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

TL;DR

This paper proves a conjecture relating the first Laplace eigenvalue and diameter for a specific class of metrics on compact simple Lie groups, confirming a boundedness property.

Contribution

It establishes the conjecture for naturally reductive left-invariant metrics on compact simple Lie groups, a subclass previously unconfirmed.

Findings

Confirmed the conjecture for naturally reductive metrics

Bounded the product of eigenvalue and diameter squared

Extended understanding of spectral geometry on Lie groups

Abstract

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.

Equations

$${\lambda }_1(M,g)\ge \frac{{\pi }^2/4}{\mathrm{diam}(M,g{)}^2}.$$

$${\lambda }_1(M,g)\le \frac{C}{\mathrm{diam}(M,g{)}^2}$$

$$⟨[Z,X{]}_{\mathfrak{p}},Y{⟩}_m+⟨X,[Z,Y{]}_{\mathfrak{p}}{⟩}_m=0$$

$$g_{\alpha ,h}(X,Y)=h(X_{\mathfrak{k}},Y_{\mathfrak{k}})+\alpha (-B_{\mathfrak{g}})(X_{\mathfrak{p}},Y_{\mathfrak{p}})\text{ for }X,Y\in \mathfrak{g}.$$

$${\lambda }_1(G,g)\le \frac{C}{\mathrm{diam}(G,g{)}^2}$$

$$\mathrm{diam}(M,g)\le \mathrm{diam}(M,\mathcal{H},s).$$

$$\mathcal{H}$$

$$\mathrm{diam}(G,g_h)\le \mathrm{diam}(G,\mathcal{H},s).$$

$$[X_{\mathfrak{k}},Y]=\big{[}[X_{\mathfrak{k}},Y_{1}],[Y_{2},\cdots,[Y_{n-1},Y_{n}]\cdots]\big{]}+\big{[}Y_{1},[X_{\mathfrak{k}},[Y_{2},\cdots,[Y_{n-1},Y_{n}]\cdots]]\big{]},$$

$$(K,h)⟶(G,g_h)\stackrel{\pi }{⟶}(G/K,-B_{\mathfrak{g}}{|}_{\mathfrak{p}}),$$

$${\lambda }_1(G,g_h)\le {\lambda }_1(G/K,-B_{\mathfrak{g}}{|}_{\mathfrak{p}}).$$

$$C=\underset{K\in \mathcal{K}\setminus \{G\}}{\max }\left\{{\lambda }_1(G/K,-B_{\mathfrak{g}}{|}_{\mathfrak{p}})\mathrm{diam}(G,\mathfrak{p},-B_{\mathfrak{g}}{|}_{\mathfrak{p}}{)}^2,{\lambda }_1(G,-B_{\mathfrak{g}})\mathrm{diam}(G,-B_{\mathfrak{g}}{)}^2\right\},$$

$${\lambda }_1(G,g)\le \frac{C}{\mathrm{diam}(G,g{)}^2}$$

$$X_1$$

$$T:=\left\{\left(\begin{array}{cc}{e}^{i\theta }& \\ & e^{-i\theta }\end{array}\right):\theta \in \mathbb{R}\right\}.$$

$$g_{\alpha ,\beta }:=\beta g_{(1,1,1)}{|}_{\mathfrak{t}}\oplus \alpha g_{(1,1,1)}{|}_{\mathfrak{p}}=g_{(\sqrt{\beta },\sqrt{\alpha },\sqrt{\alpha })}$$

Full text

This research was supported by grants from CONICET, FONCyT, SeCyT, and the Alexander von Humboldt Foundation (return fellowship)

For an arbitrary compact homogeneous Riemannian manifold $(M,g)$, Peter Li [Li80] proved that

[TABLE]

Here, $\lambda_{1}(M,g)$ denotes the smallest positive eigenvalue of the Laplace–Beltrami operator on $(M,g)$ and $\operatorname{diam}(M,g)$ is the diameter of $(M,g)$, that is, the maximum Riemannian distance between two points in $M$. 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 $\lambda_{1}(M,g)\,\operatorname{diam}(M,g)^{2}$ among all compact homogeneous Riemannian manifolds. For instance, the product $(M_{n},g_{n})$ of $n$ $d$-dimensional round spheres of constant curvature one satisfies $\lambda_{1}(M_{n},g_{n})=d$ and $\operatorname{diam}(M_{n},g_{n})=\sqrt{n}\pi$ goes to infinity when $n\to\infty$.

Eldredge, Gordina and Saloff-Coste have recently conjectured the existence of a uniform upper bound valid on special classes of homogeneous Riemannian manifolds, namely, the space of left-invariant metrics on a fixed compact connected Lie group.

[TABLE]

for all left-invariant metrics $g$ on $G$.

Among many other results, they confirm its validity for $\operatorname{SU}(2)$ in [EGS18, Thm. 8.5]. Explicit values of $C$ for $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$ 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 weaker conjecture after restricting to naturally reductive left-invariant metrics on a compact connected simple Lie group $G$ (Theorem 4 below). The reader should not consider this result as a strong evidence of Conjecture 1.

We next define naturally reductive metrics (see for instance [Be, §7.G]). Let $(M,g)$ be a homogeneous Riemannian manifold. We fix a base point $m\in M$ and $H$ a transitive group of isometries of $(M,g)$. Let $K$ be the isotropy subgroup at $m$, that is, $K=\{a\in H:a\cdot m=m\}$. The Lie algebra $\mathfrak{h}$ of $H$ decomposes into a sum $\mathfrak{h}=\mathfrak{k}\oplus\mathfrak{p}$, where $\mathfrak{k}$ denotes the Lie algebra of $K$ and $\mathfrak{p}$ is $\operatorname{Ad}(K)$-invariant. For $X\in\mathfrak{h}$, we write $X=X_{\mathfrak{k}}+X_{\mathfrak{p}}$ according to this decomposition. We have the identifications $M\equiv H/K$ and $T_{m}M\equiv\mathfrak{p}$, and the metric $g$ on $M$ corresponds to an $\operatorname{Ad}(K)$-invariant inner product $\langle{\cdot},{\cdot}\rangle_{m}$ on $\mathfrak{p}$.

[TABLE]

for all $X,Y,Z\in\mathfrak{g}$. (Here, $[\cdot,\cdot]$ denotes the bracket of the Lie algebra $\mathfrak{h}$.)

A naturally reductive space can be seen as a generalization of a symmetric space. Among their nice geometric properties, we have that every geodesic is an orbit of an one-parameter group of isometries. Normal homogeneous spaces are also naturally reductive. However, the class 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 of this paper, that is, when $M$ is a simple compact connected Lie group.

[TABLE]

D’Atri and Ziller proved that $g_{\alpha,h}$ is naturally reductive for all $\alpha,h$ as above (see [DZ79, Thm. 1]). (Note that the transitive group $H$ as in Definition 2 is $G\times K$ acting on $G$ as $(a,b)\cdot x=axb^{-1}$ for $a,x\in G$, $b\in K$.) Moreover, they also proved that any naturally reductive metric 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.

[TABLE]

for all naturally reductive metrics $g$ on $G$.

We need to introduce some notions to give an upper bound for the diameter. Recall that a sub-Riemannian manifold is a triple $(M,\mathcal{H},s)$, where $M$ is a smooth manifold, $\mathcal{H}$ is a subbundle of $TM$ and $s=(s_{m})_{m\in M}$ denotes a family of inner product on $\mathcal{H}$ which smoothly vary with the base point (see [Mo] for a general reference). A smooth curve $\gamma$ on $(M,\mathcal{H},g)$ is called horizontal if $\gamma^{\prime}(t)\in\mathcal{H}_{\gamma(t)}$ for all $t$. When $\mathcal{H}$ satisfies the bracket-generating condition (i.e. the Lie algebra generated by vector fields in $\mathcal{H}$ spans at every point the tangent space of $M$), also known as Hörmander condition, the Chow–Rashevskii Theorem ensures that $\operatorname{diam}(M,\mathcal{H},s)<\infty$ when $M$ is compact, that is, any two points in $M$ can be joined by a horizontal curve on $(M,\mathcal{H},s)$. It follows immediately that, if $g$ is a Riemannian metric on $M$ and we define the sub-Riemannian metric $s$ on $(M,\mathcal{H})$ given by $s_{m}=g_{m}|_{\mathcal{H}_{m}}$ for all $m\in M$, then

[TABLE]

We consider the sub-Riemannian manifold $(G,\mathcal{H},s)$ determined by

[TABLE]

for $X,Y\in\mathfrak{p}$ and $a\in G$. Here, $L_{a}:G\to G$ is given by $L_{a}(x)=ax$ and $\mathfrak{p}$ is seen as a subspace of $T_{e}G\equiv\mathfrak{g}$. Sub-Riemannian structures on arbitrary Lie groups of this form are called left invariant. Since $s$ is the restriction of $g_{h}$ to $\mathcal{H}$, (6) gives

[TABLE]

We next show that $\operatorname{diam}(G,\mathcal{H},s)<\infty$. It is sufficient to show that $\mathcal{H}$ satisfies the bracket-generating condition by Chow–Rashevskii Theorem. Since $\mathcal{H}$ is left invariant, this condition is equivalent to show that the Lie subalgebra of $\mathfrak{g}$ generated by $\mathfrak{p}$, say $\mathfrak{a}$, satisfies $\mathfrak{a}=\mathfrak{g}$. We next prove that $\mathfrak{a}$ is an ideal of the simple Lie algebra $\mathfrak{g}$, giving the required assertion. For $X=X_{\mathfrak{k}}+X_{\mathfrak{p}}\in\mathfrak{g}$ and $Y\in\mathfrak{a}$, we have that $[X,Y]=[X_{\mathfrak{k}},Y]+[X_{\mathfrak{p}},Y]$, so it remains to show that $[X_{\mathfrak{k}},Y]\in\mathfrak{a}$ since $[X_{\mathfrak{p}},Y]\in\mathfrak{a}$ is clear. The subspace $\mathfrak{a}$ is spanned by elements of the form $[Y_{1},[Y_{2},\cdots,[Y_{n-1},Y_{n}]\cdots]]$ with $Y_{1},\dots,Y_{n}\in\mathfrak{p}$. We next show by induction on $n$ that $[X_{\mathfrak{k}},Y]\in\mathfrak{a}$ for such element $Y$. The case $n=1$ is clear since $[X_{\mathfrak{k}},Y]=[X_{\mathfrak{k}},Y_{1}]\in\mathfrak{p}$ because $Y=Y_{1}\in\mathfrak{p}$ and $\mathfrak{p}$ is $\operatorname{ad}(\mathfrak{k})$-invariant. Furthermore, the Jacobi identity implies

[TABLE]

thus the inductive step also follows. This concludes the proof of $\mathfrak{a}=\mathfrak{g}$ and its consequence $\operatorname{diam}(G,\mathcal{H},s)<\infty$.

We now consider the Riemannian submersion with totally geodesic fibers given by

[TABLE]

which is a particular case of the general construction in [BB82, §2.2] and [Be, Thm. 9.80]. The spectral theory of Riemannian submersions with totally geodesic fibers has been intensively studied; see for instance [GLP]. It is well known that if $f$ is an eigenfunction of the Laplace–Beltrami operator $\Delta_{b}$ of the base space $(G/K,-B_{\mathfrak{g}}|_{\mathfrak{p}})$ with corresponding eigenvalue $\lambda$, then $f\circ\pi$ is an eigenfunction of the Laplace–Beltrami operator $\Delta_{g_{h}}$ of $(G,g_{h})$ with corresponding eigenvalue $\lambda$. Consequently, the smallest positive eigenvalue of $\Delta_{b}$ is an upper bound for the smallest positive eigenvalue of $\Delta_{g_{h}}$, that is

[TABLE]

We have shown that $\lambda_{1}(G,g_{h})\operatorname{diam}(G,g_{h})^{2}\leq\lambda_{1}(G/K,-B_{\mathfrak{g}}|_{\mathfrak{p}})\operatorname{diam}(G,\mathcal{H},s)^{2}$ for every left-invariant metric $h$ on $\mathfrak{k}$, when $\mathfrak{p}\neq 0$. Note that the right hand side depends on $K$ and $\mathfrak{p}$, but not on $h$. We conclude the proof by a finiteness argument on the choice of $K$ and $\mathfrak{p}$.

There is a finite collection $\mathcal{K}$ of closed subgroups of $G$ such that, for any naturally reductive left-invariant metric $g$ on $G$, there are $K\in\mathcal{K}$, $\alpha>0$, and $h$ a bi-invariant metric on $\mathfrak{k}$ such that $(G,g)$ is isometric to $(G,g_{h,\alpha})$ as in Remark 3 (see [GS10, Cor. 3.7]). Hence, by taking

[TABLE]

we conclude that (13) holds for all naturally reductive left-invariant metrics on $G$. ∎

Inside the proof, for a fixed closed subgroup $K$ of $G$ with $\dim K<\dim G$ (i.e. $\mathfrak{p}\neq 0$), it was not used that $h$ is a bi-invariant inner product on $\mathfrak{k}$. Thus, it was also proven the following statement.

[TABLE]

for every left-invariant metric $g$ on $G$ satisfying that $g(\mathfrak{k},\mathfrak{p})=0$ and $g|_{\mathfrak{p}}$ is a negative multiple of $B_{\mathfrak{g}}|_{\mathfrak{g}}$.

Theorem 4 applied to the well-known situation $G=\operatorname{SU}(2)$ returns that (2) is valid for a codimension one subspace of the space of left-invariant metrics up to isometry, as shown in the next example. For higher-dimensional compact connected simple Lie groups, the analogous codimension increases considerably.

[TABLE]

form a basis of the Lie algebra $\mathfrak{su}(2)$. For positive real numbers $a_{1},a_{2},a_{3}$, let $g_{(a_{1},a_{2},a_{3})}$ denote the left-invariant metric on $\operatorname{SU}(2)$ induced by the inner product on $\mathfrak{su}(2)$ given by $g_{(a_{1},a_{2},a_{3})}(X_{i},X_{j})=a_{i}^{2}\,\delta_{i,j}.$ Note that $g_{(1,1,1)}$ is a negative multiple of $B_{\mathfrak{su}(2)}$. Although the dimension of the space of inner products on $\mathfrak{su}(2)$ is $6$, Milnor [Mi76] proved that every left-invariant metric on $\operatorname{SU}(2)$ is isometric to $g_{(a_{1},a_{2},a_{3})}$ for some $a_{1},a_{2},a_{3}>0$. (Permutations of $(a_{1},a_{2},a_{3})$ do note change the isometry class of $g_{(a_{1},a_{2},a_{3})}$; see e.g. [EGS18, Lem. 2.8].)

The only proper closed connected subgroup of $G$ up to conjugation is the torus

[TABLE]

Up to homotheties, there is only one bi-invariant metric on $T$. Since the Lie algebra of $T$ is $\mathfrak{t}:=\operatorname{Span}_{\mathbb{R}}\{X_{1}\}$, D’Atri and Ziller’s Theorem mentioned in Remark 3 ([DZ79, Thm. 3]) ensures that every naturally reductive left-invariant metric on $\operatorname{SU}(2)$ is isometric to

[TABLE]

for some $\alpha,\beta>0$, where $\mathfrak{p}=\operatorname{Span}_{\mathbb{R}}\{X_{2},X_{3}\}$. These metrics are precisely the $3$-dimensional Berger spheres.

Acknowledgments

The author is greatly indebted to the referee for many accurate suggestions that have helped to improve significantly the presentation of the paper.