Ricci iteration on homogeneous spaces

Abstract

The Ricci iteration is a discrete analogue of the Ricci flow. We give the first study of the Ricci iteration on a class of Riemannian manifolds that are not Kähler. The Ricci iteration in the non-Kähler setting exhibits new phenomena. Among them is the existence of so-called ancient Ricci iterations. As we show, these are closely related to ancient Ricci flows and provide the first nontrivial examples of Riemannian metrics to which the Ricci operator can be applied infinitely many times. In some of the cases we study, these ancient Ricci iterations emerge (in the Gromov-Hausdorff topology) from a collapsed Einstein metric and converge smoothly to a second Einstein metric. In the case of compact homogeneous spaces with maximal isotropy, we prove a relative compactness result that excludes collapsing. Our work can also be viewed as proposing a dynamical criterion for detecting whether an ancient Ricci flow exists on a given Riemannian manifold as well as a method for predicting its limit.

Convergence of the Kähler–Ricci iteration

“…Without loss of generality we assume that ω (the reference form) is Kähler-Einstein. Using (25) we can write:…”

“…By the arguments in the proof of [15,Proposition 6.8] (see also [5,Lemma 2.7] and [15,Claim 7.11]), {f −1 k } k is contained in a bounded set of G. In particular, all derivatives up to order m, say, of f −1 k are bounded by some C m independently of k. So, to finish the proof, it suffices to estimate derivatives of h k := f k .ϕ k (since that will imply the same estimates on f [27, pp. 1539-1540] since by (25) we have…”

“…In essence, the Ricci iteration aims to reduce the Einstein equation to a sequence of prescribed Ricci curvature equations and can be thought of as a discretization of the Ricci flow. Going back to [26,27], it has been studied since by a number of authors [4,6,9,10,11,12,19,18,22,25], see also the survey [29, §6.5].…”

“…Of particular interest has been the study of the Ricci iteration on Kähler manifolds (for the non-Kähler case results are scarce, see [25]). When (M, J, g 1 ) is Kähler, the Calabi-Yau Theorem [31] guarantees the existence and uniqueness of the sequence {g i } i∈N if and only if M is Fano (i.e., has positive first Chern class c 1 (M, J)) and the Kähler class associated to g 1 is c 1 (M, J).…”

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

“…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.…”