On the cross curvature flow

Cross Curvature Solitons of Lorentzian Three-Dimensional Lie Groups

“…Thus, αx 3 = 0 and since α ̸ = 0, we obtain x 3 = 0. Also, α 2 = β 2 ; in this case, we have a left-invariant cross curvature soliton (3).…”

“…Suppose that {e 1 , e 2 , e 3 } is an orthonormal basis of signature (+ + −). We represent the Lorentzian vector product on R 3 1 , which is generated by the cross product ×, i.e.,…”

“…If α = δ, then x 3 = − 1 4 α 2 β − α 3 ; in this case, we have a left-invariant cross curvature soliton satisfying (3).…”

“…We mention that L X g indicates the Lie derivative of g with regard to X, and h is the cross curvature tensor of g. A cross curvature soliton is an interesting type of solution to the cross curvature flow. It is actually the self-similar solution of the cross curvature flow [2,3]. A cross curvature soliton is stated as being either expanding, steady, or shrinking if λ > 0, λ = 0, or λ < 0, respectively.…”

The prescribed cross curvature problem on the three-sphere

“…It has been investigated by many authors; see the survey [4]. The associated solitons were considered in [18,21].…”