Three-dimensional homogeneous Lorentzian Yamabe solitons

Homogeneous Cotton solitons

“…Therefore, it follows that x 3 = 0 and hence x 2 = 0. Finally, from the last two equations in (8) one obtains x 1 = 0, and this ends the proof.…”

On warped product gradient Yamabe solitons

“…Proof of Theorem 1.6 : Consider the change f = v 2 d+1 . Then by Proposition 1.5 we have that (16) ∆ w v − 1 dp (scal gB − ρ)v − 1 dp λ F v 1−p = 0, where w = − h 2d , p = 4 d + 1 .…”

“…Write g B = ·, · = | · | 2 for simplicity. Let v a positive solution to (16), then u = log v satisfies the equation ∆ w u = (β − 1)|∇u| 2 − L, where L = β|∇u| 2 − 1 dp (scal gB − ρ) − λ F dp e −pu , β ∈ (0, 1). Now, consider a cut-off function ξ satisfying…”

Invariant Ricci collineations on three-dimensional Lie groups

“…Left-invariant Yamabe solitons on three-dimensional Lorentzian Lie groups have been investigated in [14], showing that any such Lie group must correspond to a Type IV.3 non-unimodular Lie group with γ = 0 and α = δ 2 ̸ = 0. Moreover, ξ = (ξ 1 , ξ 2 , ξ 3 ) is a left-invariant Yamabe soliton if and only if…”

“…Hence every Killing vector field is an affine Killing vector field and a Ricci collineation but the converse may not be true, so we will emphasize the existence of the non-affine Killing ones; for this purpose we will take advantage of the subspaces of Killing and affine Killing vector fields which have been determined in [13]. Also, note that any homothetic vector field is a Ricci collineation (and hence so is any Yamabe soliton with constant scalar curvature [14]). Hence, we will focus on the proper case, i.e., Ricci collineations which are neither Killing nor homothetic.…”