Abstract
We calculate an upper bound for the second non-zero eigenvalue of the scalar Laplacian, \(\lambda _{2}\), for toric-Kähler–Einstein metrics in terms of the polytope data. We also give a similar upper bound for Koiso–Sakane type Kähler–Einstein metrics. We provide some detailed examples in complex dimensions 1, 2 and 3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abreu, M.: Kähler geometry of toric varieties and extremal metrics. Internat. J. Math. 9(6), 641–651 (1998)
Abreu, M., Freitas, P.: On the invariant spectrum of \({S}^{1}\)-invariant metrics on \(\mathbb{S}^{2}\). Proc. London Math. Soc. 84(1), 213–230 (2002)
Besse, A.: Einstein Manifolds. Springer Classics in Mathematics, Springer-Verlag, Berlin (2008)
Braun, V., Brelidze, T., Douglas, M., Ovrut, B.: Eigenvalues and eigenfunctions of the scalar Laplace operator on Calabi–Yau manifolds. J. High Energy Phys. 120(7) (2008)
Dammerman, B.: Metrics of special curvature with symmetry, DPhil thesis, University of Oxford (2006), arXiv:math/0610738v1
Dancer, A., Wang, M.: On Ricci solitons of Cohomogeneity one. Ann. Global Anal. Geom. 39(3), 259–292 (2011)
Donaldson, S.K.: Some numerical results in complex differential geometry. Pure Appl. Math. Q. 5(2), 571–618 (2009)
Doran, C., Headrick, M., Herzog, C., Kantor, J., Wiseman, T.: Numerical Kähler–Einstein metric on the third del Pezzo. Comm. Math. Phys. 282(2), 357–393 (2008)
Dryden, E., Guillemin, V., Senas-Dias, R.: Hearing Delzant polytopes from the equivalent spectrum. Trans. Amer. Math. Soc. 364(2), 887–910 (2012)
Guillemin, V.: Kaehler structures on toric varieties. J. Differ. Geom. 40(2), 285–309 (1994)
Hall, S.J., Murphy, T.: On the spectrum of the Page and Chen–LeBrun–Weber metrics, to appear in Ann. Global Anal. Geom
Headrick, M., Wiseman, T.: Numerical Ricci-flat metrics on \(K3\). Class. Quantum Gravity 22(3), 4931–4960 (2005)
Keller, J.: Ricci iterations on Kähler classes. J. Inst. Math. Jussieu 8(4), 743–768 (2009)
Koiso, N., Sakane, Y.: Nonhomogeneous Kähler–Einstein metrics on compact complex manifolds II. Osaka J. Math. 25(4), 933–959 (1988)
Matsushima, Y.: Remarks on Kähler–Einstein manifolds. Nagoya Math. J. 46, 161–173 (1972)
Sakane, Y.: Examples of compact Einstein Kähler manifolds with positive Ricci tensor. Osaka Math. J. 23, 585–617 (1986)
Siu, Y.-T.: The existence of Kähler–Einstein metrics on manifolds with positive anti canonical line bundle and a suitable finite symmetry group. Ann. Math. (2) 127(3), 585–627 (1988)
Wang, X., Zhu, X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188(1), 87–103 (2004)
Acknowledgments
We would like to thank Emily Dryden for useful comments and suggestions. SH would like to thank Karl Sternberg and Richard Jarman for their hospitality whilst much of this paper was written. We would like to thank the anonymous referee for useful comments and corrections. This work was supported by a Dennison research grant from the University of Buckingham. TM was supported by an ARC grant.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hall, S.J., Murphy, T. Bounding \(\lambda _{2}\) for Kähler–Einstein metrics with large symmetry groups. Ann Glob Anal Geom 46, 145–158 (2014). https://doi.org/10.1007/s10455-014-9416-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-014-9416-2