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DIAMETER AND LAPLACE EIGENVALUE ESTIMATES FOR COMPACT HOMOGENEOUS RIEMANNIAN MANIFOLDS

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Abstract

Let G be a compact connected Lie group and let K be a closed subgroup of G. In this paper, we study whether the functional 𝔤 ↦ ⋋1 (G/K, 𝔤) diam (G/K, 𝔤)2 is bounded among G-invariant metrics 𝔤 on G/K. Eldredge, Gordina, and Saloff-Coste conjectured in 2018 that this assertion holds when K is trivial; the only particular cases known so far are when G is abelian, SU(2), and SO(3). In this article, we prove the existence of the mentioned upper bound for every compact homogeneous space G/K having multiplicity-free isotropy representation.

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References

  1. A. Agrachev, D. Barilari, U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge Stud. Adv. Math., Vol. 181, Cambridge University Press, Cambridge, 2019.

  2. Arvanitoyeorgos, A.: Geometry of flag manifolds. Int. J. Geom. Methods Mod. Phys. 3(5–6), 957–974 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Bettiol, E. A. Lauret, P. Piccione, The first eigenvalue of a homogeneous CROSS, J. Geom. Anal. (2021), https://doi.org/10.1007/s12220-021-00826-7, arXiv:2001.08471 (2020).

  4. D. Burago, Yu. Burago, S. Ivanov, A Course in Metric Geometry, Grad. Stud. Math., Vol. 33, Amer. Math. Soc., Providence, RI, 2001.

  5. Chen, Z., Kang, Y., Liang, K.: Invariant Einstein metrics on three-locally-symmetric spaces. Comm. Anal. Geom. 24(4), 769–792 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dickinson, W., Kerr, M.M.: The geometry of compact homogeneous spaces with two isotropy summands. Ann. Global Anal. Geom. 34(4), 329–350 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Eldredge, N., Gordina, M., Saloff-Coste, L.: Left-invariant geometries on SU(2) are uniformly doubling. Geom. Funct. Anal. 28(5), 1321–1367 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  9. Judge, C., Lyons, R.: Upper bounds for the spectral function on homogeneous spaces via volume growth. Rev. Mat. Iberoam. 35(6), 1835–1858 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. D. N. Kupeli, Singular Semi-Riemannian Geometry, Mathematics and its Applications, Vol. 366, Kluwer, Dordrecht, 1996.

  11. Lauret, E.A.: The smallest Laplace eigenvalue of homogeneous 3-spheres. Bull. Lond. Math. Soc. 51(1), 49–69 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lauret, E.A.: On the smallest Laplace eigenvalue for naturally reductive metrics on compact simple Lie groups. Proc. Amer. Math. Soc. 148(8), 3375–3380 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. E. A. Lauret, Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups, Potential Anal. (2021), in press.

  14. E. Le Donne, Lecture Notes on Sub-Riemannian Geometry, https://www.dropbox.com/s/d08lpyg6slf3z0b/sub-Riem notes2021 4.pdf?dl=0.

  15. Li, P.: Eigenvalue estimates on homogeneous manifolds. Comment. Math. Helvetici. 55, 347–363 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. Macias Virgós, E.: Non-closed Lie subgroups of Lie groups. Ann. Global Anal. Geom. 11(1), 35–40 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Montgomery, A Tour of Subriemannian Geometries, their Geodesics and Applications, Math. Surveys Monogr, Vol. 91, Amer. Math. Soc., Providence, RI, 2002.

  18. Mutô, H., Urakawa, H.: On the least positive eigenvalue of Laplacian for compact homogeneous spaces. Osaka J. Math. 17(2), 471–484 (1980)

    MathSciNet  MATH  Google Scholar 

  19. Nikonorov, Y.G.: Classification of generalized Wallach spaces. Geom. Dedicata. 181, 193–212 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Takeuchi, Modern Spherical Functions, Translations of Mathematical Monographs, Vol. 135, American Mathematical Society, Providence, RI, 1994.

  21. Wang, M.Y.-K., Ziller, W.: On isotropy irreducible Riemannian manifolds. Acta Math. 166(3–4), 223–261 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ziller, W.: Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259(3), 351–358 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Instituto de Matemtica (INMABB), Departamento de Matemtica, Universidad Nacional del Sur, (UNS)-CONICET, Baha Blanca, Argentina

    EMILIO A. LAURET

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  1. EMILIO A. LAURET
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Correspondence to EMILIO A. LAURET.

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Emilio A. Lauret is research was supported by grants from FONCyT (PICT-2018-02073) and SGCYT–UNS.

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LAURET, E.A. DIAMETER AND LAPLACE EIGENVALUE ESTIMATES FOR COMPACT HOMOGENEOUS RIEMANNIAN MANIFOLDS. Transformation Groups 28, 1629–1650 (2023). https://doi.org/10.1007/s00031-022-09693-0

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