Convergence conjecture for Ricci iterations on homogeneous spaces

Let $M=G/H$ be a homogeneous space, let $H$ be maximal in $G$, and let $\mathcal{M}$ denote the space of Riemannian metrics under consideration. Let $g_1\in\mathcal{M}$, and let $r(g_1)$ be the maximal number of backward Ricci-iteration steps available from $g_1$. A sequence $\{g_i\}_{i\in\mathbb{N}}\subset\mathcal{M}$ satisfies the Ricci iteration when it obeys the relation

; a backward sequence $\{g_{-i}\}_{i=-1}^{\infty}$ satisfies

. Convergence conjecture for Ricci iterations.

(i) If a sequence $\{g_i\}_{i\in\mathbb{N}}\subset\mathcal{M}$ starting with $g_1$ and satisfying

\exists and is unique, then this sequence converges to an Einstein metric. (ii) If $r(g_1)=\infty$, then the sequence $\{g_{-i}\}_{i=-1}^{\infty}\subset\mathcal{M}$ given by

converges to an Einstein metric.

The conjecture predicts convergence of both forward and indefinitely extendible backward Ricci iterations under the maximality assumption. The surrounding results establish convergence in several special homogeneous settings and compactness/subsequential convergence under related hypotheses, but the general assertions remain open.