A note on gradient Bach solitons

Revisiting Gradient Bach Solitons via Maximum Principles

“…Let us observe that if the Weyl curvature tensor is harmonic then Ric(∇f, •) = 0 (see [29,Lemma 2.1]). Consequently, from Theorem 2.1 we derive the following consequence, also obtained in [8, Corollary 5.3]: Corollary 2.1.…”

“…Furthermore, Ho investigated the Bach flow on a four-dimensional Lie group, in which he considered the convergence of the Bach flow. More recently, Shin [29] proved that, under a finite weighted Dirichlet integral condition, any complete noncompact gradient Bach soliton with harmonic Weyl curvature (which means that the divergence of W vanishes) is Bach-flat. In particular, he also studied complete fourdimensional gradient Bach solitons.…”