Quasi-Einstein structures and Hitchin's equations

Alex Colling; Maciej Dunajski

arxiv: 2504.18475 · v2 · submitted 2025-04-25 · 🧮 math.DG · gr-qc· hep-th· nlin.SI

Quasi-Einstein structures and Hitchin's equations

Alex Colling , Maciej Dunajski This is my paper

Pith reviewed 2026-05-22 18:38 UTC · model grok-4.3

classification 🧮 math.DG gr-qchep-thnlin.SI

keywords quasi-Einstein structuresKilling vector fieldclosed manifoldsHitchin equationsEinstein-Weyl structuresrigidityclassification

0 comments

The pith

Quasi-Einstein structures in a given class on closed manifolds must have a Killing vector field.

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

This paper proves that quasi-Einstein structures belonging to a particular class on closed manifolds necessarily possess a Killing vector field. Such a field represents a continuous symmetry of the geometry. The proof extends an earlier rigidity result that applied to extremal black hole horizons. It thereby finishes the full classification of all compact quasi-Einstein surfaces in the class under consideration. Additional analysis produces new explicit constructions on non-compact surfaces and on the product manifold consisting of a two-sphere and a circle.

Core claim

We prove that a class of quasi-Einstein structures on closed manifolds must admit a Killing vector field. This extends the rigidity theorem obtained for the extremal black hole horizons and completes the classification of compact quasi-Einstein 2-manifolds in this class. We also explore special cases of the quasi-Einstein equations related to integrability and the Hitchin equations, as well as to Einstein-Weyl structures and Kazdan-Warner type PDEs. This leads to novel explicit examples of quasi-Einstein structures on non-compact 2-manifolds and on S² × S¹.

What carries the argument

The quasi-Einstein condition on the Ricci tensor together with the closed manifold assumption, which forces the existence of a Killing vector field through rigidity methods.

If this is right

This completes the classification of compact quasi-Einstein 2-manifolds in this class.
Special cases of the equations relate to integrability and the Hitchin equations.
Explicit examples arise on non-compact 2-manifolds and on the manifold S² × S¹.

Where Pith is reading between the lines

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

The rigidity may extend to other curvature conditions on compact manifolds.
The examples on S² × S¹ could model periodic solutions in related geometric equations.
Links to Kazdan-Warner equations may produce existence results for prescribed curvature problems.

Load-bearing premise

The quasi-Einstein structures belong to the specific class where the rigidity methods from the referenced prior theorem extend directly.

What would settle it

A closed manifold carrying a quasi-Einstein structure from this class but possessing no Killing vector field would disprove the claim.

read the original abstract

We prove (Theorem 1.1.) that a class of quasi-Einstein structures on closed manifolds must admit a Killing vector field. This extends the rigidity theorem obtained in \cite{DL23} for the extremal black hole horizons and completes the classification of compact quasi-Einstein 2-manifolds in this class. We also explore special cases of the quasi-Einstein equations related to integrability and the Hitchin equations, as well as to Einstein-Weyl structures and Kazdan-Warner type PDEs. This leads to novel explicit examples of quasi-Einstein structures on (non-compact) 2-manifolds and on $S^2 \times S^1$.

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

The paper extends the DL23 rigidity result to show that a restricted class of quasi-Einstein structures on closed manifolds admits a Killing vector field, while adding explicit examples on non-compact surfaces and S^2 x S^1.

read the letter

The main takeaway is that the authors prove a class of quasi-Einstein structures on closed manifolds must admit a Killing vector field. This extends the DL23 theorem for extremal black hole horizons and completes the classification of compact quasi-Einstein 2-manifolds in this setting. They adapt the integrated curvature identities from the prior work without introducing boundary terms or changing the sign conditions that produce the Killing field, which keeps the argument direct and avoids obvious gaps. The paper also connects the equations to integrability, Hitchin equations, Einstein-Weyl structures, and Kazdan-Warner PDEs, yielding new explicit examples on non-compact 2-manifolds and on S^2 times S^1. Those constructions are fresh and give concrete cases that were not in the earlier literature. The main limitation is the narrow class: the result holds where the DL23 methods carry over directly, which is stated upfront but restricts how far the rigidity goes. The Hitchin and Weyl sections stand apart from the central proof and do not feed back into it, so they broaden the paper without deepening the theorem. No circularity or fitting issues appear in how they build on the cited result. This is for differential geometers working on quasi-Einstein metrics, low-dimensional rigidity, or geometry tied to general relativity. Readers focused on 2-manifold classifications or looking for verifiable examples will get the most from it. The core claim is a straightforward extension of established techniques with clear thinking, so it deserves a serious referee. I would send it for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves Theorem 1.1 asserting that a class of quasi-Einstein structures on closed manifolds must admit a Killing vector field. This extends the rigidity theorem from DL23 for extremal black hole horizons and completes the classification of compact quasi-Einstein 2-manifolds in this class. Additionally, the paper explores special cases of the quasi-Einstein equations related to integrability, the Hitchin equations, Einstein-Weyl structures, and Kazdan-Warner type PDEs, providing novel explicit examples on non-compact 2-manifolds and on S^2 × S^1.

Significance. Assuming the central result holds, the paper makes a meaningful contribution by extending known rigidity results to a broader class of quasi-Einstein structures, thereby completing classifications in the compact 2-dimensional case. The explorations of connections to Hitchin equations and the provision of explicit examples enhance the understanding of these geometric objects and may inspire further research in conformal geometry and integrable systems.

minor comments (3)

[Introduction] The definition of the specific class of quasi-Einstein structures (via the equation and parameter range) could be stated more prominently early in the introduction to clarify the precise scope of Theorem 1.1.
[Proof of Theorem 1.1] The adaptation of integrated curvature identities from DL23 would benefit from an explicit sentence confirming that no new boundary terms arise under the closed-manifold assumption and that sign conditions are preserved.
[Examples] The explicit examples on S^2 × S^1 and non-compact 2-manifolds would be strengthened by a short direct verification that they satisfy the quasi-Einstein equation with the stated parameters.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its contribution to rigidity results for quasi-Einstein structures, and the recommendation for minor revision. We appreciate the summary highlighting the extension of the DL23 theorem and the new examples provided.

Circularity Check

1 steps flagged

Minor self-citation extending prior rigidity result without load-bearing reduction

specific steps

self citation load bearing [Abstract and Theorem 1.1]
"This extends the rigidity theorem obtained in cite{DL23} for the extremal black hole horizons and completes the classification of compact quasi-Einstein 2-manifolds in this class."

The proof of the main result is described as a direct extension of techniques from DL23 (likely overlapping authorship given Dunajski), but the extension itself adapts identities to the closed-manifold setting without reducing the conclusion to the citation alone.

full rationale

The paper's central Theorem 1.1 extends the rigidity theorem of DL23 to closed manifolds by adapting integrated curvature identities, with explicit assumptions on the quasi-Einstein class and manifold topology stated independently. No self-definitional equations, fitted parameters renamed as predictions, or ansatz smuggling appear in the provided excerpts. The self-citation to DL23 is present but not the sole justification, as the adaptation to closed manifolds adds independent content without altering sign conditions or introducing circular reductions. Subsequent sections on Hitchin equations and examples are separate and do not affect the main claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the definition of the quasi-Einstein class and extension of DL23; no free parameters, new entities, or ad hoc axioms are apparent from the abstract.

axioms (1)

standard math Standard differential geometry facts about closed manifolds, Killing vector fields, and quasi-Einstein equations.
Invoked implicitly in the statement of Theorem 1.1 and its extension of DL23.

pith-pipeline@v0.9.0 · 5640 in / 1327 out tokens · 70473 ms · 2026-05-22T18:38:36.444889+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/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1.1. Let (M,g) be a closed Riemannian n-manifold admitting a non-gradient vector field X such that the QEE (1.1) hold with either (i) m≥2 or (ii) m≤2−n. Then (M,g) admits a Killing vector field K. Moreover, [K,X]=0.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Proposition 2.1. For any solution to (2.2) with m≠2 the following identity holds [divergence identity involving Γ^{m/2-1}∇_{(a}K_b)∇^a K^b + …]

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

45 extracted references · 45 canonical work pages · 2 internal anchors

[1]

Angella, S

D. Angella, S. Calamai and C. Spotti. On the Chern–Yamabe problem. Math. Res. Lett. 24(3) (2017) 645-677

work page 2017
[2]

Bahuaud, S

E. Bahuaud, S. Gunasekaran, H. K. Kunduri and E. Woolgar. Static near-horizon geometries and rigidity of quasi-Einstein manifolds. Lett. Math. Phys. 112, 6 (2022) 116

work page 2022
[3]

Bahuaud, S

E. Bahuaud, S. Gunasekaran, H. K. Kunduri and E. Woolgar. Deformations of the Kerr-(A)dS near horizon geometry. Class. Quant. Grav. 41 (2024) 065001

work page 2024
[4]

C. B¨ ohm. Inhomogeneous Einstein metrics on low-dimens ional spheres and other low-dimensional spaces. Invent. Math. 134(1) (1998) 145-176

work page 1998
[5]

R. L. Bryant, M. Dunajski, and M. G. Eastwood. Metrisabil ity of two-dimensional projective structures, J. Dif- ferential Geometry 83 (2009) 465–499

work page 2009
[6]

E. Cartan. Sur une classe d’espaces de Weyl. ´Ecole Norm. Sup. 60 (1943) 1-16

work page 1943
[7]

Case, Y-J

J. Case, Y-J. Shu and G. Wei. Rigidity of quasi-Einstein m etrics. Diﬀ. Geom. Appl. 29 (2011) 93–100

work page 2011
[8]

J. Case. Smooth metric measure spaces and quasi-Einstei n metrics. Int. J. Math. 23 (2012) 1250110

work page 2012
[9]

Chac´ on and H

E. Chac´ on and H. Garc ´ ıa-Compe´ an. Self-dual gravity via Hitchin’s equations J. Math. Phys. 60 (2019) 052502

work page 2019
[10]

E. S. Cheb-Terrab and A. D. Roche. Hypergeometric solut ions for third order linear ODEs. arXiv:0803.3474 (2008)

work page internal anchor Pith review Pith/arXiv arXiv 2008
[11]

P. T. Chru´ sciel, S. J. Szybka and K. P. Tod. Towards a cla ssiﬁcation of vacuum near-horizons geometries. Class. Quant. Grav. 35 (2018) 015002

work page 2018
[12]

E. Cochran. Killing vector ﬁelds on compact m-Quasi-Ei nstein manifolds. Proc. Amer. Math. Soc. 153 (2025) 841-849

work page 2025
[13]

New quasi-Einstein metrics on a two-sphere

A. Colling, M. Dunajski, H. K. Kunduri and J. Lucietti. N ew quasi-Einstein metrics on a two-sphere. arXiv:2403.04117 (2024)

work page internal anchor Pith review Pith/arXiv arXiv 2024
[14]

Colling, D

A. Colling, D. Katona and J. Lucietti. Rigidity of the ex tremal Kerr-Newman horizon. Lett. Math. Phys. 115, 19 (2025)

work page 2025
[15]

Derdzinski

A. Derdzinski. Connections with skew-symmetric Ricci tensor on surfaces. Result. Math. 52 (2008) 223–245

work page 2008
[16]

Dobkowski-Ry/suppress lko, W

D. Dobkowski-Ry/suppress lko, W. Kami´ nski, J. Lewandowski andA. Szereszewski. The Near Horizon Geometry equation on compact 2-manifolds including the general solution for g > 0, Phys. Lett. B 785 (2018) 381-385

work page 2018
[17]

Dunajski, L

M. Dunajski, L. J. Mason, N M. J. Woodhouse. From 2D Integ rable Systems to Self-Dual gravity’, J. Phys. A: Math. Gen 31 (1998) 6019

work page 1998
[18]

Dunajski

M. Dunajski. Nonlinear graviton from the sine-Gordon e quation. Twistor Newletter 40 43-5; In Mason, L.J. et al. Further advances in twistor theory, 3 85-87

work page
[19]

Dunajski

M. Dunajski. Solitons, Instantons, and Twistors. 2nd e dn. Oxford Graduate Texts in Mathematics 31. Oxford University Press, Oxford (2024)

work page 2024
[20]

Dunajski, and J

M. Dunajski and J. Lucietti. Intrinsic rigidity of extr emal horizons. arXiv:2306.17512 (2023). To appear in the Journal of Diﬀerential Geometry

work page arXiv 2023
[21]

Gauduchon

P. Gauduchon. Structures de Weyl-Einstein, espaces de twisteurs et vari´ et´ es de typeS1 × S3. J. reine angew. Math. 469 (1995) 1–50

work page 1995
[22]

G. W. Gibbons, and S. W. Hawking. Gravitational multi-i nstantons, Phys. Lett. 78B (1978) 430-432

work page 1978
[23]

N. Hitchin. The Self-Duality Equations on a Riemann sur face. Proc. Lon. Math. Soc. 55 (1987) 59-126

work page 1987
[24]

N. J. Hitchin. Higgs bundles and diﬀeomorphism groups. In H.-D. Cao and S.-T. Yau, editors, Advances in geometry and mathematical physics. Surveys in diﬀerential geometry , pages 139–163. International Press (2016) 139–163

work page 2016
[25]

Jezierski

J. Jezierski. On the existence of Kundt’s metrics and de generate (or extremal) Killing horizons. Class. Quant. Grav. 26 (2009) 035011

work page 2009
[26]

Kami´ nski and J

W. Kami´ nski and J. Lewandowski. Extreme horizon equat ion. arXiv:2406.20068 (2024)

work page arXiv 2024
[27]

J. L. Kazdan and F. W. Warner. Curvature functions for co mpact 2-manifolds. Ann. Math. 99 (1974) 14-47

work page 1974
[28]

J. L. Kazdan. Applications of Partial diﬀerential Equa tions to Problems in Geometry (1993)

work page 1993
[29]

Kim and Y

D.-S. Kim and Y. H. Kim. Compact Einstein warped product spaces with nonpositive scalar curvature. Proc. Amer. Math. Soc. 131 (2003) 2573-2576

work page 2003
[30]

Kry´ nski

W. Kry´ nski. Webs and projective structures on a plane. Diﬀ. Geom. Appl. 37 (2014) 133

work page 2014
[31]

H. K. Kunduri, J. Lucietti and H. S. Reall. Near-horizon symmetries of extremal black holes. Class. Quant. Grav. 24 (2007) 4169-4189

work page 2007
[32]

H. K. Kunduri and J. Lucietti, Classiﬁcation of near-ho rizon geometries of extremal black holes. Living Rev. Rel. 16 (2013) 8

work page 2013
[33]

Lewandowski and T

J. Lewandowski and T. Pawlowski. Extremal isolated hor izons: a Local uniqueness theorem. Class. Quant. Grav. 20 (2003) 587-606. 26 ALEX COLLING AND MACIEJ DUNAJSKI

work page 2003
[34]

M. Limoncu. Modiﬁcations of the Ricci tensor and applic ations. Arch. Math. 95 (2010) 191-199

work page 2010
[35]

H. Lu, D. N. Page and C. N. Pope. New inhomogeneous Einste in metrics on sphere bundles over Einstein-K¨ ahler manifolds. Phys. Lett. B 593 (2004) 218-226

work page 2004
[36]

L. J. Mason, and N. M. J. Woodhouse. Integrability, Self -duality and Twistor theory, LMS Monograph, OUP. (1996)

work page 1996
[37]

J. Milnor. Topology from the diﬀerentiable viewpoint. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ (1997) reprint of the 1965 original

work page 1997
[38]

Nurowski and M

P. Nurowski and M. Randall. Generalized Ricci solitons . J. Geom. Anal. 26 (2016) 1280–1345

work page 2016
[39]

Pedersen and K

H. Pedersen and K. P. Tod. Three-dimensional Einstein- Weyl geometry. Adv. Math. 97 (1993) 74–109

work page 1993
[40]

Z. Qian. Estimates for weighted volumes and applicatio ns. Quart. J. Math. 48 (1997) 235-242

work page 1997
[41]

M. Randall. Local obstructions to projective surfaces admitting skew-symmetric Ricci tensor. Jour. Geom. Phys. 76 (2014) 192

work page 2014
[42]

K. P. Tod. Compact 3-dimensional Einstein-Weyl struct ures. J. London. Math. Soc. 45 (1992) 341

work page 1992
[43]

W. Wylie. Rigidity of compact static near-horizon geom etries with negative cosmological constant. Lett. Math. Phys. 113, 29 (2023)

work page 2023
[44]

W. Yu. A note on Kazdan–Warner type equations on compact Riemannian manifolds. Non-linear Anal. 246 (2024) 113596

work page 2024
[45]

A. V. Zhiber and V. V. Sokolov. Exactly integrable hyper bolic equations of Liouville type. Russian Math. Surveys 56 (2001) 61–101. Department of Applied Mathematics and Theoretical Physics , University of Cambridge, Wilber- force Road, Cambridge CB3 0W A, UK. Email address : aec200@cam.ac.uk Department of Applied Mathematics and Theoretical Physics , Uni...

work page 2001

[1] [1]

Angella, S

D. Angella, S. Calamai and C. Spotti. On the Chern–Yamabe problem. Math. Res. Lett. 24(3) (2017) 645-677

work page 2017

[2] [2]

Bahuaud, S

E. Bahuaud, S. Gunasekaran, H. K. Kunduri and E. Woolgar. Static near-horizon geometries and rigidity of quasi-Einstein manifolds. Lett. Math. Phys. 112, 6 (2022) 116

work page 2022

[3] [3]

Bahuaud, S

E. Bahuaud, S. Gunasekaran, H. K. Kunduri and E. Woolgar. Deformations of the Kerr-(A)dS near horizon geometry. Class. Quant. Grav. 41 (2024) 065001

work page 2024

[4] [4]

C. B¨ ohm. Inhomogeneous Einstein metrics on low-dimens ional spheres and other low-dimensional spaces. Invent. Math. 134(1) (1998) 145-176

work page 1998

[5] [5]

R. L. Bryant, M. Dunajski, and M. G. Eastwood. Metrisabil ity of two-dimensional projective structures, J. Dif- ferential Geometry 83 (2009) 465–499

work page 2009

[6] [6]

E. Cartan. Sur une classe d’espaces de Weyl. ´Ecole Norm. Sup. 60 (1943) 1-16

work page 1943

[7] [7]

Case, Y-J

J. Case, Y-J. Shu and G. Wei. Rigidity of quasi-Einstein m etrics. Diﬀ. Geom. Appl. 29 (2011) 93–100

work page 2011

[8] [8]

J. Case. Smooth metric measure spaces and quasi-Einstei n metrics. Int. J. Math. 23 (2012) 1250110

work page 2012

[9] [9]

Chac´ on and H

E. Chac´ on and H. Garc ´ ıa-Compe´ an. Self-dual gravity via Hitchin’s equations J. Math. Phys. 60 (2019) 052502

work page 2019

[10] [10]

E. S. Cheb-Terrab and A. D. Roche. Hypergeometric solut ions for third order linear ODEs. arXiv:0803.3474 (2008)

work page internal anchor Pith review Pith/arXiv arXiv 2008

[11] [11]

P. T. Chru´ sciel, S. J. Szybka and K. P. Tod. Towards a cla ssiﬁcation of vacuum near-horizons geometries. Class. Quant. Grav. 35 (2018) 015002

work page 2018

[12] [12]

E. Cochran. Killing vector ﬁelds on compact m-Quasi-Ei nstein manifolds. Proc. Amer. Math. Soc. 153 (2025) 841-849

work page 2025

[13] [13]

New quasi-Einstein metrics on a two-sphere

A. Colling, M. Dunajski, H. K. Kunduri and J. Lucietti. N ew quasi-Einstein metrics on a two-sphere. arXiv:2403.04117 (2024)

work page internal anchor Pith review Pith/arXiv arXiv 2024

[14] [14]

Colling, D

A. Colling, D. Katona and J. Lucietti. Rigidity of the ex tremal Kerr-Newman horizon. Lett. Math. Phys. 115, 19 (2025)

work page 2025

[15] [15]

Derdzinski

A. Derdzinski. Connections with skew-symmetric Ricci tensor on surfaces. Result. Math. 52 (2008) 223–245

work page 2008

[16] [16]

Dobkowski-Ry/suppress lko, W

D. Dobkowski-Ry/suppress lko, W. Kami´ nski, J. Lewandowski andA. Szereszewski. The Near Horizon Geometry equation on compact 2-manifolds including the general solution for g > 0, Phys. Lett. B 785 (2018) 381-385

work page 2018

[17] [17]

Dunajski, L

M. Dunajski, L. J. Mason, N M. J. Woodhouse. From 2D Integ rable Systems to Self-Dual gravity’, J. Phys. A: Math. Gen 31 (1998) 6019

work page 1998

[18] [18]

Dunajski

M. Dunajski. Nonlinear graviton from the sine-Gordon e quation. Twistor Newletter 40 43-5; In Mason, L.J. et al. Further advances in twistor theory, 3 85-87

work page

[19] [19]

Dunajski

M. Dunajski. Solitons, Instantons, and Twistors. 2nd e dn. Oxford Graduate Texts in Mathematics 31. Oxford University Press, Oxford (2024)

work page 2024

[20] [20]

Dunajski, and J

M. Dunajski and J. Lucietti. Intrinsic rigidity of extr emal horizons. arXiv:2306.17512 (2023). To appear in the Journal of Diﬀerential Geometry

work page arXiv 2023

[21] [21]

Gauduchon

P. Gauduchon. Structures de Weyl-Einstein, espaces de twisteurs et vari´ et´ es de typeS1 × S3. J. reine angew. Math. 469 (1995) 1–50

work page 1995

[22] [22]

G. W. Gibbons, and S. W. Hawking. Gravitational multi-i nstantons, Phys. Lett. 78B (1978) 430-432

work page 1978

[23] [23]

N. Hitchin. The Self-Duality Equations on a Riemann sur face. Proc. Lon. Math. Soc. 55 (1987) 59-126

work page 1987

[24] [24]

N. J. Hitchin. Higgs bundles and diﬀeomorphism groups. In H.-D. Cao and S.-T. Yau, editors, Advances in geometry and mathematical physics. Surveys in diﬀerential geometry , pages 139–163. International Press (2016) 139–163

work page 2016

[25] [25]

Jezierski

J. Jezierski. On the existence of Kundt’s metrics and de generate (or extremal) Killing horizons. Class. Quant. Grav. 26 (2009) 035011

work page 2009

[26] [26]

Kami´ nski and J

W. Kami´ nski and J. Lewandowski. Extreme horizon equat ion. arXiv:2406.20068 (2024)

work page arXiv 2024

[27] [27]

J. L. Kazdan and F. W. Warner. Curvature functions for co mpact 2-manifolds. Ann. Math. 99 (1974) 14-47

work page 1974

[28] [28]

J. L. Kazdan. Applications of Partial diﬀerential Equa tions to Problems in Geometry (1993)

work page 1993

[29] [29]

Kim and Y

D.-S. Kim and Y. H. Kim. Compact Einstein warped product spaces with nonpositive scalar curvature. Proc. Amer. Math. Soc. 131 (2003) 2573-2576

work page 2003

[30] [30]

Kry´ nski

W. Kry´ nski. Webs and projective structures on a plane. Diﬀ. Geom. Appl. 37 (2014) 133

work page 2014

[31] [31]

H. K. Kunduri, J. Lucietti and H. S. Reall. Near-horizon symmetries of extremal black holes. Class. Quant. Grav. 24 (2007) 4169-4189

work page 2007

[32] [32]

H. K. Kunduri and J. Lucietti, Classiﬁcation of near-ho rizon geometries of extremal black holes. Living Rev. Rel. 16 (2013) 8

work page 2013

[33] [33]

Lewandowski and T

J. Lewandowski and T. Pawlowski. Extremal isolated hor izons: a Local uniqueness theorem. Class. Quant. Grav. 20 (2003) 587-606. 26 ALEX COLLING AND MACIEJ DUNAJSKI

work page 2003

[34] [34]

M. Limoncu. Modiﬁcations of the Ricci tensor and applic ations. Arch. Math. 95 (2010) 191-199

work page 2010

[35] [35]

H. Lu, D. N. Page and C. N. Pope. New inhomogeneous Einste in metrics on sphere bundles over Einstein-K¨ ahler manifolds. Phys. Lett. B 593 (2004) 218-226

work page 2004

[36] [36]

L. J. Mason, and N. M. J. Woodhouse. Integrability, Self -duality and Twistor theory, LMS Monograph, OUP. (1996)

work page 1996

[37] [37]

J. Milnor. Topology from the diﬀerentiable viewpoint. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ (1997) reprint of the 1965 original

work page 1997

[38] [38]

Nurowski and M

P. Nurowski and M. Randall. Generalized Ricci solitons . J. Geom. Anal. 26 (2016) 1280–1345

work page 2016

[39] [39]

Pedersen and K

H. Pedersen and K. P. Tod. Three-dimensional Einstein- Weyl geometry. Adv. Math. 97 (1993) 74–109

work page 1993

[40] [40]

Z. Qian. Estimates for weighted volumes and applicatio ns. Quart. J. Math. 48 (1997) 235-242

work page 1997

[41] [41]

M. Randall. Local obstructions to projective surfaces admitting skew-symmetric Ricci tensor. Jour. Geom. Phys. 76 (2014) 192

work page 2014

[42] [42]

K. P. Tod. Compact 3-dimensional Einstein-Weyl struct ures. J. London. Math. Soc. 45 (1992) 341

work page 1992

[43] [43]

W. Wylie. Rigidity of compact static near-horizon geom etries with negative cosmological constant. Lett. Math. Phys. 113, 29 (2023)

work page 2023

[44] [44]

W. Yu. A note on Kazdan–Warner type equations on compact Riemannian manifolds. Non-linear Anal. 246 (2024) 113596

work page 2024

[45] [45]

A. V. Zhiber and V. V. Sokolov. Exactly integrable hyper bolic equations of Liouville type. Russian Math. Surveys 56 (2001) 61–101. Department of Applied Mathematics and Theoretical Physics , University of Cambridge, Wilber- force Road, Cambridge CB3 0W A, UK. Email address : aec200@cam.ac.uk Department of Applied Mathematics and Theoretical Physics , Uni...

work page 2001