Extremal metrics involving scalar curvature

Shota Hamanaka

arxiv: 2504.06547 · v4 · submitted 2025-04-09 · 🧮 math.DG

Extremal metrics involving scalar curvature

Shota Hamanaka This is my paper

Pith reviewed 2026-05-22 21:18 UTC · model grok-4.3

classification 🧮 math.DG

keywords extremal metricsscalar curvaturerigidity theoremsRiemannian manifoldsnon-rigiditycurvature rigidity

0 comments

The pith

Sufficient conditions show when extremal metrics fail to be rigid under scalar curvature theorems.

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

The paper studies extremal metrics in relation to rigidity theorems that tie properties of the metric to its scalar curvature. It supplies explicit sufficient conditions under which a metric satisfying those theorems is nevertheless not rigid. Concrete examples of Riemannian manifolds meeting the conditions are constructed to illustrate the failure of rigidity. The work therefore maps out cases where the rigidity conclusions do not hold. Readers interested in the scope of curvature rigidity results would use these conditions to separate rigid from non-rigid examples.

Core claim

We investigate extremal metrics at which various types of rigidity theorems involving scalar curvatures hold. The rigidity we discuss here is related to the rigidity theorems presented by Mario Listing in his previous preprint. More specifically, we give some sufficient conditions for metrics not to be rigid in this sense. We also give several examples of Riemannian manifolds that satisfy such sufficient conditions.

What carries the argument

Sufficient conditions for non-rigidity of extremal metrics with respect to scalar-curvature rigidity theorems, together with explicit manifold examples that meet those conditions.

If this is right

Extremal metrics on certain manifolds escape the rigidity conclusions previously established.
The set of rigid extremal metrics is properly smaller than the full class of extremal metrics.
Concrete manifold examples now exist where scalar curvature does not force uniqueness or rigidity of the metric.
The boundary between rigid and non-rigid cases can be located by checking the supplied sufficient conditions.

Where Pith is reading between the lines

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

The same style of sufficient conditions might be tested on other curvature functionals beyond scalar curvature.
The examples could serve as test cases for numerical or computational searches for additional non-rigid metrics.
Classification efforts for extremal metrics may now need to incorporate these non-rigidity criteria as a standard check.

Load-bearing premise

The rigidity theorems from the earlier preprint remain valid and apply directly to the extremal metrics examined in this work.

What would settle it

An explicit Riemannian manifold that meets all the stated sufficient conditions yet still satisfies the rigidity conclusion of the scalar-curvature theorems.

read the original abstract

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

Hamanaka gives sufficient conditions for non-rigidity of extremal metrics tied to Listing's theorems plus some manifold examples, but the work stays tightly coupled to that prior preprint.

read the letter

The main takeaway is that this paper takes Mario Listing's rigidity theorems for extremal metrics involving scalar curvature and turns them around to supply sufficient conditions under which the metrics fail to be rigid. It also lists several Riemannian manifold examples that satisfy those conditions. The contribution is the explicit non-rigidity side and the concrete cases rather than new rigidity proofs from scratch. That adds some useful illustrations for people already tracking this narrow slice of scalar-curvature work. The examples appear to be the clearest addition, since they give readers something specific to check against the rigidity statements. The citation pattern is straightforward and appropriate for an incremental extension. The main limitation is the heavy dependence on the earlier preprint. The sufficient conditions and the examples are defined in terms of Listing's results, so any gap in how the extremal metrics fit the hypotheses carries straight over. There is no sign of an independent applicability check or a self-contained derivation that would stand if the prior theorems needed adjustment. This keeps the overall strength tied to the quality of that earlier work. The paper is aimed at specialists in Riemannian geometry who already know Listing's preprint. A reader outside that small group will find little to engage with. It is narrow in scope and does not reorganize broader ideas. I recommend sending it for peer review. The topic is well-defined and the examples could be worth documenting once a referee verifies the derivations and the fit with the cited theorems.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates extremal metrics at which various rigidity theorems involving scalar curvature hold, in the sense of Mario Listing's prior preprint. It supplies sufficient conditions under which such metrics fail to be rigid and provides several examples of Riemannian manifolds satisfying those conditions.

Significance. If the sufficient conditions are rigorously derived and the examples correctly satisfy the hypotheses of the referenced rigidity theorems, the work would clarify the scope of rigidity results for extremal metrics. The contribution is primarily conditional on the prior framework; no independent derivations, parameter-free constructions, or machine-checked elements are indicated.

major comments (1)

[Abstract and main text (no numbered sections or equations supplied)] The central claims (sufficient conditions for non-rigidity and the listed examples) rest entirely on the unverified applicability of the rigidity theorems from Listing's previous preprint. No section derives or checks that the extremal metrics and curvature functionals considered here lie inside the hypotheses of those theorems; any gap in the prior work transfers directly to the new results.

minor comments (1)

The manuscript is extremely brief; consider adding explicit statements of the curvature functionals, the precise definition of extremal metrics used, and at least one fully worked example with curvature computations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting the dependence on the prior framework. We address the major comment below and will incorporate revisions to improve clarity.

read point-by-point responses

Referee: The central claims (sufficient conditions for non-rigidity and the listed examples) rest entirely on the unverified applicability of the rigidity theorems from Listing's previous preprint. No section derives or checks that the extremal metrics and curvature functionals considered here lie inside the hypotheses of those theorems; any gap in the prior work transfers directly to the new results.

Authors: We agree that the manuscript would be strengthened by an explicit verification step. The sufficient conditions and examples are constructed to lie within the hypotheses of the rigidity theorems from Listing's preprint, as the extremal metrics are defined using the same scalar curvature functionals and the manifolds are selected to satisfy the relevant curvature and metric assumptions stated there. Nevertheless, the current text does not include a dedicated check or cross-reference for each example. In the revised version we will add a short subsection (or appendix) that lists the hypotheses of the referenced theorems and confirms, for each example, that they hold. This addresses the concern without altering the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results extend external rigidity theorems without internal reduction

full rationale

The manuscript defines its notion of rigidity by explicit reference to Mario Listing's separate prior preprint and then supplies sufficient conditions plus examples for non-rigidity in that sense. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear. The derivation chain consists of conditional statements whose validity rests on the external theorems' hypotheses being met; this is a standard dependency on prior literature rather than any reduction of the paper's claims to its own inputs by construction. The provided abstract and framing contain no equations or steps that equate outputs to inputs within the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; the sole identifiable assumption is reliance on the validity of the rigidity theorems from the referenced prior preprint. No free parameters, invented entities, or additional axioms are visible.

axioms (1)

domain assumption Rigidity theorems from Mario Listing's previous preprint hold and apply to the extremal metrics under study.
The paper's investigation of non-rigidity is defined relative to those theorems.

pith-pipeline@v0.9.0 · 5561 in / 1189 out tokens · 34344 ms · 2026-05-22T21:18:08.563804+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 2 internal anchors

[1]

M. T. Anderson, On uniqueness and differentiability in the space of Yamabe metrics, CCM7, no. 3 (2005), 299–310

work page 2005
[2]

Andrews and C

B. Andrews and C. Hopper, The Ricci flow in Riemannian geometry: a complete proof of the differentiable 1/4-pinching sphere theorem, Lecture Notes in Mathematics2011, Berlin, Heidelberg: Springer Nature (2010)

work page 2010
[3]

B¨ ar, Dirac eigenvalues and the hyperspherical radius, arXiv:2407.21704v2 (2024)

C. B¨ ar, Dirac eigenvalues and the hyperspherical radius, arXiv:2407.21704v2 (2024)

work page arXiv 2024
[4]

A. L. Besse, Einstein manifolds, Classics in Mathematics, Springer, Berlin (2007). (Reprint of the 1987 edition)

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Brendle, F

S. Brendle, F. C. Marques and A. Neves, Deformations of the hemisphere that increase scalar curvature, Invent. Math.185, no. 1 (2011), 175–197

work page 2011
[6]

Burkhardt-Guim, Pointwise lower scalar curvature bounds forC0 metrics via regularizing Ricci flow, Geom

P. Burkhardt-Guim, Pointwise lower scalar curvature bounds forC0 metrics via regularizing Ricci flow, Geom. Funct. Anal.29(2019), 1703–1772

work page 2019
[7]

Cecchini, B

S. Cecchini, B. Hanke and T. Schick, Lipschitz rigidity for scalar curvature, J. Eur. Math. Soc. (2024). to appear

work page 2024
[8]

Dahl and K

M. Dahl and K. Kr¨ oncke, Local and global scalar curvature rigidity of Einstein manifolds, Math. Ann.388, no. 1 (2024), 453–510

work page 2024
[9]

A. E. Fischer and J. E. Marsden, The manifold of conformally equivalent metrics, Can. J. Math.29, no. 1 (1977), 193–209

work page 1977
[10]

Goette and U

S. Goette and U. Semmelmann, Scalar curvature estimates for compact symmetric spaces, Differ. Geom. Appl.16, no. 1 (2002), 65–78

work page 2002
[11]

M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, InFunctional Analysis on the Eve of the 21st Century Volume II: In Honor of the Eightieth Birthday of IM Gelfand, Progr. Math. 136(1996), 1–213. 27

work page 1996
[12]

Hamanaka, Non-Einstein relative Yamabe metrics, Kodai Math

S. Hamanaka, Non-Einstein relative Yamabe metrics, Kodai Math. J.44, no. 2 (2021), 265–272

work page 2021
[13]

Hamanaka and P

S. Hamanaka and P. T. Ho, Notes on the uniqueness of Type II Yamabe metrics, Nonlinear Differential Equations and Applications NoDEA32, no. 5 (2025), 81 pp

work page 2025
[14]

Kato, Examples of non-Einstein Yamabe metrics with positive scalar curvature, Tokyo J

S. Kato, Examples of non-Einstein Yamabe metrics with positive scalar curvature, Tokyo J. Math.17, no. 1 (1994), 187–189

work page 1994
[15]

Kobayashi, K

O. Kobayashi, K. Akutagawa and H. Izeki, The Yamabe problem (MSJ Memoirs vol. 7), Mathematical Society of Japan, 2013. (in Japanese)

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[16]

Koiso, A decomposition of the spaceMof Riemannian metrics on a manifold, Osaka Math

N. Koiso, A decomposition of the spaceMof Riemannian metrics on a manifold, Osaka Math. J.16, no. 2 (1979), 423–429

work page 1979
[17]

Kr¨ oncke and U

K. Kr¨ oncke and U. Semmelmann, On stability and scalar curvature rigidity of quaternion-K¨ ahler manifolds, arXiv:2412.13351 (2024)

work page arXiv 2024
[18]

Lee and L.-F

M.-C. Lee and L.-F. Tam, Rigidity of Lipschitz map using harmonic map heat flow, arXiv:2207.11017 (2022)

work page arXiv 2022
[19]

Li, Geometric Analysis, Cambridge University Press134, (2012)

P. Li, Geometric Analysis, Cambridge University Press134, (2012)

work page 2012
[20]

Scalar Curvature on Compact Symmetric Spaces

M. Listing, Scalar curvature on compact symmetric spaces, arXiv:1007.1832 (2010)

work page internal anchor Pith review Pith/arXiv arXiv 2010
[21]

Scalar curvature and vector bundles

M. Listing, Scalar curvature and vector bundles, arXiv:1202.4325 (2012)

work page internal anchor Pith review Pith/arXiv arXiv 2012
[22]

Llarull, Sharp estimates and the Dirac operator, Math

M. Llarull, Sharp estimates and the Dirac operator, Math. Ann.310, no. 1 (1998), 55–71

work page 1998
[23]

Lohkamp, Scalar curvature and hammocks, Math

J. Lohkamp, Scalar curvature and hammocks, Math. Ann.313(1999), 385–407

work page 1999
[24]

Lott, Index theory for scalar curvature on manifolds with boundary, Proc

J. Lott, Index theory for scalar curvature on manifolds with boundary, Proc. Am. Math. Soc.149, no. 10 (2021), 4451–4459

work page 2021
[25]

Matsuo, The prescribed scalar curvature problem for metrics with unit total volume, Math

S. Matsuo, The prescribed scalar curvature problem for metrics with unit total volume, Math. Ann.360, no. 3 (2014), 675–680

work page 2014
[26]

Matsuzawa, Einstein metrics and fibred Riemannian structures, Kodai Math

T. Matsuzawa, Einstein metrics and fibred Riemannian structures, Kodai Math. J.6, no. 3 (1983), 340–345

work page 1983
[27]

Milnor, Curvatures of left invariant metrics on Lie groups, Adv

J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21(1976), 293–329

work page 1976
[28]

Obata, The conjectures on conformal transformations of Riemannian manifolds, Bull

M. Obata, The conjectures on conformal transformations of Riemannian manifolds, Bull. Amer. Math. Soc.77(1971), 265–270. 28

work page 1971
[29]

R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Varia- tions: Lectures given at the 2nd 1987 Session of the Centro Internazionale Matematico Estivo (CIME) held at Montecatini Terme, Italy, July 20–28, (1987), 120–154. E-mail adress: hamanaka1311558@gmail.com Depart...

work page 1987

[1] [1]

M. T. Anderson, On uniqueness and differentiability in the space of Yamabe metrics, CCM7, no. 3 (2005), 299–310

work page 2005

[2] [2]

Andrews and C

B. Andrews and C. Hopper, The Ricci flow in Riemannian geometry: a complete proof of the differentiable 1/4-pinching sphere theorem, Lecture Notes in Mathematics2011, Berlin, Heidelberg: Springer Nature (2010)

work page 2010

[3] [3]

B¨ ar, Dirac eigenvalues and the hyperspherical radius, arXiv:2407.21704v2 (2024)

C. B¨ ar, Dirac eigenvalues and the hyperspherical radius, arXiv:2407.21704v2 (2024)

work page arXiv 2024

[4] [4]

A. L. Besse, Einstein manifolds, Classics in Mathematics, Springer, Berlin (2007). (Reprint of the 1987 edition)

work page 2007

[5] [5]

Brendle, F

S. Brendle, F. C. Marques and A. Neves, Deformations of the hemisphere that increase scalar curvature, Invent. Math.185, no. 1 (2011), 175–197

work page 2011

[6] [6]

Burkhardt-Guim, Pointwise lower scalar curvature bounds forC0 metrics via regularizing Ricci flow, Geom

P. Burkhardt-Guim, Pointwise lower scalar curvature bounds forC0 metrics via regularizing Ricci flow, Geom. Funct. Anal.29(2019), 1703–1772

work page 2019

[7] [7]

Cecchini, B

S. Cecchini, B. Hanke and T. Schick, Lipschitz rigidity for scalar curvature, J. Eur. Math. Soc. (2024). to appear

work page 2024

[8] [8]

Dahl and K

M. Dahl and K. Kr¨ oncke, Local and global scalar curvature rigidity of Einstein manifolds, Math. Ann.388, no. 1 (2024), 453–510

work page 2024

[9] [9]

A. E. Fischer and J. E. Marsden, The manifold of conformally equivalent metrics, Can. J. Math.29, no. 1 (1977), 193–209

work page 1977

[10] [10]

Goette and U

S. Goette and U. Semmelmann, Scalar curvature estimates for compact symmetric spaces, Differ. Geom. Appl.16, no. 1 (2002), 65–78

work page 2002

[11] [11]

M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, InFunctional Analysis on the Eve of the 21st Century Volume II: In Honor of the Eightieth Birthday of IM Gelfand, Progr. Math. 136(1996), 1–213. 27

work page 1996

[12] [12]

Hamanaka, Non-Einstein relative Yamabe metrics, Kodai Math

S. Hamanaka, Non-Einstein relative Yamabe metrics, Kodai Math. J.44, no. 2 (2021), 265–272

work page 2021

[13] [13]

Hamanaka and P

S. Hamanaka and P. T. Ho, Notes on the uniqueness of Type II Yamabe metrics, Nonlinear Differential Equations and Applications NoDEA32, no. 5 (2025), 81 pp

work page 2025

[14] [14]

Kato, Examples of non-Einstein Yamabe metrics with positive scalar curvature, Tokyo J

S. Kato, Examples of non-Einstein Yamabe metrics with positive scalar curvature, Tokyo J. Math.17, no. 1 (1994), 187–189

work page 1994

[15] [15]

Kobayashi, K

O. Kobayashi, K. Akutagawa and H. Izeki, The Yamabe problem (MSJ Memoirs vol. 7), Mathematical Society of Japan, 2013. (in Japanese)

work page 2013

[16] [16]

Koiso, A decomposition of the spaceMof Riemannian metrics on a manifold, Osaka Math

N. Koiso, A decomposition of the spaceMof Riemannian metrics on a manifold, Osaka Math. J.16, no. 2 (1979), 423–429

work page 1979

[17] [17]

Kr¨ oncke and U

K. Kr¨ oncke and U. Semmelmann, On stability and scalar curvature rigidity of quaternion-K¨ ahler manifolds, arXiv:2412.13351 (2024)

work page arXiv 2024

[18] [18]

Lee and L.-F

M.-C. Lee and L.-F. Tam, Rigidity of Lipschitz map using harmonic map heat flow, arXiv:2207.11017 (2022)

work page arXiv 2022

[19] [19]

Li, Geometric Analysis, Cambridge University Press134, (2012)

P. Li, Geometric Analysis, Cambridge University Press134, (2012)

work page 2012

[20] [20]

Scalar Curvature on Compact Symmetric Spaces

M. Listing, Scalar curvature on compact symmetric spaces, arXiv:1007.1832 (2010)

work page internal anchor Pith review Pith/arXiv arXiv 2010

[21] [21]

Scalar curvature and vector bundles

M. Listing, Scalar curvature and vector bundles, arXiv:1202.4325 (2012)

work page internal anchor Pith review Pith/arXiv arXiv 2012

[22] [22]

Llarull, Sharp estimates and the Dirac operator, Math

M. Llarull, Sharp estimates and the Dirac operator, Math. Ann.310, no. 1 (1998), 55–71

work page 1998

[23] [23]

Lohkamp, Scalar curvature and hammocks, Math

J. Lohkamp, Scalar curvature and hammocks, Math. Ann.313(1999), 385–407

work page 1999

[24] [24]

Lott, Index theory for scalar curvature on manifolds with boundary, Proc

J. Lott, Index theory for scalar curvature on manifolds with boundary, Proc. Am. Math. Soc.149, no. 10 (2021), 4451–4459

work page 2021

[25] [25]

Matsuo, The prescribed scalar curvature problem for metrics with unit total volume, Math

S. Matsuo, The prescribed scalar curvature problem for metrics with unit total volume, Math. Ann.360, no. 3 (2014), 675–680

work page 2014

[26] [26]

Matsuzawa, Einstein metrics and fibred Riemannian structures, Kodai Math

T. Matsuzawa, Einstein metrics and fibred Riemannian structures, Kodai Math. J.6, no. 3 (1983), 340–345

work page 1983

[27] [27]

Milnor, Curvatures of left invariant metrics on Lie groups, Adv

J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21(1976), 293–329

work page 1976

[28] [28]

Obata, The conjectures on conformal transformations of Riemannian manifolds, Bull

M. Obata, The conjectures on conformal transformations of Riemannian manifolds, Bull. Amer. Math. Soc.77(1971), 265–270. 28

work page 1971

[29] [29]

R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Varia- tions: Lectures given at the 2nd 1987 Session of the Centro Internazionale Matematico Estivo (CIME) held at Montecatini Terme, Italy, July 20–28, (1987), 120–154. E-mail adress: hamanaka1311558@gmail.com Depart...

work page 1987