Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups

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 S of the space of left-invariant metrics M on G such that there exists a positive real number C depending on G and S such that λ 1 (G, g) diam(G, g) 2 ≤ C for all g ∈ S. The existence of the constant C for S = M is the original conjecture.

About left-invariant geometry and homogeneous pseudo-Riemannian Einstein structures on the Lie group SU(3)

“…and the scalar curvature of the family of metrics d 1/8 × h (for which the determinant stays equal to 1 when the parameters vary) is τ /d 1/8 , where the expression of τ in terms of the parameters η, γ, , β, α was given in (12). We shall only display a few curves that illustrate the stationarity property by giving plots of ∂ ∂u τ (−d) 1/8 , for u ∈ {η, γ, , β}, in a neighborhood of the found solution 15 . 7.…”

“…where α runs over the set of all roots (use the identity α |α α| = 2g to relate ( 14) and (15) as in ( 13)).…”

“…Let us conclude this section with a slightly more involved example: we consider the SU(3) representation of highest weight w = (2, 1), which is of dimension [15]. Using the restriction matrix on the weight system 22 of this highest weight of SU(3), we obtain the weights 23 appearing in the branching to U(2).…”

“…Laplacian of left-invariant metrics. Let h be an arbitrary left-invariant metric on the Lie group G, the spectrum of the corresponding Laplacian ∆ is discussed in a number of places (see for instance [13], [15], and references therein). Using the Peter-Weyl theorem together with left-invariance of the metric one can replace a difficult problem of analysis on manifolds by a simpler algebraic problem: as in the bi-invariant case, the eigenvalues of the Laplace operator can be obtained, up to sign, as eigenvalues of some appropriate metric-dependent modified Casimir operator (consider for instance the expression (19) below) evaluated in irreducible representations of G. One should be careful with this terminology because the associated modified Casimir elements (that can be defined in the enveloping algebra of Lie(G)) are not, in general, central.…”

“…4.2, the eigenvalue of −∆0 is 1, and those of −∆, calculated using (20), coincide, of course, for each of the four terms of the branching decomposition, with the components of the diagonal 8 × 8 matrix C(1, 1) obtained previously. Let us finally consider the SU(3) representation of highest weight w = (2, 1), of dimension [15], whose branching to U(2) was also considered in sect. 4.2.…”