Local stability of Einstein metrics under the Ricci iteration

The Prescribed Ricci Curvature Problem for Homogeneous Metrics

“…One of the results of [15] establishes the existence and the convergence of Ricci iterations by means of the inverse function theorem. This approach turns out to be effective even on spaces without a homogeneous structure, as the first-named author and Hallgren demonstrate in the forthcoming work [14].…”

The prescribed cross curvature problem on the three-sphere

“…where σ 0 is constructed in Section 6.3. Lemma 6.5 and the global Lipschitz continuity of E imply the existence of a unique solution on the interval − 1, 7 8 to the initial-value problem…”

“…It is easy to see that l(r) remains negative for all r if α ≤ 0. To prove property 5, assume that l 7 8 = 0 and l 7 8 ≥ 5 7 . Then α > 0 and, by property 3, l (r) > 0.…”

“…Ergo, l(r) ∈ [−1, 0] for r ∈ − 7 8 , 7 8 . The ODE in (6.15) implies l (r) = l 7 8 − Formula (6.7) implies…”

“…In line with statement 1, set σ(r) = (r + 1) 2 for r ∈ −1, − 7 8 . Denote by l sm : − 1, − 7 8 → R the solution to equation (6.4) satisfying…”