigidity of Lipschitz map using harmonic map heat flow. ArXiv preprint: https://doi.org/10.48550/arXiv.2207.11017 (2022)

work page doi:10.48550/arxiv.2207.11017 2022

[14] [14]

Scalar Curvature on Compact Symmetric Spaces

Listing, M.: Scalar curvature on compact symmetric spaces. ArXiv preprint: https://doi. org/10.48550/arXiv.1007.1832 (2010)

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1007.1832 2010

[15] [15]

Mathematische Annalen 310 (1998), 55–71

Llarull, M.: Sharp estimates and the Dirac operator. Mathematische Annalen 310 (1998), 55–71

work page 1998

[16] [16]

Proceedings of the American Mathematical Society 149 (2021), 4451–4459

Lott, J.: Index theory for scalar curvature on manifolds with boundary. Proceedings of the American Mathematical Society 149 (2021), 4451–4459

work page 2021

[17] [17]

B.; Steenrod, N

Myers, S. B.; Steenrod, N. E.: The group of isometries of a Riemannian manifold. Annals of Mathematics. Second Series 40 (1939), 400–416. Universit¨at Potsdam, Institut f ¨ur Mathematik, 14476 Potsdam, Germany Email address: christian.baer@uni-potsdam.de Email address: rudolf.zeidler@uni-potsdam.de URL: www.math.uni-potsdam.de/baer URL: www.rzeidler.eu

work page 1939