Abstract
We consider Riemannian functionals defined by \(L^2\)-norms of Ricci curvature, scalar curvature, Weyl curvature, and Riemannian curvature. Rigidity, stability, and local minimizing properties of Einstein metrics as critical metrics of these quadratic functionals have been studied in 8. In this paper, we study the same for products of Einstein metrics with Einstein constants of possibly opposite signs. In particular, we show that the Riemannian product of closed Einstein manifolds \((M_0,g_0)\) and \((M_1,g_1)\) with respective Einstein constants \(\lambda (>0)\) and \(-\lambda \), is unstable for certain quadratic functionals if the first eigenvalue of the Laplacian of \((M_1,g_1)\) is sufficiently small. We also prove the stability of \(L^{\frac{n}{2}}\)-norm of Weyl curvature at compact quotients of \(S^n\times {\mathbb {H}}^m\).
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References
Besse A L, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (1987) (Berlin: Springer-Verlag)
Besson G, Courtois G and Gallot S, Volume et entropie minimale des espaces localement symétriques, Invent. Math. 103 (1991) 417–445
Berger M, Quelques formules de variation pour une structure riemannienne, Ann. Sci. Ecole Norm. Sup. 4e série 3 (1970) 285–294
Bhattacharya A and Maity S, Some unstable critical metrics for the \(L^{\frac{n}{2}}\)-norm of the curvature tensor, Math. Res. Lett.21 (2014) 235–240
Chen Q and He C, On Bach flat warped product Einstein manifolds, Pacific J. Math. 265 (2013) 313–326
Guo X and Li H, The stability of Fubini-Study metric on \({\mathbb{P}}P^n\), Proc. Amer. Math. Soc. 146 (2018) 325–333
Guo X, Li H and Wei G, On variational formulas of a conformally invariant functional, Results Math. 67 (2015) 49–70
Gursky M J and Viaclovsky J A, Rigidity and stability of Einstein metrics for quadratic curvature functionals, J. Reine Angew. Math. 700 (2015) 37–91
Kobayashi O, On a conformally invariant functional of the space of Riemannian metrics, J. Math. Soc. Japan 37 (1985) 373–389
Koiso N, Nondeformability of Einstein metrics, Osaka J. Math. 15 (1978) 419–433
Koiso N, Rigidity and infinitesimal deformability of Einstein metrics, Osaka J. Math. 19 (1982) 643–668
Koiso N, Rigidity and stability of Einstein metrics-the case of compact symmetric spaces, Osaka J. Math. 17 (1980) 51–73
Maity S, On the stability of \(L^p\)-norms of curvature tensor at rank one symmetrics spaces, Manuscripta Math. 159 (2019) 183–202
Schoen R, A lower bound for the first eigenvalue of a negatively curved manifold, J. Differential Geom. 17 (1982) 232–238
Schoen R and Yau S T, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988) 47–71
Zhu S, The classification of complete locally conformally flat manifolds of nonnegative ricci curvature, Pacific J. Math. 163 (1994) 189–199
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Communicated by Mj Mahan.
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Bhattacharya, A., Maity, S. Stability of quadratic curvature functionals at product of Einstein manifolds. Proc Math Sci 131, 42 (2021). https://doi.org/10.1007/s12044-021-00636-5
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DOI: https://doi.org/10.1007/s12044-021-00636-5