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Toric Aspects of the First Eigenvalue

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Abstract

In this paper we study the smallest non-zero eigenvalue \(\lambda _1\) of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for \(\lambda _1\) in terms of moment polytope data. We show that this bound can only be attained for \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric and therefore \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric is spectrally determined among all toric Kähler metrics. We also study the equivariant counterpart of \(\lambda _1\) which we denote by \(\lambda _1^T\). It is the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that \(\lambda _1^T\) is not bounded among toric Kähler metrics thus generalizing a result of Abreu–Freitas on \(S^2\). In particular, \(\lambda _1^T\) and \(\lambda _1\) do not coincide in general.

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Notes

  1. It is possible that this result was previously known but the authors did not find a reference for it in the literature and thus state it and prove it.

  2. To recover the original convention introduced by Lerman and Tolman in the rational case, take \(m_k\in \mathbb Z\) such that \(\frac{1}{m_k}\nu _ k\) is primitive in \(\Lambda \) so \((P, m_1,\ldots m_d,\Lambda )\) is a rational labelled polytope.

References

  1. Abreu, M.: Kähler geometry of toric varieties and extremal metrics. Int. J. Math. 9, 641–651 (1998)

    Article  MATH  Google Scholar 

  2. Abreu, M.: Kähler metrics on toric orbifolds. J. Differ. Geom. 58, 151–187 (2001)

    Article  MATH  Google Scholar 

  3. Abreu, M., Freitas, P.: On the invariant spectrum of $S^1$-invariant metrics on $S^2$. Proc. Lond. Math. Soc 84, 213–230 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abreu, M.: Kähler geometry of toric manifolds in symplectic coordinates. In: Eliashberg, Y., Khesin, B., Lalonde, F. (eds.) Symplectic and Contact Topology: Interactions and Perspectives. Fields Institute Communications, vol. 35, pp. 1–24. American Mathematical Society, Providence (2003)

    Google Scholar 

  5. Apostolov, V., Jakobson, D., Konkarev, G.: An extremal eigenvalue problem in Kähler geometry. Conformal and complex geometry in honor of Paul Gauduchon. J. Geom. Phys. 91, 108–116 (2015)

    Article  MathSciNet  Google Scholar 

  6. Arezzo, C., Ghigi, A., Loi, A.: Stable bundles and the first eigenvalue of the Laplacian. J. Geom. Anal. 17(3), 375–386 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berger, M., Gauduchon, P., Mazet, E.: Le Spectre d’une Variété Riemannienne, vol. 194. Lecture Notes in Mathematics. Springer, Berlin (1994)

  8. Biliotti, L., Ghigi, A.: Satake–Furstenberg compactifications, the moment map and $\lambda _1$. Am. J. Math. 135(1), 237–274 (2013)

    Article  MATH  Google Scholar 

  9. Bourguignon, J.P., Li, P., Yau, S.T.: Upper bound for the first eigenvalue of algebraic submanifolds. Comment. Math. Helv. 69, 199–207 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. AMS, Providence (2011)

  11. Delzant, T.: Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. Fr. 116, 315–339 (1988)

    Article  MATH  Google Scholar 

  12. Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62, 289–349 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Donaldson, S.K.: Interior estimates for solutions of Abreu’s equation. Collect. Math. 56, 103–142 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Donaldson, S.K.: Kähler geometry on toric manifolds, and some other manifolds with large symmetry. In: Handbook of Geometric Analysis No. 1. Advanced Lectures in Mathematics (ALM) 7, 29–75, International Press, Somerville (2008)

  15. Donaldson, S.K.: Some numerical results in complex differential geometry. Pure Appl. Math. Q. 5(2, Special issue: In honor of Friedrich Hirzebruch), 571–618

  16. Dryden, E., Guillemin, V., Sena-Dias, R.: Equivariant spectrum on toric orbifolds. Adv. Math. 231(3–4), 1271–1290 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. El Soufi, A., Ilias, S.: Immersions minimales, première valeur propre du Laplacien et volume conforme. Math. Ann. 275(2), 257–267 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guan, D.: On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles. Math. Res. Lett. 6, 547–555 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guillemin, V.: Moment Maps and Combinatorial Invariants of Hamiltonian $T^n$-Spaces, vol. 122 of Progress in Mathematics. Birkhäuser Boston, Boston (1994)

  20. Guillemin, V.: Kähler structures on toric varieties. J. Differ. Geom 40, 285–309 (1994)

    Article  MATH  Google Scholar 

  21. Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math. 67, 491–513 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hall, S., Murphy, T.: Bounding the first invariant eigenvalue of toric Kähler manifolds. Preprint (2015)

  23. Hersch, J.: Quatre propriétés isopérimétriques de membranes sphériques homogénes , C.R. Acad. Sci. Paris Sér. A–B 270, A1645–A1648 (1970)

  24. Mabuchi, T.: Einstein–Kähler forms, Futaki invariants and convex geometry on toric Fano varieties. Osaka J. Math. 24(4), 705–737 (1987)

    MathSciNet  MATH  Google Scholar 

  25. Matsushima, Y.: Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété Kählérienne. Nagoya Math. J. 11, 145–150 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lejmi, M.: Extremal almost-Kähler metrics. Int. J. Math. 21(12), 1639–1662 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lerman, E., Tolman, S.: Hamiltonian torus actions on symplectic orbifolds and toric varieties. Trans. Am. Math. Soc. 349, 4201–4230 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Polterovich, L.: Symplectic aspects of the first eigenvalue. J. Reine Angew. Math. 502, 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tanno, S.: Eigenvalues of the Laplacian of Riemannian manifolds. Tohoku Math. J. (2) 25, 391–403 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

R. Sena-Dias was partially supported by FCT/Portugal through projects PEst-OE/EEI/LAOO9/2013, EXCL/MAT-GEO/0222/2012, PTDC/MAT/117762/2010 and PTDC/MAT-GEO/1608/2014 and E. Legendre is partially supported by the ANR French grant EMARKS. We would also like to thank CAST for a travel grant that allowed EL to visit Lisbon. The authors would like to thank Emily Dryden and Julien Keller for interesting conversations concerning the topic of this paper and also Stuart Hall and Tommy Murphy for sharing their preprint. Finally we are also indebted to the anonymous referee for pointing out a small mistake in a first version of the draft.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France

    Eveline Legendre

  2. Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Rosa Sena-Dias

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  1. Eveline Legendre
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Correspondence to Rosa Sena-Dias.

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Legendre, E., Sena-Dias, R. Toric Aspects of the First Eigenvalue. J Geom Anal 28, 2395–2421 (2018). https://doi.org/10.1007/s12220-017-9908-y

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