Einstein metrics at conical limits: stability, disconnected moduli, and entropy rigidity

Nifa, Anass

doi:10.5281/zenodo.22914134

Preprint · v1

Einstein metrics at conical limits: stability, disconnected moduli, and entropy rigidity

Anass Nifa18 September 2026

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Abstract

We study three variational problems for Einstein metrics with conical limits. For every $k\ge5$ , the complete Ricci-flat Böhm metric on $\mathbb{R}^{k+1}\times S^k$ has holonomy $SO(2k+1)$ and a homogeneous coercive estimate on all symmetric two-tensors. It attracts sufficiently small smooth $L^2\cap C_b^{2,\eta}$ perturbations under the fixed-background Ricci–DeTurck flow, with convergence in $L^2$ and every uniform $C^j$ norm. Uniformly small ancient perturbations with bounded $L^2$ norm are stationary. For $p,q\ge2$ and $p+q\le8$ , we construct distinct normalized Einstein volumes on $S^{p+q+1}$ , on $S^{p+1}\times Q$ for every closed connected positive Einstein $q$ -manifold $Q$ , and on the specified identity doubles of disk bundles with vanishing O’Neill tensor. Their full Einstein moduli spaces have infinitely many connected components. The volumes approach the sine-cone value from above; specified subsequences have geometric asymptotics. On a marked conical end of rate greater than $(n-2)/2$ , with absolutely integrable scalar curvature, a large-scale upper bound for Perelman’s $\mu$ -functional has coefficient $\lambda_{\mathrm{nc}}-\mathfrak m_C$ . No spin or scalar-curvature sign is assumed in this comparison. A parallel spinor compatible with the end induced in the universal cover gives mass–energy rigidity and characterizes equality in the global cone entropy bound by Ricci-flatness. Equality also makes the end homomorphism surjective with finite image and extends the compatible cone spinors. An ancient Ricci flow with one such slice is stationary when its finite backward Nash entropy is at least the cone value.

MSC 2020

53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C27: Spin and Spinᶜ geometry
53E20: Ricci flows
58J50: Spectral problems; spectral geometry; scattering theory on manifolds
58D27: Moduli problems for differential geometric structures

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22914134
All versions: 10.5281/zenodo.22914133
Subjects: Einstein metric; Ricci-flat metric; conical end; nonlinear stability; conical mass; Perelman entropy; parallel spinor; Einstein moduli
Language: English

Cite this paper

Anass Nifa. Einstein metrics at conical limits: stability, disconnected moduli, and entropy rigidity. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22914134

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References

As listed in the manuscript bibliography.

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DOI
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DOI · arXiv
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R. Haslhofer, A renormalized Perelman-functional and a lower bound for the ADM-mass, J. Geom. Phys. 61 (2011), no. 11, 2162–2167. doi:10.1016/j.geomphys.2011.06.016.
DOI
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H.-J. Hein and C. LeBrun, Mass in Kähler geometry, Comm. Math. Phys. 347 (2016), no. 1, 183–221. doi:10.1007/s00220-016-2661-4.
DOI
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M. W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1976. doi:10.1007/978-1-4684-9449-5.
DOI
[28]
N. Koiso, Rigidity and infinitesimal deformability of Einstein metrics, Osaka J. Math. 19 (1982), no. 3, 643–668.
[29]
K. Kröncke, Stable and unstable Einstein warped products, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6537–6563; arXiv:1507.01782.
arXiv
[30]
K. Kröncke and Á. Szabó, Optimal coordinates for Ricci-flat conifolds, Calc. Var. Partial Differential Equations 63 (2024), article 188. doi:10.1007/s00526-024-02780-y.
DOI
[31]
H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989.
[32]
Y. Liu, T. Sano and L. Tasin, Infinitely many families of Sasaki–Einstein metrics on spheres, J. Differential Geom. 130 (2025), no. 1, 1–26. doi:10.4310/jdg/1747062009; arXiv:2203.08468v2.
DOI · arXiv v2
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DOI · arXiv v1
[34]
A. Nifa, Dirichlet deformations, Morse index, and nonlinear symmetry of Ricci-flat Böhm metrics, preprint (2026), 129 pp. doi:10.5281/zenodo.22810800.
DOI
[35]
A. Nifa, Negative Lichnerowicz modes on Einstein warped products and spheres, preprint (2026), 21 pp. doi:10.5281/zenodo.22681024.
DOI
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P. Schwahn and U. Semmelmann, Einstein metrics, their moduli spaces and stability, arXiv:2507.18463v3 (15 June 2026).
arXiv v3
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G. Verger, Instability of Böhm’s Einstein metrics, arXiv:2608.25865v1 (26 August 2026).
arXiv v1
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E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), no. 3, 381–402.
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Q. Yao, Mass and expansion of asymptotically conical Kähler metrics, Math. Ann. 393 (2025), nos. 3–4, 3479–3512. doi:10.1007/s00208-025-03305-4; arXiv:2206.06505v1 fixes the lettered theorem numbering cited here.
DOI · arXiv v1
[40]
A. Deruelle and K. Kröncke, Stability of ALE Ricci-flat manifolds under Ricci flow, J. Geom. Anal. 31 (2021), 2829–2870. doi:10.1007/s12220-020-00376-4. The local result used here is Lemma 3.2, not the global ALE stability theorem.
DOI
[41]
K. Kröncke and O. L. Petersen, Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity, Adv. Math. 502 (2026), Part B, article 111139. doi:10.1016/j.aim.2026.111139; arXiv:2009.11854v2 (21 July 2026).
DOI · arXiv v2
[42]
M. Stolarski and A. Waldron, Integrable deformations and stability of the Ricci flow, arXiv:2604.15198 (2026), Theorems 1.1 and 1.3.
arXiv
[43]
T. Ozuch, Stability of gravitational instantons with a bounded Killing vector field, arXiv:2608.31008v1 (31 August 2026).
arXiv v1
[44]
A. Kristály, Sharp Sobolev inequalities on noncompact Riemannian manifolds with Ric ≥0 via optimal transport theory, Calc. Var. Partial Differential Equations 63 (2024), article 200. doi:10.1007/s00526-024-02810-9.
DOI
[45]
P.-Y. Chan, Z. Ma and Y. Zhang, A local Sobolev inequality on Ricci flow and its applications, J. Funct. Anal. 285 (2023), article 109995. Theorem numbering is that of arXiv:2111.05517v2.
arXiv v2
[46]
B. Kotschwar, Backwards uniqueness of the Ricci flow, Int. Math. Res. Not. IMRN 2010 (2010), no. 21, 4064–4097. doi:10.1093/imrn/rnq022; arXiv:0906.4920v1, Theorem 1.
DOI · arXiv v1
[47]
C. LeBrun, Counter-examples to the generalized positive action conjecture, Comm. Math. Phys. 118 (1988), 591–596.
[48]
J. A. Viaclovsky, An index theorem on anti-self-dual orbifolds, Int. Math. Res. Not. IMRN 2013 (2013), no. 17, 3911–3930; arXiv:1202.0578v2, Section 1.3.
arXiv v2
[49]
A. Alaee, M. Khuri and H. Kunduri, A comparison theorem for the mass of ALE and ALF toric 4-manifolds, arXiv:2605.12352v2 (15 July 2026).
arXiv v2
[50]
Y. Li, Ricci flow on asymptotically Euclidean manifolds, Geom. Topol. 22 (2018), 1837–1891. doi:10.2140/gt.2018.22.1837; arXiv:1603.05336v4 fixes the theorem numbering cited here.
DOI · arXiv v4
[51]
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159v1 (2002).
arXiv v1
[52]
A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser, Basel, 1995. doi:10.1007/978-3-0348-9234-6.
DOI
[53]
E. Issoglio and F. Russo, Stochastic differential equations with singular coefficients: the martingale problem view and the stochastic dynamics view, J. Theoret. Probab. 37 (2024), 2352–2393. doi:10.1007/s10959-024-01325-5.
DOI

[reference-1] [1]
B. S. Acharya, Supersymmetry, Ricci flat manifolds and the string landscape, J. High Energy Phys. 2020 (2020), no. 8, article 128; arXiv:1906.06886v3.
arXiv v3

[reference-2] [2]
W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428–443.

[reference-3] [3]
S. B. Angenent and D. Knopf, Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions, arXiv:2503.12210v1 (2025).
arXiv v1

[reference-4] [4]
D. Artacho, On the stability of Einstein metrics carrying a special twisted spinor, arXiv:2512.01620v1 (1 December 2025).
arXiv v1

[reference-5] [5]
R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), no. 5, 661–693. doi:10.1002/cpa.3160390505.
DOI

[reference-6] [6]
A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10, Springer-Verlag, Berlin, 1987.

[reference-7] [7]
O. Biquard and T. Ozuch, Instability of conformally Kähler, Einstein metrics, J. Differential Geom. 133 (2026), no. 3, 335–346. doi:10.4310/jdg/1779980240; arXiv:2310.10109v3 fixes the numbered question cited here.
DOI · arXiv v3

[reference-8] [8]
C. Böhm, Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces, Invent. Math. 134 (1998), 145–176. doi:10.1007/s002220050261.
DOI

[reference-9] [9]
C. Böhm, Non-compact cohomogeneity one Einstein manifolds, Bull. Soc. Math. France 127 (1999), no. 1, 135–177. doi:10.24033/bsmf.2345.
DOI

[reference-10] [10]
J.-P. Bourguignon and P. Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys. 144 (1992), no. 3, 581–599. doi:10.1007/BF02099184.
DOI

[reference-11] [11]
S. Cecchini, D. Räde and R. Zeidler, Nonnegative scalar curvature on manifolds with at least two ends, J. Topol. 16 (2023), no. 3, 855–876. doi:10.1112/topo.12303; arXiv:2205.12174v2.
DOI · arXiv v2

[reference-12] [12]
J. Chen, P. Liu, Y. Shi and J. Zhu, Incompressible hypersurface, positive scalar curvature and positive mass theorem, Math. Ann. 393 (2025), no. 1, 1241–1320. doi:10.1007/s00208-025-03276-6; arXiv:2112.14442.
DOI · arXiv

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M. Dahl, The positive mass theorem for ALE manifolds, Banach Center Publ. 41 (1997), 133–142. doi:10.4064/-41-1-133-142.
DOI

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X. Dai, Y. Sun and C. Wang, Positive mass theorem for asymptotically flat spin manifolds with isolated conical singularities, Trans. Amer. Math. Soc. 378 (2025), no. 4, 2617–2642. doi:10.1090/tran/9331; arXiv:2310.13285v2 fixes the theorem numbering cited here.
DOI · arXiv v2

[reference-15] [15]
X. Dai, X. Wang and G. Wei, On the stability of Riemannian manifold with parallel spinors, Invent. Math. 161 (2005), no. 1, 151–176.

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A. Deruelle, Asymptotic estimates and compactness of expanding gradient Ricci solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 2, 485–530. doi:10.2422/2036-2145.201502_004; arXiv:1411.2366v1 fixes the theorem numbering used here.
DOI · arXiv v1

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J.-H. Eschenburg and M. Y. Wang, The initial value problem for cohomogeneity one Einstein metrics, J. Geom. Anal. 10 (2000), no. 1, 109–137. doi:10.1007/BF02921808.
DOI

[reference-18] [18]
L. Foscolo and M. Haskins, New G₂-holonomy cones and exotic nearly Kähler structures on S⁶ and S³ × S³, Ann. of Math. (2) 185 (2017), no. 1, 59–130. doi:10.4007/annals.2017.185.1.2.
DOI

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P. Ghosh, Mass and rigidity in almost Kähler geometry, Math. Ann. 396 (2026), article 39, published online 16 September 2026. doi:10.1007/s00208-026-03583-6; arXiv:2603.08627v2. Theorem numbering refers to the preprint.
DOI · arXiv v2

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G. W. Gibbons, S. A. Hartnoll and C. N. Pope, Böhm and Einstein–Sasaki metrics, black holes, and cosmological event horizons, Phys. Rev. D 67 (2003), 084024. doi:10.1103/PhysRevD.67.084024; arXiv:hep-th/0208031v1.
DOI · arXiv v1

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D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Classics in Mathematics, Springer, Berlin, 2001.

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M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, vol. 14, Springer-Verlag, New York, 1973. doi:10.1007/978-1-4615-7904-5.
DOI

[reference-23] [23]
S. J. Hall, R. Haslhofer and M. Siepmann, The stability inequality for Ricci-flat cones, J. Geom. Anal. 24 (2014), no. 1, 472–494. doi:10.1007/s12220-012-9343-z; arXiv:1111.4981.
DOI · arXiv

[reference-24] [24]
R. Haslhofer, A renormalized Perelman-functional and a lower bound for the ADM-mass, J. Geom. Phys. 61 (2011), no. 11, 2162–2167. doi:10.1016/j.geomphys.2011.06.016.
DOI

[reference-25] [25]
A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.

[reference-26] [26]
H.-J. Hein and C. LeBrun, Mass in Kähler geometry, Comm. Math. Phys. 347 (2016), no. 1, 183–221. doi:10.1007/s00220-016-2661-4.
DOI

[reference-27] [27]
M. W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1976. doi:10.1007/978-1-4684-9449-5.
DOI

[reference-28] [28]
N. Koiso, Rigidity and infinitesimal deformability of Einstein metrics, Osaka J. Math. 19 (1982), no. 3, 643–668.

[reference-29] [29]
K. Kröncke, Stable and unstable Einstein warped products, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6537–6563; arXiv:1507.01782.
arXiv

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K. Kröncke and Á. Szabó, Optimal coordinates for Ricci-flat conifolds, Calc. Var. Partial Differential Equations 63 (2024), article 188. doi:10.1007/s00526-024-02780-y.
DOI

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H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989.

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Y. Liu, T. Sano and L. Tasin, Infinitely many families of Sasaki–Einstein metrics on spheres, J. Differential Geom. 130 (2025), no. 1, 1–26. doi:10.4310/jdg/1747062009; arXiv:2203.08468v2.
DOI · arXiv v2

[reference-33] [33]
I. M. Lopez and T. Ozuch, Ancient and expanding spin ALE Ricci flows, J. Funct. Anal. 289 (2025), no. 9, article 111062. doi:10.1016/j.jfa.2025.111062. The expander theorem is Theorem 1.4 in the published article; the first preprint version is arXiv:2407.18438v1.
DOI · arXiv v1

[reference-34] [34]
A. Nifa, Dirichlet deformations, Morse index, and nonlinear symmetry of Ricci-flat Böhm metrics, preprint (2026), 129 pp. doi:10.5281/zenodo.22810800.
DOI

[reference-35] [35]
A. Nifa, Negative Lichnerowicz modes on Einstein warped products and spheres, preprint (2026), 21 pp. doi:10.5281/zenodo.22681024.
DOI

[reference-36] [36]
P. Schwahn and U. Semmelmann, Einstein metrics, their moduli spaces and stability, arXiv:2507.18463v3 (15 June 2026).
arXiv v3

[reference-37] [37]
G. Verger, Instability of Böhm’s Einstein metrics, arXiv:2608.25865v1 (26 August 2026).
arXiv v1

[reference-38] [38]
E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), no. 3, 381–402.

[reference-39] [39]
Q. Yao, Mass and expansion of asymptotically conical Kähler metrics, Math. Ann. 393 (2025), nos. 3–4, 3479–3512. doi:10.1007/s00208-025-03305-4; arXiv:2206.06505v1 fixes the lettered theorem numbering cited here.
DOI · arXiv v1

[reference-40] [40]
A. Deruelle and K. Kröncke, Stability of ALE Ricci-flat manifolds under Ricci flow, J. Geom. Anal. 31 (2021), 2829–2870. doi:10.1007/s12220-020-00376-4. The local result used here is Lemma 3.2, not the global ALE stability theorem.
DOI

[reference-41] [41]
K. Kröncke and O. L. Petersen, Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity, Adv. Math. 502 (2026), Part B, article 111139. doi:10.1016/j.aim.2026.111139; arXiv:2009.11854v2 (21 July 2026).
DOI · arXiv v2

[reference-42] [42]
M. Stolarski and A. Waldron, Integrable deformations and stability of the Ricci flow, arXiv:2604.15198 (2026), Theorems 1.1 and 1.3.
arXiv

[reference-43] [43]
T. Ozuch, Stability of gravitational instantons with a bounded Killing vector field, arXiv:2608.31008v1 (31 August 2026).
arXiv v1

[reference-44] [44]
A. Kristály, Sharp Sobolev inequalities on noncompact Riemannian manifolds with Ric ≥0 via optimal transport theory, Calc. Var. Partial Differential Equations 63 (2024), article 200. doi:10.1007/s00526-024-02810-9.
DOI

[reference-45] [45]
P.-Y. Chan, Z. Ma and Y. Zhang, A local Sobolev inequality on Ricci flow and its applications, J. Funct. Anal. 285 (2023), article 109995. Theorem numbering is that of arXiv:2111.05517v2.
arXiv v2

[reference-46] [46]
B. Kotschwar, Backwards uniqueness of the Ricci flow, Int. Math. Res. Not. IMRN 2010 (2010), no. 21, 4064–4097. doi:10.1093/imrn/rnq022; arXiv:0906.4920v1, Theorem 1.
DOI · arXiv v1

[reference-47] [47]
C. LeBrun, Counter-examples to the generalized positive action conjecture, Comm. Math. Phys. 118 (1988), 591–596.

[reference-48] [48]
J. A. Viaclovsky, An index theorem on anti-self-dual orbifolds, Int. Math. Res. Not. IMRN 2013 (2013), no. 17, 3911–3930; arXiv:1202.0578v2, Section 1.3.
arXiv v2

[reference-49] [49]
A. Alaee, M. Khuri and H. Kunduri, A comparison theorem for the mass of ALE and ALF toric 4-manifolds, arXiv:2605.12352v2 (15 July 2026).
arXiv v2

[reference-50] [50]
Y. Li, Ricci flow on asymptotically Euclidean manifolds, Geom. Topol. 22 (2018), 1837–1891. doi:10.2140/gt.2018.22.1837; arXiv:1603.05336v4 fixes the theorem numbering cited here.
DOI · arXiv v4

[reference-51] [51]
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159v1 (2002).
arXiv v1

[reference-52] [52]
A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser, Basel, 1995. doi:10.1007/978-3-0348-9234-6.
DOI

[reference-53] [53]
E. Issoglio and F. Russo, Stochastic differential equations with singular coefficients: the martingale problem view and the stochastic dynamics view, J. Theoret. Probab. 37 (2024), 2352–2393. doi:10.1007/s10959-024-01325-5.
DOI