Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces

Abstract

Consider a compact Lie group G and a closed Lie subgroup H < G. Let M be the set of G-invariant Riemannian metrics on the homogeneous space M = G/H. By studying variational properties of the scalar curvature functional on M, we obtain an existence theorem for solutions to the prescribed Ricci curvature problem on M . To illustrate the applicability of this result, we explore cases where M is a generalised Wallach space and a generalised flag manifold.Keywords: Prescribed Ricci curvature, homogeneous space, generalised Wallach space, generalised flag manifold for some c > 0, where T is a given G-invariant (0, 2)-tensor field. The study of (1.1) in the framework of homogeneous spaces was initiated in [22] and continued in [16]; see also [17,15,11]. It is on the basis of [22] that the first results about the Ricci iteration in the non-Kähler setting were obtained in [23]. These results provided a new approach to uniformisation on homogeneous spaces. Moreover, they led to the discovery of several dynamical properties of the Ricci curvature.Assume the (0, 2)-tensor field T lies in M. As [22, Lemma 2.1] demonstrates, a metric g ∈ M satisfies (1.1) for some c ∈ R if and only if it is (up to scaling) a critical point of the functional S onThis parallels the well-known variational interpretation of the Eisntein equation. Indeed, a metric g ∈ M satisfies Ric g = λg

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

“…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 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].…”

On the variational properties of the prescribed Ricci curvature functional

“…Remark 7.2. (a) The sufficient condition for the attainment of the global maximum in part (a) of the proposition was first obtained in [32]. Maple experiments suggest that the critical points in part (b) are typically non-degenerate and hence saddles.…”

“…The goal of this paper is to study the global behavior of the scalar curvature functional. In [19,32] one finds a general theorem for the existence of a global maximum, under certain assumptions on the isotropy representation of G/H. Our first result removes these assumptions, thereby expanding greatly the class of spaces to which the theorem applies.…”

Prescribing Ricci curvature on a Product of Spheres

“…For example, if M is a homogeneous space, and g and T are assumed to be invariant under the transitive action of a group G, (1.2) becomes a system of algebraic equations. Existence results were obtained in, for example, [6,7,21,22].…”