RICCI CURVATURE AND NORMALIZED RICCI FLOW ON GENERALIZED WALLACH SPACES

The paper proves two main claims: (I) when a1+a2+a3 ≤ 1/2, the normalized Ricci flow (NRF) evolves some Ric>0 metrics to mixed Ricci curvature; and (II) when a1+a2+a3 > 1/2, positivity is preserved for all Ric>0 metrics if θ ≥ max{θ1,θ2,θ3}, with θ := a1+a2+a3 − 1/2 and θi := ai − 1/2 + 1/2 sqrt((1−2ai)(1+2ai)−1), and even if this fails, some Ric>0 metrics remain positive along the flow (Theorems 5 and 6) . The reduction to a 3D ODE for invariant metrics and the explicit principal Ricci components r_i (hence the λ_i=0 boundary of the Ric>0 region) are given in the paper’s preliminaries . The paper proves the inward/outward pointing criterion by computing d/dt λ_i = (∇λ_i)·V along the boundary curves and analyzing a quartic polynomial h(t) whose sign is controlled by θ and θ_i (Lemmas 6–7 and ensuing arguments) . The candidate solution reaches the same threshold conditions and conclusions by a barrier argument phrased for the unnormalized flow (URF); this is consistent because the extra “radial” term in NRF is orthogonal to ∇λ_i on λ_i=0 by homogeneity, so the sign test matches the paper’s analysis. The only gap is that the candidate’s existence of an invariant compact subregion inside R+ when θ < max θ_i is asserted but not justified; the paper instead gives a precise sign-analysis showing trajectories can both exit and re-enter R in that regime (Subcase B2) . Overall, the results coincide; the proofs differ in presentation.