Eldredge, Gordina and Saloff-Coste recently conjectured that, for a given compact connected Lie group $G$, there is a positive real number $C$ such that $\lambda_1(G,g)\operatorname{diam}(G,g)^2\leq C$ for all left-invariant metrics $g$ on $G$. In this short note, we establish the conjecture for the small subclass of naturally reductive left-invariant metrics on a compact simple Lie group.Fil: Lauret, Emilio Agustin. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentin

Lauret, Emilio Agustin

Proceedings of the American Mathematical Society

English

arXiv

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arXiv:1906.03325v2  [math.DG]  13 Dec 2019ON THE SMALLEST LAPLACE EIGENVALUE FOR NATURALLYREDUCTIVE METRICS ON COMPACT SIMPLE LIE GROUPSEMILIO A. LAURETAbstract. Eldredge, Gordina and Saloff-Coste recently conjectured that, for a given compactconnected Lie group G, there is a positive real number C such that λ1(G, g) diam(G, g)2 ≤ Cfor all left-invariant metrics g on G. In this short note, we establish the conjecture for the smallsubclass of naturally reductive left-invariant metrics on a compact simple Lie group.For an arbitrary compact homogeneous Riemannian manifold (M, g), Peter Li [Li80] provedthat(1) λ1(M, g) ≥π2/4diam(M, g)2.Here, λ1(M, g) denotes the smallest positive eigenvalue of the Laplace–Beltrami operator on(M, g) and diam(M, g) is the diameter of (M, g), that is, the maximum Riemannian distance be-tween two points inM . This lower bound has been recently improved by Judge and Lyons [JL19,Thm. 1.3].In contrast, there is no uniform upper bound for the term λ1(M, g) diam(M, g)2 amongall compact homogeneous Riemannian manifolds. For instance, the product (Mn, gn) of n d-dimensional round spheres of constant curvature one satisfies λ1(Mn, gn) = d and diam(Mn, gn) =√nπ goes to infinity when n → ∞.Eldredge, Gordina and Saloff-Coste have recently conjectured the existence of a uniformupper bound valid on special classes of homogeneous Riemannian manifolds, namely, the spaceof left-invariant metrics on a fixed compact connected Lie group.Conjecture 1. [EGS18, (1.2)] Given G a compact connected Lie group, there exists C > 0(depending only on G) such that(2) λ1(M, g) ≤Cdiam(M, g)2for all left-invariant metrics g on G.Among many other results, they confirm its validity for SU(2) in [EGS18, Thm. 8.5]. Explicitvalues of C for SU(2) and SO(3) can be found in [La19, Thm. 1.4].The main goal of this article is to give a simple and short proof of the validity of the weakerconjecture after restricting to naturally reductive left-invariant metrics on a compact connectedsimple Lie group G (Theorem 4 below). The reader should not consider this result as a strongevidence of Conjecture 1.We next define naturally reductive metrics (see for instance [Be, §7.G]). Let (M, g) be ahomogeneous Riemannian manifold. We fix a base point m ∈ M and H a transitive group ofisometries of (M, g). Let K be the isotropy subgroup at m, that is, K = {a ∈ H : a ·m = m}.Date: December 2019.2010 Mathematics Subject Classification. Primary 35P15, Secondary 58C40, 53C30.Key words and phrases. Naturally reductive metric, Laplace eigenvalue, diameter, compact simple Lie group.This research was supported by grants from CONICET, FONCyT, SeCyT, and the Alexander von HumboldtFoundation (return fellowship).12 EMILIO A. LAURETThe Lie algebra h of H decomposes into a sum h = k⊕ p, where k denotes the Lie algebra of Kand p is Ad(K)-invariant. For X ∈ h, we write X = Xk +Xp according to this decomposition.We have the identifications M ≡ H/K and TmM ≡ p, and the metric g on M corresponds toan Ad(K)-invariant inner product 〈·, ·〉m on p.Definition 2. A Riemannian manifold (M, g) is said to be naturally reductive if it admits atransitive action by isometries by a Lie group H and an Ad(K)-invariant complement p asabove such that(3) 〈[Z,X ]p, Y 〉m + 〈X, [Z, Y ]p〉m = 0for all X, Y, Z ∈ g. (Here, [·, ·] denotes the bracket of the Lie algebra h.)A naturally reductive space can be seen as a generalization of a symmetric space. Amongtheir nice geometric properties, we have that every geodesic is an orbit of an one-parametergroup of isometries. Normal homogeneous spaces are also naturally reductive. However, theclass of naturally reductive spaces is much broader and contains many other interesting cases.We now give a simple construction of naturally reductive metrics in the case of interest ofthis paper, that is, when M is a simple compact connected Lie group.Remark 3. LetG be a semisimple compact connected Lie group. It is well known that the spaceof left-invariant metrics on G is in 1-to-1 correspondence with the space of inner products onits Lie algebra g. Let K be a closed subgroup of G and let p denote the orthogonal complementof k in g with respect to the Killing form Bg of g. (We recall that Bg is an Ad(G)-invariantnegative definite bilinear form on g.) Given h a bi-invariant metric on K and α a positive realnumber, we define the left-invariant metric gh,α on G induced by the inner product on g givenby(4) gα,h(X, Y ) = h(Xk, Yk) + α (−Bg)(Xp, Yp) for X, Y ∈ g.D’Atri and Ziller proved that gα,h is naturally reductive for all α, h as above (see [DZ79,Thm. 1]). (Note that the transitive group H as in Definition 2 is G × K acting on G as(a, b) ·x = axb−1 for a, x ∈ G, b ∈ K.) Moreover, they also proved that any naturally reductivemetric on G is isometric to one of the above form when G is assumed simple (see [DZ79,Thm. 3]).We are now in position to prove the main theorem.Theorem 4. Let G be a compact connected simple Lie group. There exists C = C(G) > 0 suchthat(5) λ1(G, g) ≤Cdiam(G, g)2for all naturally reductive metrics g on G.Proof. Let G be a compact connected simple Lie group. We pick any naturally reductive metricon G, that is, gα,h as in (4), for some choice of K, α, and h as in Remark 3. Since the termλ1(M, g) diam(M, g)2 is invariant under positive scaling of g, we can assume without loosinggenerality that α = 1. We abbreviate gh = gh,1. Moreover, we avoid the case when p = 0 forbeing trivial since gh is necessarily a negative multiple of the Killing form Bg of g. Under thisnew assumption, p 6= 0, we will not use that h is a bi-invariant metric on K. More precisely,gh is defined as in (4) with α = 1 and h any inner product on k.We need to introduce some notions to give an upper bound for the diameter. Recall that asub-Riemannian manifold is a triple (M,H, s), where M is a smooth manifold, H is a subbundleof TM and s = (sm)m∈M denotes a family of inner product on H which smoothly vary with thebase point (see [Mo] for a general reference). A smooth curve γ on (M,H, g) is called horizontalON THE SMALLEST LAPLACE EIGENVALUE 3if γ′(t) ∈ Hγ(t) for all t. When H satisfies the bracket-generating condition (i.e. the Lie algebragenerated by vector fields in H spans at every point the tangent space of M), also known asHörmander condition, the Chow–Rashevskii Theorem ensures that diam(M,H, s) < ∞ whenM is compact, that is, any two points in M can be joined by a horizontal curve on (M,H, s). Itfollows immediately that, if g is a Riemannian metric on M and we define the sub-Riemannianmetric s on (M,H) given by sm = gm|Hm for all m ∈ M , then(6) diam(M, g) ≤ diam(M,H, s).We consider the sub-Riemannian manifold (G,H, s) determined byH =⋃a∈GdLa(p), sa(dLa(X), dLa(Y ))= −Bg(X, Y ),(7)for X, Y ∈ p and a ∈ G. Here, La : G → G is given by La(x) = ax and p is seen as a subspaceof TeG ≡ g. Sub-Riemannian structures on arbitrary Lie groups of this form are called leftinvariant. Since s is the restriction of gh to H, (6) gives(8) diam(G, gh) ≤ diam(G,H, s).We next show that diam(G,H, s) < ∞. It is sufficient to show that H satisfies the bracket-generating condition by Chow–Rashevskii Theorem. Since H is left invariant, this conditionis equivalent to show that the Lie subalgebra of g generated by p, say a, satisfies a = g. Wenext prove that a is an ideal of the simple Lie algebra g, giving the required assertion. ForX = Xk + Xp ∈ g and Y ∈ a, we have that [X, Y ] = [Xk, Y ] + [Xp, Y ], so it remains to showthat [Xk, Y ] ∈ a since [Xp, Y ] ∈ a is clear. The subspace a is spanned by elements of theform [Y1, [Y2, · · · , [Yn−1, Yn] · · · ]] with Y1, . . . , Yn ∈ p. We next show by induction on n that[Xk, Y ] ∈ a for such element Y . The case n = 1 is clear since [Xk, Y ] = [Xk, Y1] ∈ p becauseY = Y1 ∈ p and p is ad(k)-invariant. Furthermore, the Jacobi identity implies(9) [Xk, Y ] =[[Xk, Y1], [Y2, · · · , [Yn−1, Yn] · · · ]]+[Y1, [Xk, [Y2, · · · , [Yn−1, Yn] · · · ]]],thus the inductive step also follows. This concludes the proof of a = g and its consequencediam(G,H, s) < ∞.We now consider the Riemannian submersion with totally geodesic fibers given by(10) (K, h) −→ (G, gh) π−→ (G/K,−Bg|p),which is a particular case of the general construction in [BB82, §2.2] and [Be, Thm. 9.80]. Thespectral theory of Riemannian submersions with totally geodesic fibers has been intensivelystudied; see for instance [GLP]. It is well known that if f is an eigenfunction of the Laplace–Beltrami operator ∆b of the base space (G/K,−Bg|p) with corresponding eigenvalue λ, thenf ◦ π is an eigenfunction of the Laplace–Beltrami operator ∆gh of (G, gh) with correspondingeigenvalue λ. Consequently, the smallest positive eigenvalue of ∆b is an upper bound for thesmallest positive eigenvalue of ∆gh, that is(11) λ1(G, gh) ≤ λ1(G/K,−Bg|p).We have shown that λ1(G, gh) diam(G, gh)2 ≤ λ1(G/K,−Bg|p) diam(G,H, s)2 for every left-invariant metric h on k, when p 6= 0. Note that the right hand side depends on K and p, butnot on h. We conclude the proof by a finiteness argument on the choice of K and p.There is a finite collection K of closed subgroups of G such that, for any naturally reductiveleft-invariant metric g on G, there are K ∈ K, α > 0, and h a bi-invariant metric on k suchthat (G, g) is isometric to (G, gh,α) as in Remark 3 (see [GS10, Cor. 3.7]). Hence, by taking(12) C = maxK∈Kr{G}{λ1(G/K,−Bg|p) diam(G, p,−Bg|p)2, λ1(G,−Bg) diam(G,−Bg)2},we conclude that (13) holds for all naturally reductive left-invariant metrics on G. 4 EMILIO A. LAURETInside the proof, for a fixed closed subgroup K of G with dimK < dimG (i.e. p 6= 0), it wasnot used that h is a bi-invariant inner product on k. Thus, it was also proven the followingstatement.Theorem 5. Let G be a compact connected simple Lie group, let K be a closed subgroup ofG of dimension strictly less than dimG, and let p denote the orthogonal complement of k in gwith respect to the Killing form Bg of g. There exists C = C(G,K) > 0 such that(13) λ1(G, g) ≤Cdiam(G, g)2for every left-invariant metric g on G satisfying that g(k, p) = 0 and g|p is a negative multipleof Bg|g.Theorem 4 applied to the well-known situation G = SU(2) returns that (2) is valid for acodimension one subspace of the space of left-invariant metrics up to isometry, as shown inthe next example. For higher-dimensional compact connected simple Lie groups, the analogouscodimension increases considerably.Example 6. We consider G = SU(2), which is diffeomorphic to the 3-sphere S3. The elementsX1 =(i 00 −i), X2 =(0 1−1 0), X3 =(0 ii 0),(14)form a basis of the Lie algebra su(2). For positive real numbers a1, a2, a3, let g(a1,a2,a3) denote theleft-invariant metric on SU(2) induced by the inner product on su(2) given by g(a1,a2,a3)(Xi, Xj) =a2i δi,j. Note that g(1,1,1) is a negative multiple of Bsu(2). Although the dimension of the space ofinner products on su(2) is 6, Milnor [Mi76] proved that every left-invariant metric on SU(2) isisometric to g(a1,a2,a3) for some a1, a2, a3 > 0. (Permutations of (a1, a2, a3) do note change theisometry class of g(a1,a2,a3); see e.g. [EGS18, Lem. 2.8].)The only proper closed connected subgroup of G up to conjugation is the torus(15) T :={(eiθe−iθ): θ ∈ R}.Up to homotheties, there is only one bi-invariant metric on T . Since the Lie algebra of T ist := SpanR{X1}, D’Atri and Ziller’s Theorem mentioned in Remark 3 ([DZ79, Thm. 3]) ensuresthat every naturally reductive left-invariant metric on SU(2) is isometric to(16) gα,β := β g(1,1,1)|t ⊕ α g(1,1,1)|p = g(√β,√α,√α)for some α, β > 0, where p = SpanR{X2, X3}. These metrics are precisely the 3-dimensionalBerger spheres.Acknowledgments. The author is greatly indebted to the referee for many accurate sugges-tions that have helped to improve significantly the presentation of the paper.References[BB82] L. Bérard-Bergery, J.-P. Bourguignon. Laplacians and Riemannian submersions with totallygeodesic fibres. Illinois J. Math. 26:2 (1982), 181–200.[Be] A. Besse. Einstein manifolds. Class. Math. Springer, Berlin, 2008.[DZ79] J.E. D’Atri, W. Ziller. Naturally reductive metrics and Einstein metrics on compact Lie groups.Mem. Amer. Math. Soc. 18:215, 1979.[EGS18] N. Eldredge, M. Gordina, L. Saloff-Coste. Left-invariant geometries on SU(2) are uniformlydoubling. Geom. Funct. Anal. 28:5 (2018), 1321–1367. DOI: 10.1007/s00039-018-0457-8.[GLP] P.B. Gilkey, J.V. Leahy, J. Park. Spectral geometry, Riemannian submersions, and the Gromov-Lawson conjecture. Stud. Adv. Math., Chapman & Hall/CRC, Boca Raton FL, 1999.ON THE SMALLEST LAPLACE EIGENVALUE 5[GS10] C. Gordon, C. Sutton. Spectral isolation of naturally reductive metrics on simple Lie groups. Math.Z. 266:4 (2010), 979–995. DOI: 10.1007/s00209-009-0640-6.[JL19] C. Judge, R. Lyons. Upper bounds for the spectral function on homogeneous spaces via volumegrowth. Rev. Mat. Iberoam. 35:6, (2019), 1835–1858. DOI: 10.4171/rmi/1103.[La19] E.A. Lauret. The smallest Laplace eigenvalue of homogeneous 3-spheres. Bull. Lond. Math. Soc.51:1, (2019), 49–69. DOI: 10.1112/blms.12213.[Li80] P. Li. Eigenvalue estimates on homogeneous manifolds. Comment. Math. Helvetici 55 (1980), 347–363.DOI: 10.1007/BF02566692.[Mi76] J. Milnor. Curvatures of left invariant metrics on lie groups. Adv. Math. 21:3 (1976), 293–329. DOI:10.1016/S0001-8708(76)80002-3.[Mo] R. Montgomery. A tour of subriemannian geometries, their geodesics and applications. Math. Sur-veys Monogr 91. Amer. Math. Soc., Providence, 2002.Instituto de Matemática (INMABB), Departamento de Matemática, Universidad Nacionaldel Sur (UNS)-CONICET, Bah́ıa Blanca, Argentina.E-mail address : emilio.lauret@uns.edu.ar

On the smallest Laplace eigenvalue for naturally reductive metrics on compact simple Lie groups

CONICET Digital

arXiv:1906.03325v2  [math.DG]  13 Dec 2019ON THE SMALLEST LAPLACE EIGENVALUE FOR NATURALLYREDUCTIVE METRICS ON COMPACT SIMPLE LIE GROUPSEMILIO A. LAURETAbstract. Eldredge, Gordina and Saloff-Coste recently conjectured that, for a given compactconnected Lie group G, there is a positive real number C such that λ1(G, g) diam(G, g)2 ≤ Cfor all left-invariant metrics g on G. In this short note, we establish the conjecture for the smallsubclass of naturally reductive left-invariant metrics on a compact simple Lie group.For an arbitrary compact homogeneous Riemannian manifold (M, g), Peter Li [Li80] provedthat(1) λ1(M, g) ≥ pi2/4diam(M, g)2.Here, λ1(M, g) denotes the smallest positive eigenvalue of the Laplace–Beltrami operator on(M, g) and diam(M, g) is the diameter of (M, g), that is, the maximum Riemannian distance be-tween two points inM . This lower bound has been recently improved by Judge and Lyons [JL19,Thm. 1.3].In contrast, there is no uniform upper bound for the term λ1(M, g) diam(M, g)2 amongall compact homogeneous Riemannian manifolds. For instance, the product (Mn, gn) of n d-dimensional round spheres of constant curvature one satisfies λ1(Mn, gn) = d and diam(Mn, gn) =√npi goes to infinity when n→∞.Eldredge, Gordina and Saloff-Coste have recently conjectured the existence of a uniformupper bound valid on special classes of homogeneous Riemannian manifolds, namely, the spaceof left-invariant metrics on a fixed compact connected Lie group.Conjecture 1. [EGS18, (1.2)] Given G a compact connected Lie group, there exists C > 0(depending only on G) such that(2) λ1(M, g) ≤ Cdiam(M, g)2for all left-invariant metrics g on G.Among many other results, they confirm its validity for SU(2) in [EGS18, Thm. 8.5]. Explicitvalues of C for SU(2) and SO(3) can be found in [La19, Thm. 1.4].The main goal of this article is to give a simple and short proof of the validity of the weakerconjecture after restricting to naturally reductive left-invariant metrics on a compact connectedsimple Lie group G (Theorem 4 below). The reader should not consider this result as a strongevidence of Conjecture 1.We next define naturally reductive metrics (see for instance [Be, §7.G]). Let (M, g) be ahomogeneous Riemannian manifold. We fix a base point m ∈ M and H a transitive group ofisometries of (M, g). Let K be the isotropy subgroup at m, that is, K = {a ∈ H : a ·m = m}.Date: December 2019.2010 Mathematics Subject Classification. Primary 35P15, Secondary 58C40, 53C30.Key words and phrases. Naturally reductive metric, Laplace eigenvalue, diameter, compact simple Lie group.This research was supported by grants from CONICET, FONCyT, SeCyT, and the Alexander von HumboldtFoundation (return fellowship).12 EMILIO A. LAURETThe Lie algebra h of H decomposes into a sum h = k⊕ p, where k denotes the Lie algebra of Kand p is Ad(K)-invariant. For X ∈ h, we write X = Xk +Xp according to this decomposition.We have the identifications M ≡ H/K and TmM ≡ p, and the metric g on M corresponds toan Ad(K)-invariant inner product 〈·, ·〉m on p.Definition 2. A Riemannian manifold (M, g) is said to be naturally reductive if it admits atransitive action by isometries by a Lie group H and an Ad(K)-invariant complement p asabove such that(3) 〈[Z,X ]p, Y 〉m + 〈X, [Z, Y ]p〉m = 0for all X, Y, Z ∈ g. (Here, [·, ·] denotes the bracket of the Lie algebra h.)A naturally reductive space can be seen as a generalization of a symmetric space. Amongtheir nice geometric properties, we have that every geodesic is an orbit of an one-parametergroup of isometries. Normal homogeneous spaces are also naturally reductive. However, theclass of naturally reductive spaces is much broader and contains many other interesting cases.We now give a simple construction of naturally reductive metrics in the case of interest ofthis paper, that is, when M is a simple compact connected Lie group.Remark 3. LetG be a semisimple compact connected Lie group. It is well known that the spaceof left-invariant metrics on G is in 1-to-1 correspondence with the space of inner products onits Lie algebra g. Let K be a closed subgroup of G and let p denote the orthogonal complementof k in g with respect to the Killing form Bg of g. (We recall that Bg is an Ad(G)-invariantnegative definite bilinear form on g.) Given h a bi-invariant metric on K and α a positive realnumber, we define the left-invariant metric gh,α on G induced by the inner product on g givenby(4) gα,h(X, Y ) = h(Xk, Yk) + α (−Bg)(Xp, Yp) for X, Y ∈ g.D’Atri and Ziller proved that gα,h is naturally reductive for all α, h as above (see [DZ79,Thm. 1]). (Note that the transitive group H as in Definition 2 is G × K acting on G as(a, b) ·x = axb−1 for a, x ∈ G, b ∈ K.) Moreover, they also proved that any naturally reductivemetric on G is isometric to one of the above form when G is assumed simple (see [DZ79,Thm. 3]).We are now in position to prove the main theorem.Theorem 4. Let G be a compact connected simple Lie group. There exists C = C(G) > 0 suchthat(5) λ1(G, g) ≤ Cdiam(G, g)2for all naturally reductive metrics g on G.Proof. Let G be a compact connected simple Lie group. We pick any naturally reductive metricon G, that is, gα,h as in (4), for some choice of K, α, and h as in Remark 3. Since the termλ1(M, g) diam(M, g)2 is invariant under positive scaling of g, we can assume without loosinggenerality that α = 1. We abbreviate gh = gh,1. Moreover, we avoid the case when p = 0 forbeing trivial since gh is necessarily a negative multiple of the Killing form Bg of g. Under thisnew assumption, p 6= 0, we will not use that h is a bi-invariant metric on K. More precisely,gh is defined as in (4) with α = 1 and h any inner product on k.We need to introduce some notions to give an upper bound for the diameter. Recall that asub-Riemannian manifold is a triple (M,H, s), whereM is a smooth manifold, H is a subbundleof TM and s = (sm)m∈M denotes a family of inner product on H which smoothly vary with thebase point (see [Mo] for a general reference). A smooth curve γ on (M,H, g) is called horizontalON THE SMALLEST LAPLACE EIGENVALUE 3if γ′(t) ∈ Hγ(t) for all t. When H satisfies the bracket-generating condition (i.e. the Lie algebragenerated by vector fields in H spans at every point the tangent space of M), also known asHo¨rmander condition, the Chow–Rashevskii Theorem ensures that diam(M,H, s) < ∞ whenM is compact, that is, any two points inM can be joined by a horizontal curve on (M,H, s). Itfollows immediately that, if g is a Riemannian metric on M and we define the sub-Riemannianmetric s on (M,H) given by sm = gm|Hm for all m ∈M , then(6) diam(M, g) ≤ diam(M,H, s).We consider the sub-Riemannian manifold (G,H, s) determined byH =⋃a∈GdLa(p), sa(dLa(X), dLa(Y ))= −Bg(X, Y ),(7)for X, Y ∈ p and a ∈ G. Here, La : G→ G is given by La(x) = ax and p is seen as a subspaceof TeG ≡ g. Sub-Riemannian structures on arbitrary Lie groups of this form are called leftinvariant. Since s is the restriction of gh to H, (6) gives(8) diam(G, gh) ≤ diam(G,H, s).We next show that diam(G,H, s) <∞. It is sufficient to show that H satisfies the bracket-generating condition by Chow–Rashevskii Theorem. Since H is left invariant, this conditionis equivalent to show that the Lie subalgebra of g generated by p, say a, satisfies a = g. Wenext prove that a is an ideal of the simple Lie algebra g, giving the required assertion. ForX = Xk + Xp ∈ g and Y ∈ a, we have that [X, Y ] = [Xk, Y ] + [Xp, Y ], so it remains to showthat [Xk, Y ] ∈ a since [Xp, Y ] ∈ a is clear. The subspace a is spanned by elements of theform [Y1, [Y2, · · · , [Yn−1, Yn] · · · ]] with Y1, . . . , Yn ∈ p. We next show by induction on n that[Xk, Y ] ∈ a for such element Y . The case n = 1 is clear since [Xk, Y ] = [Xk, Y1] ∈ p becauseY = Y1 ∈ p and p is ad(k)-invariant. Furthermore, the Jacobi identity implies(9) [Xk, Y ] =[[Xk, Y1], [Y2, · · · , [Yn−1, Yn] · · · ]]+[Y1, [Xk, [Y2, · · · , [Yn−1, Yn] · · · ]]],thus the inductive step also follows. This concludes the proof of a = g and its consequencediam(G,H, s) <∞.We now consider the Riemannian submersion with totally geodesic fibers given by(10) (K, h) −→ (G, gh) pi−→ (G/K,−Bg|p),which is a particular case of the general construction in [BB82, §2.2] and [Be, Thm. 9.80]. Thespectral theory of Riemannian submersions with totally geodesic fibers has been intensivelystudied; see for instance [GLP]. It is well known that if f is an eigenfunction of the Laplace–Beltrami operator ∆b of the base space (G/K,−Bg|p) with corresponding eigenvalue λ, thenf ◦ pi is an eigenfunction of the Laplace–Beltrami operator ∆gh of (G, gh) with correspondingeigenvalue λ. Consequently, the smallest positive eigenvalue of ∆b is an upper bound for thesmallest positive eigenvalue of ∆gh, that is(11) λ1(G, gh) ≤ λ1(G/K,−Bg|p).We have shown that λ1(G, gh) diam(G, gh)2 ≤ λ1(G/K,−Bg|p) diam(G,H, s)2 for every left-invariant metric h on k, when p 6= 0. Note that the right hand side depends on K and p, butnot on h. We conclude the proof by a finiteness argument on the choice of K and p.There is a finite collection K of closed subgroups of G such that, for any naturally reductiveleft-invariant metric g on G, there are K ∈ K, α > 0, and h a bi-invariant metric on k suchthat (G, g) is isometric to (G, gh,α) as in Remark 3 (see [GS10, Cor. 3.7]). Hence, by taking(12) C = maxK∈Kr{G}{λ1(G/K,−Bg|p) diam(G, p,−Bg|p)2, λ1(G,−Bg) diam(G,−Bg)2},we conclude that (13) holds for all naturally reductive left-invariant metrics on G. 4 EMILIO A. LAURETInside the proof, for a fixed closed subgroup K of G with dimK < dimG (i.e. p 6= 0), it wasnot used that h is a bi-invariant inner product on k. Thus, it was also proven the followingstatement.Theorem 5. Let G be a compact connected simple Lie group, let K be a closed subgroup ofG of dimension strictly less than dimG, and let p denote the orthogonal complement of k in gwith respect to the Killing form Bg of g. There exists C = C(G,K) > 0 such that(13) λ1(G, g) ≤ Cdiam(G, g)2for every left-invariant metric g on G satisfying that g(k, p) = 0 and g|p is a negative multipleof Bg|g.Theorem 4 applied to the well-known situation G = SU(2) returns that (2) is valid for acodimension one subspace of the space of left-invariant metrics up to isometry, as shown inthe next example. For higher-dimensional compact connected simple Lie groups, the analogouscodimension increases considerably.Example 6. We consider G = SU(2), which is diffeomorphic to the 3-sphere S3. The elementsX1 =(i 00 −i), X2 =(0 1−1 0), X3 =(0 ii 0),(14)form a basis of the Lie algebra su(2). For positive real numbers a1, a2, a3, let g(a1,a2,a3) denote theleft-invariant metric on SU(2) induced by the inner product on su(2) given by g(a1,a2,a3)(Xi, Xj) =a2i δi,j. Note that g(1,1,1) is a negative multiple of Bsu(2). Although the dimension of the space ofinner products on su(2) is 6, Milnor [Mi76] proved that every left-invariant metric on SU(2) isisometric to g(a1,a2,a3) for some a1, a2, a3 > 0. (Permutations of (a1, a2, a3) do note change theisometry class of g(a1,a2,a3); see e.g. [EGS18, Lem. 2.8].)The only proper closed connected subgroup of G up to conjugation is the torus(15) T :={(eiθe−iθ): θ ∈ R}.Up to homotheties, there is only one bi-invariant metric on T . Since the Lie algebra of T ist := SpanR{X1}, D’Atri and Ziller’s Theorem mentioned in Remark 3 ([DZ79, Thm. 3]) ensuresthat every naturally reductive left-invariant metric on SU(2) is isometric to(16) gα,β := β g(1,1,1)|t ⊕ α g(1,1,1)|p = g(√β,√α,√α)for some α, β > 0, where p = SpanR{X2, X3}. These metrics are precisely the 3-dimensionalBerger spheres.Acknowledgments. The author is greatly indebted to the referee for many accurate sugges-tions that have helped to improve significantly the presentation of the paper.References[BB82] L. Be´rard-Bergery, J.-P. Bourguignon. Laplacians and Riemannian submersions with totallygeodesic fibres. Illinois J. Math. 26:2 (1982), 181–200.[Be] A. Besse. Einstein manifolds. Class. Math. Springer, Berlin, 2008.[DZ79] J.E. D’Atri, W. Ziller. Naturally reductive metrics and Einstein metrics on compact Lie groups.Mem. Amer. Math. Soc. 18:215, 1979.[EGS18] N. Eldredge, M. Gordina, L. Saloff-Coste. Left-invariant geometries on SU(2) are uniformlydoubling. Geom. Funct. Anal. 28:5 (2018), 1321–1367. DOI: 10.1007/s00039-018-0457-8.[GLP] P.B. Gilkey, J.V. Leahy, J. Park. Spectral geometry, Riemannian submersions, and the Gromov-Lawson conjecture. Stud. Adv. Math., Chapman & Hall/CRC, Boca Raton FL, 1999.ON THE SMALLEST LAPLACE EIGENVALUE 5[GS10] C. Gordon, C. Sutton. Spectral isolation of naturally reductive metrics on simple Lie groups. Math.Z. 266:4 (2010), 979–995. DOI: 10.1007/s00209-009-0640-6.[JL19] C. Judge, R. Lyons. Upper bounds for the spectral function on homogeneous spaces via volumegrowth. Rev. Mat. Iberoam. 35:6, (2019), 1835–1858. DOI: 10.4171/rmi/1103.[La19] E.A. Lauret. The smallest Laplace eigenvalue of homogeneous 3-spheres. Bull. Lond. Math. Soc.51:1, (2019), 49–69. DOI: 10.1112/blms.12213.[Li80] P. Li. Eigenvalue estimates on homogeneous manifolds. Comment. Math. Helvetici 55 (1980), 347–363.DOI: 10.1007/BF02566692.[Mi76] J. Milnor. Curvatures of left invariant metrics on lie groups. Adv. Math. 21:3 (1976), 293–329. DOI:10.1016/S0001-8708(76)80002-3.[Mo] R. Montgomery. A tour of subriemannian geometries, their geodesics and applications. Math. Sur-veys Monogr 91. Amer. Math. Soc., Providence, 2002.Instituto de Matema´tica (INMABB), Departamento de Matema´tica, Universidad Nacionaldel Sur (UNS)-CONICET, Bah´ıa Blanca, Argentina.E-mail address : emilio.lauret@uns.edu.ar

On the smallest Laplace eigenvalue for naturally reductive metrics on compact simple Lie groups

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