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Diameter and Laplace Eigenvalue Estimates for Left-invariant Metrics on Compact Lie Groups

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Abstract

Let G be a compact connected Lie group of dimension m. Once a bi-invariant metric on G is fixed, left-invariant metrics on G are in correspondence with m × m positive definite symmetric matrices. We estimate the diameter and the smallest positive eigenvalue of the Laplace-Beltrami operator associated to a left-invariant metric on G in terms of the eigenvalues of the corresponding positive definite symmetric matrix. As a consequence, we give partial answers to a conjecture by Eldredge, Gordina and Saloff-Coste; namely, we give large subsets \(\mathcal {S}\) of the space of left-invariant metrics \({\mathscr{M}}\) on G such that there exists a positive real number C depending on G and \(\mathcal {S}\) such that λ1(G,g)diam(G,g)2 ≤ C for all \(g\in \mathcal {S}\). The existence of the constant C for \(\mathcal {S}={\mathscr{M}}\) is the original conjecture.

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Acknowledgements

The author is grateful for helpful and motivating conversations with Renato Bettiol, Yves de Cornulier, Nate Eldredge, Lenny Fukshansky, Fernando Galaz-García, Jorge Lauret, Enrico Le Donne, Juan Pablo Rossetti, Michael Ruzhansky, Dorothee Schueth, and Ovidiu Cristinel Stoica. The author is greatly indebted to the referee for a careful reading and for providing a counterexample of a conjecture in the first submitted version of the article.

Funding

This research was supported by grants from CONICET, FonCyT (BID-PICT 2018-02073), SeCyT, and the Alexander von Humboldt Foundation (return fellowship).

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

    Emilio A. Lauret

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Correspondence to Emilio A. Lauret.

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Lauret, E.A. Diameter and Laplace Eigenvalue Estimates for Left-invariant Metrics on Compact Lie Groups. Potential Anal 58, 37–70 (2023). https://doi.org/10.1007/s11118-021-09932-1

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