The prescribed Ricci curvature problem for naturally reductive metrics on non-compact simple Lie groups

The study of the prescribed Ricci curvature problem is an important part of modern geometry with ties to flows, relativity and other subjects. The first wave of interest in this problem came in the 1980s; see [Bes87, Chapter 5] and [Aub98, Section 6.5]. Particularly extensive contributions were made at that time by DeTurck and his collaborators. For a discussion of the subsequent advances, including the recent progress in the framework on homogeneous spaces, see the survey [BP19].

Let $M$ be a smooth manifold. In its original interpretation, the prescribed Ricci curvature problem comes down to the equation

where the Riemannian metric $g$ on $M$ is the unknown and the (0,2)-tensor field $T$ is given. The paper [DeT81] proved, for nondegenerate $T$, the existence of $g$ satisfying this equation in a neighbourhood of a point on $M$; see also [DG99, Pul13, Pul16a]. However, subsequent research into the solvability of (1.1) on all of $M$ revealed the need for a more nuanced interpretation of the prescribed Ricci curvature problem. Specifically, suppose $M$ is closed and $T$ is positive-definite. The results of [Ham84, DeT85, Delan03] and other papers suggest that there exists at most one $c\in\mathbb{R}$ such that the equation

can be solved for $g$ on all of $M$. This is certainly the case if $M$ is the 2-dimensional sphere and $T$ is positive-definite; see [WW70, DeT85, Ham84]. Thus, on a closed manifold, one customarily interprets the prescribed Ricci curvature problem as the question of finding $g$ and $c$ such that (1.2) holds. This paradigm was originally proposed by DeTurck and Hamilton in [Ham84, DeT85]. As it turns out, (1.2) arises in applications, such as the construction of the Ricci iteration; see [PR19, BPRZ19] and also [BP19, Section 3.10]. On the other hand, if $M$ is open, it may be possible to obtain compelling existence theorems for (1.1) without the additional constant $c$. We refer to [Delay02, Delay18] for examples of such theorems.

In recent years, the third-named author and his collaborators produced a series of results, surveyed in [BP19], on the prescribed Ricci curvature problem in the class of homogeneous metrics. More precisely, suppose $G$ is a connected Lie group. Let $M$ be a homogeneous space with respect to $G$. Assume that the metric $g$ and the tensor field $T$ are $G$-invariant. Then (1.1) reduces to an overdetermined system of algebraic equations, whereas (1.2) reduces to a determined one. For compact $M$ and positive-semidefinite $T$, the third-named author showed in [Pul16b] that homogeneous metrics satisfying (1.2) are, up to scaling, critical points of the scalar curvature functional subject to the constraint $\operatorname{tr}_{g}T=1$. This observation led to the discovery of several sufficient conditions for the solvability of (1.2) in [Pul16b, GP17, Pul20]. It parallels the well-known variational approach to the Einstein equation; see, e.g., [WZ86, §1]. In the case where $M$ is compact but $T$ is not positive-semidefinite, the question of solvability of (1.2) remains largely open. We hope that the present paper will stimulate its investigation; see Remark 5.1.

As for the prescribed Ricci curvature problem for homogeneous metrics on non-compact spaces, progress has been scarce so far. Buttsworth conducted in [But19] a comprehensive study of this problem on unimodular three-dimensional Lie groups. In most of the cases he considered, there is at most one constant $c\in\mathbb{R}$ such that a metric $g$ satisfying (1.2) exists. Several questions related to, but distinct from, the solvability of (1.1) and (1.2) on non-compact Lie groups have been studied by Milnor, Kowalski–Nikcevic, Eberlein, Kremlev–Nikonorov, Ha–Lee, Pina–dos Santos, the first-named author in collaboration with Lafuente (forthcoming work), and others. For a discussion of those results and a collection of references, see [BP19, Sections 2 and 4.1].

Left-invariant naturally reductive metrics on a Lie group form an important family nested between the set of all left-invariant metrics and the set of bi-invariant ones. The investigation of this family has led to several significant advances in geometry. For instance, it yielded new solutions to the Einstein equation and new insights into the spectral theory of the Laplacian on manifolds; see [DZ79, GS10, Lau19]. In the recent paper [APZ20], Ziller and two of the authors studied (1.2) for naturally reductive metrics on a compact Lie group using variational methods. That work exposed several interesting and previously unseen patterns of behaviour of the scalar curvature functional. For instance, one of the main theorems of [APZ20] is a necessary condition for the existence of a critical point subject to the constraint $\operatorname{tr}_{g}T=1$. No results of this kind had appeared in the literature before.

The present paper studies the prescribed Ricci curvature problem for naturally reductive metrics on a non-compact Lie group $G$. We assume that $G$ is simple. The more general case of semisimple $G$ seems to be much more difficult analytically—we intend to consider it elsewhere. Since $G$ is an open manifold, it is reasonable for us to view the prescribed Ricci curvature problem as the question of finding solutions to (1.1). On the other hand, the fact that naturally reductive metrics are homogeneous suggests that a “better” interpretation of this problem may be given by (1.2). The present paper studies both equations. We show that (1.1) reduces to an overdetermined algebraic system. Even so, we are able to obtain a comprehensive existence and uniqueness theorem. Equation (1.2) reduces to a determined system. In order to find conditions for solvability, we characterise metrics satisfying (1.2) as critical points of the scalar curvature functional subject to one of three $T$-dependent constraints. While this characterisation is similar in spirit to the one obtained for compact Lie groups in [APZ20], it bears some conceptual distinctions and requires a different proof. We obtain existence theorems for global maxima and classify some of the other critical points. The development and application of our variational methods presents many interesting analytical challenges and provides a wealth of insight for the investigation of (1.2) on compact homogeneous spaces in the case of mixed-signature $T$ (see, e.g., Remark 5.1).

The paper is organised as follows. In Section 2, we recall the characterisation, originally obtained by Gordon, of naturally reductive metrics on a non-compact simple Lie group. This characterisation underpins all of our results. In Section 3, we compute the Ricci curvature of a naturally reductive metric on $G$. We believe that the formulas we obtain are of independent interest. Section 4 is devoted to equation (1.1). We produce a necessary and sufficient condition for the existence of a solution. We also establish uniqueness up to scaling. Section 5 focuses on (1.2). We develop the variational approach to this equation and describe several types of critical points of the scalar curvature functional. At the end, we summarise the implications for the existence and the number of solutions. Section 6 examines the case where the metrics we consider are naturally reductive with respect to $G\times K$ for a simple subgroup $K<G$. Here, we find conditions for the solvability of (1.2) that are both necessary and sufficient. We also determine the precise number of solutions. Finally, Section 7 offers a series of examples.

Consider a connected non-compact simple Lie group $G$ with Lie algebra $\mathfrak{g}$. The results of [Gor85, Section 5] yield a convenient characterisation of left-invariant naturally reductive metrics on $G$. We present this characterisation in Theorem 2.1 below. For the basic theory of naturally reductive metrics, see [DZ79, Section 1] and [Gor85, Section 2]. In what follows, we identify every left-invariant (0,2)-tensor field on $G$ with the bilinear form it induces on $\mathfrak{g}$.

Let $K$ be a maximal compact subgroup of $G$ with Lie algebra $\mathfrak{k}$. Suppose $B$ is the Killing form of $\mathfrak{g}$. Denote by $\mathfrak{p}$ the $B$-orthogonal complement of $\mathfrak{k}$ in $\mathfrak{g}$. Then

is a Cartan decomposition. We have the inclusions

The Killing form $B$ is positive-definite on $\mathfrak{p}$ and negative-definite on $\mathfrak{k}$. Thus,

is an inner product on $\mathfrak{g}$. Clearly, $Q$ is $\operatorname{ad}(\mathfrak{k})$-invariant, and

The quotient $G/K$ is a symmetric space. Because $G$ is simple, this space is irreducible. Consequently, the pair $(\mathfrak{g},\mathfrak{k})$ must appear in Table 3 or 4 of [Bes87, Section 7.H]. Let $\mathfrak{k}_{1},\ldots,\mathfrak{k}_{r}$ be the simple ideals of $[\mathfrak{k},\mathfrak{k}]$. Denote by $\mathfrak{k}_{r+1}$ the centre of $\mathfrak{k}$. Then

where $s=0$ if $\mathfrak{k}_{r+1}$ is trivial and $s=1$ otherwise. Analysing the tables in [Bes87, Section 7.H], we conclude that $\mathfrak{k}_{r+1}$ is at most 1-dimensional.

The direct product $G\times K$ acts on $G$ in accordance with the formula

The isotropy subgroup at the identity element of $G$ is

Denote by $\mathcal{M}_{K}$ the set of left-invariant metrics on $G$ that are naturally reductive with respect to $G\times K$ and some decomposition of the Lie algebra of $G\times K$. The main purpose of this paper is to study the prescribed Ricci curvature problem in $\mathcal{M}_{K}$. Gordon showed in [Gor85, Section 5] that every left-invariant naturally reductive metric on $G$ must lie in $\mathcal{M}_{K}$ for some choice of $K$. Moreover, she obtained the following characterisation result.

Our main objective in this section is to produce formulas for the Ricci curvature and the scalar curvature of a metric $g$ given by (2.3). To do so, we need to introduce an array of constants, $\kappa_{1},\ldots,\kappa_{r+s}$, associated with the pair $(\mathfrak{g},\mathfrak{k})$. Throughout the paper,

As we explained above, $d_{r+1}=1$ if the centre of $\mathfrak{k}$ is non-trivial.

Suppose $B_{i}$ is the Killing form of $\mathfrak{k}_{i}$ for $i=1,\ldots,r+s$. There exists $\kappa_{i}\in\mathbb{R}$ such that

Using the assumption that $G$ is simple, one can easily check that $0<\kappa_{i}<1$ for $i=1,\ldots,r$. If the centre of $\mathfrak{k}$ is non-trivial, then $\kappa_{r+1}=0$.

Next, we state a proposition that provides a way of computing $\kappa_{i}$ for a specific pair $(\mathfrak{g},\mathfrak{k})$ and $i=1,\ldots,r$. In what follows, superscript $\mathbb{C}$ means complexification. Clearly, the algebra $\mathfrak{g}^{\mathbb{C}}$ is semisimple. We preserve the notation $\operatorname{ad}$ for the adjoint representation of $\mathfrak{g}^{\mathbb{C}}$. Choose Cartan subalgebras $H$ in $\mathfrak{g}^{\mathbb{C}}$ and $H_{i}$ in $\mathfrak{k}_{i}^{\mathbb{C}}$. It is easy to verify that $\mathfrak{g}^{\mathbb{C}}$ is a completely reducible $\mathfrak{k}_{i}^{\mathbb{C}}$-module under the action given by $\operatorname{ad}$. This observation implies that every element of $H_{i}$ must be semisimple in $\mathfrak{g}^{\mathbb{C}}$. Consequently, we may assume that $H$ contains $H_{i}$. Let $\Phi^{+}$ and $\Phi_{i}^{+}$ be sets of positive roots of $\mathfrak{g}^{\mathbb{C}}$ and $\mathfrak{k}_{i}^{\mathbb{C}}$. The notation $\operatorname{tr}$ stands for the trace of a linear operator.

If $r=1$, one can calculate $\kappa_{i}$ using formula (3.3) below; see Examples 7.1 and 7.2.

We are now ready to state the main result of this section.

To prove Theorem 3.6, we apply the strategy developed in [DZ79, Section 5]. In what follows, $\operatorname{tr}_{h}$ stands for the trace of a bilinear form with respect to an inner product $h$. The notation $\pi_{\mathfrak{u}}$ is used for the $Q$-orthogonal projection onto $\mathfrak{u}\subset\mathfrak{g}$. For $j=1,\ldots,r+s$, define a bilinear form $A_{j}$ on $\mathfrak{p}$ by setting

Fix $Q$-orthonormal bases $(v_{i})_{i=1}^{n}$ of $\mathfrak{p}$ and $\big{(}v_{k}^{j}\big{)}_{k=1}^{d_{j}}$ of $\mathfrak{k}_{j}$. We need the following auxiliary result; cf. [Jen73, pages 609–610] and [DZ79, pages 32–34].

Denote by $S$ the scalar curvature functional on $\mathcal{M}_{K}$. Our next goal is to produce a formula for $S$. Taking the trace of the second equality in Lemma 3.7 and using the first one, we obtain

Theorem 3.6 and (3.3) imply the following result.

Consider a (0,2)-tensor field $T$ on the simple group $G$. In this section, we state a necessary and sufficient condition for the solvability of the equation

in the class $\mathcal{M}_{K}$. If a metric $g\in\mathcal{M}_{K}$ satisfying (4.1) exists, then $T$ must be left-invariant. Moreover, by Theorem 3.6, the formula

holds with $T_{\mathfrak{p}},T_{1},\ldots,T_{r+s}>0$.

It is tempting to use Theorem 4.1 to study the solvability of the equation

in the class $\mathcal{M}_{K}$. Indeed, suppose $\Xi$ is the set of left-invariant tensor fields on $G$ satisfying (4.2), (4.3) and (4.4). Theorem 4.1 states that (4.1) has a solution if and only if $T$ lies in $\Xi$. Using this result, we can easily obtain a description of the set of tensor fields that coincide with Ricci curvatures of metrics in $\mathcal{M}_{K}$ up to scaling. Namely, a pair $(g,c)\in\mathcal{M}_{K}\times(0,\infty)$ satisfying (4.5) exists if and only if

However, in general, it is difficult to determine whether a specific $T$ given by (4.2) lies in $\Xi^{\prime}$. To do so, one has to check whether (4.3) and (4.4) hold with $T_{\mathfrak{p}},T_{1},\ldots,T_{r+s}$ replaced by $cT_{\mathfrak{p}},cT_{1},\ldots,cT_{r+s}$ for some $c>0$. Already when $r+s=2$, this involves the tricky task of understanding if a polynomial of degree $4$ has roots that obey several constraints; when $r+s\geq 3$, the question seems to be substantially harder. In the present paper, we take a different approach to the analysis of (4.5). We are able to show that the existence of a pair $(g,c)\in\mathcal{M}_{K}\times(0,\infty)$ satisfying (4.5) follows from simple inequalities for the components of $T$. Moreover, we draw interesting conclusions regarding the non-uniqueness of such a pair.

Suppose $T$ is a left-invariant symmetric (0,2)-tensor field on $G$. Our next goal is to study the solvability of equation (4.5) in the class $\mathcal{M}_{K}$. As above, we assume the group $G$ is simple. This implies, in particular, that $\kappa_{i}<1$ for all $i$.

First, we re-state the problem in variational terms. More precisely, define

Each of these three spaces carries a manifold structure induced from $\mathcal{M}_{K}$. In Section 5.1, we show that $g$ satisfies (4.5) if and only if it is (up to scaling) a critical point of $S|_{\mathcal{M}^{+}_{T}}$, $S|_{\mathcal{M}^{-}_{T}}$ or $S|_{\mathcal{M}^{0}_{T}}$. This resembles the variational interpretation of (4.5) on compact Lie groups for positive-semidefinite $T$ (see [APZ20, Proposition 3.1]); however, in that case, only $S|_{\mathcal{M}^{+}_{T}}$ needs to be considered. In Sections 5.2 and 5.3, we obtain sufficient conditions for the existence of global maxima of $S|_{\mathcal{M}^{+}_{T}}$ and $S|_{\mathcal{M}^{-}_{T}}$, respectively. This requires complex estimates on the scalar curvature obtained in Lemmas 5.4 and 5.6. In Section 5.4, we classify completely the critical points of $S|_{\mathcal{M}^{0}_{T}}$ assuming $r+s\leq 2$. The analysis here can be reduced, as Lemma 5.9 demonstrates, to the study of a cubic polynomial in one variable. Finally, in Section 5.5, we summarise the implications of our results for the prescribed Ricci curvature problem.