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Transactions of the American Mathematical Society

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Stable and unstable Einstein warped products


Author: Klaus Kröncke
Journal: Trans. Amer. Math. Soc. 369 (2017), 6537-6563
MSC (2010): Primary 58J05, 53C25, 53C21
DOI: https://doi.org/10.1090/tran/6959
Published electronically: May 16, 2017
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Abstract: In this article, we systematically investigate the stability properties of certain warped product Einstein manifolds. We characterize stability of these metrics in terms of an eigenvalue condition of the Einstein operator on the base manifold. In particular, we prove that all complete manifolds carrying imaginary Killing spinors are strictly stable. Moreover, we show that Ricci-flat and hyperbolic cones over Kähler-Einstein Fano manifolds and over nonnegatively curved Einstein manifolds are stable if the cone has dimension $ n\geq 10$.


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  • [AM11] Lars Andersson and Vincent Moncrief, Einstein spaces as attractors for the Einstein flow, J. Differential Geom. 89 (2011), no. 1, 1-47. MR 2863911
  • [Bam15] Richard H. Bamler, Stability of symmetric spaces of noncompact type under Ricci flow, Geom. Funct. Anal. 25 (2015), no. 2, 342-416. MR 3334231
  • [Bar93] Urs Barmettler,
    On the Lichnerowicz Laplacian, PhD thesis, ETH Zürich, 1993.
  • [Bau89] Helga Baum, Complete Riemannian manifolds with imaginary Killing spinors, Ann. Global Anal. Geom. 7 (1989), no. 3, 205-226. MR 1039119
  • [Ber65] M. Berger, Sur les variétés d'Einstein compactes, Comptes Rendus de la IIIe Réunion du Groupement des Mathématiciens d'Expression Latine (Namur, 1965) Librairie Universitaire, Louvain, 1966, pp. 35-55 (French). MR 0238226
  • [Bes08] Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR 2371700
  • [CHI04] Huai-Song Cao, Richard Hamilton, and Tom Ilmanen,
    Gaussian densities and stability for some Ricci solitons,
    preprint,
    arXiv:math/0404165.
  • [CH15] Huai-Dong Cao and Chenxu He, Linear stability of Perelman's $ \nu $-entropy on symmetric spaces of compact type, J. Reine Angew. Math. 709 (2015), 229-246. MR 3430881
  • [Dai07] Xianzhe Dai,
    Stability of Einstein metrics and spin structures,
    Proceedings of the 4th International Congress of Chinese Mathematicians, Vol. II (2007), pp. 59-72.
  • [DWW05] Xianzhe Dai, Xiaodong Wang, and Guofang Wei, On the stability of Riemannian manifold with parallel spinors, Invent. Math. 161 (2005), no. 1, 151-176. MR 2178660
  • [FIK03] Mikhail Feldman, Tom Ilmanen, and Dan Knopf, Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons, J. Differential Geom. 65 (2003), no. 2, 169-209. MR 2058261
  • [GIK02] Christine Guenther, James Isenberg, and Dan Knopf, Stability of the Ricci flow at Ricci-flat metrics, Comm. Anal. Geom. 10 (2002), no. 4, 741-777. MR 1925501
  • [GK04] Andreas Gastel and Manfred Kronz, A family of expanding Ricci solitons, Variational problems in Riemannian geometry, Progr. Nonlinear Differential Equations Appl., vol. 59, Birkhäuser, Basel, 2004, pp. 81-93. MR 2076268
  • [GH02] Gary Gibbons and Sean A. Hartnoll, Gravitational instability in higher dimensions, Phys. Rev. D (3) 66 (2002), no. 6, 064024, 17. MR 1945130
  • [GHP03] G. W. Gibbons, Sean A. Hartnoll, and C. N. Pope, Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons, Phys. Rev. D (3) 67 (2003), no. 8, 084024, 24. MR 1995313
  • [GPY82] David J. Gross, Malcolm J. Perry, and Laurence G. Yaffe, Instability of flat space at finite temperature, Phys. Rev. D (3) 25 (1982), no. 2, 330-355. MR 643602
  • [HHS14] Stuart Hall, Robert Haslhofer, and Michael Siepmann, The stability inequality for Ricci-flat cones, J. Geom. Anal. 24 (2014), no. 1, 472-494. MR 3145931
  • [HM14] Robert Haslhofer and Reto Müller, Dynamical stability and instability of Ricci-flat metrics, Math. Ann. 360 (2014), no. 1-2, 547-553. MR 3263173
  • [Koi83] N. Koiso, Einstein metrics and complex structures, Invent. Math. 73 (1983), no. 1, 71-106. MR 707349
  • [Krö13] Klaus Kröncke,
    Ricci flow, Einstein metrics and the Yamabe invariant,
    preprint, arXiv:1312.2224.
  • [Krö15a] Klaus Kröncke, On infinitesimal Einstein deformations, Differential Geom. Appl. 38 (2015), 41-57. MR 3304669
  • [Krö15b] Klaus Kröncke, Stability and instability of Ricci solitons, Calc. Var. Partial Differential Equations 53 (2015), no. 1-2, 265-287. MR 3336320
  • [Oba62] Morio Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340. MR 0142086
  • [PP84a] Don N. Page and C. N. Pope, Stability analysis of compactifications of $ D=11$ supergravity with $ {\textup {SU}}(3)\times {\textup {SU}}(2)\times {\textup {U}}(1)$ symmetry, Phys. Lett. B 145 (1984), no. 5-6, 337-341. MR 760601
  • [PP84b] Don N. Page and C. N. Pope, Which compactifications of $ D=11$ supergravity are stable?, Phys. Lett. B 144 (1984), no. 5-6, 346-350. MR 758411
  • [Ses06] Natasa Sesum, Linear and dynamical stability of Ricci-flat metrics, Duke Math. J. 133 (2006), no. 1, 1-26. MR 2219268
  • [Sie13] Michael Siepmann,
    Ricci flows of Ricci flat cones,
    PhD thesis, ETH Zürich, 2013.
  • [SSS08] Oliver C. Schnürer, Felix Schulze, and Miles Simon, Stability of Euclidean space under Ricci flow, Comm. Anal. Geom. 16 (2008), no. 1, 127-158. MR 2411470
  • [SS13] Felix Schulze and Miles Simon, Expanding solitons with non-negative curvature operator coming out of cones, Math. Z. 275 (2013), no. 1-2, 625-639. MR 3101823
  • [Wan91] McKenzie Y. Wang, Preserving parallel spinors under metric deformations, Indiana Univ. Math. J. 40 (1991), no. 3, 815-844. MR 1129331
  • [War06] Claude Warnick, Semi-classical stability of AdS NUT instantons, Classical Quantum Gravity 23 (2006), no. 11, 3801-3817. MR 2235434
  • [Zhu11] Meng Zhu, The second variation of the Ricci expander entropy, Pacific J. Math. 251 (2011), no. 2, 499-510. MR 2811045

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Additional Information

Klaus Kröncke
Affiliation: Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
Email: klaus.kroencke@uni-hamburg.de

DOI: https://doi.org/10.1090/tran/6959
Keywords: Einstein metrics, linear stability, Ricci-flat cones, hyperbolic cones
Received by editor(s): September 14, 2015
Received by editor(s) in revised form: April 13, 2016
Published electronically: May 16, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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