Volume asymptotics and Einstein moduli on spheres

Nifa, Anass

doi:10.5281/zenodo.23484018

Preprint · v2

Volume asymptotics and Einstein moduli on spheres

Anass NifaInitial manuscript date: 18 September 2026Version 2 posted: 19 October 2026

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Abstract

The moduli spaces of positive Einstein metrics of volume one on $S^6$ and $S^8$ , modulo all diffeomorphisms, each have countably infinitely many connected components. More generally, this holds on $S^{p+q+1}$ and $S^{p+1}\times Q$ whenever $p,q\ge2$ , $d=p+q\le8$ , and $Q$ is a closed connected positive Einstein $q$ -manifold. For sequences obtained from Böhm’s cap construction and normalized by $\operatorname{Ric}=dg$ , we determine the leading volume difference from the limiting sine-cone volume. It is a positive definite quadratic form in the data on a minimal hypersurface. For every such pair, suitable spherical sequences and reflection doubles have volumes converging from above to a positive limit, with successive volume excesses in limiting ratio $\exp(-2\pi\sqrt{(d-1)/(9-d)})$ .

MSC 2020

53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
58D27: Moduli problems for differential geometric structures

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.23484018
All versions: 10.5281/zenodo.22914133
Subjects: Einstein metric; Einstein moduli; normalized volume; sine-cone; cohomogeneity one
Language: English

Cite this paper

Anass Nifa. Volume asymptotics and Einstein moduli on spheres. Preprint, Zenodo, v2 (2026). doi:10.5281/zenodo.23484018

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References

As listed in the manuscript bibliography.

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M. T. Anderson, Extension of symmetries on Einstein manifolds with boundary, Selecta Math. (N.S.) 16 (2010), 343–375; corrected author version, arXiv:0704.3373v6.
arXiv v6

[reference-2] [2]
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-3] [3]
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-4] [4]
C. Böhm, M. Y. Wang and W. Ziller, A variational approach for compact homogeneous Einstein manifolds, Geom. Funct. Anal. 14 (2004), no. 4, 681–733. doi:10.1007/s00039-004-0471-x; author manuscript.
DOI · Source

[reference-5] [5]
T. Buttsworth and L. Hodgkinson, Computationally assisted proof of a novel O(3) × O(10)-invariant Einstein metric on S¹², J. Lond. Math. Soc. 113 (2026), no. 2, e70477. doi:10.1112/jlms.70477; arXiv:2412.01184v1.
DOI · arXiv v1

[reference-6] [6]
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DOI · arXiv v1

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DOI · arXiv v5

[reference-8] [8]
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; author manuscript.
DOI · Source

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DOI

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DOI · arXiv v1

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arXiv v2

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DOI · arXiv v2

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J. Nienhaus and M. Wink, Einstein metrics on the ten-sphere, J. Eur. Math. Soc. (2025), published online first. doi:10.4171/JEMS/1668; arXiv:2303.04832v3.
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DOI · arXiv v2

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DOI

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arXiv v3

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A. Siffert, Detecting Einstein metrics on products of spheres, arXiv:2610.00066v1 (2026).
arXiv v1

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A. Siffert, Quantized Einstein metrics on S⁷ with SU(3)-symmetry, arXiv:2609.02788v1 (2026).
arXiv v1

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G. Teschl, Ordinary differential equations and dynamical systems, Graduate Studies in Mathematics, vol. 140, American Mathematical Society, Providence, RI, 2012; author’s preliminary version.
Source

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G. Verger, Instability of Böhm’s Einstein metrics, arXiv:2608.25865v1 (2026).
arXiv v1

[reference-23] [23]
M. Y. Wang and W. Ziller, Einstein metrics on principal torus bundles, J. Differential Geom. 31 (1990), no. 1, 215–248. doi:10.4310/jdg/1214444095.
DOI

[reference-24] [24]
Q. S. Wang, Doubly warped product Einstein metrics on spheres, arXiv:2606.05519v3 (2026).
arXiv v3