Pseudo-Ricci—Bourguignon solitons in the complex hyperbolic two-plane Grassmannians

By using the notion of pseudo-anti-commuting Ricci tensor, we have investigated a Hopf real hypersurface in the complex hyperbolic two-plane Grassmannian $G_2^{*}(\mathbb{C}^{m+2})$ which admits a pseudo-Ricci—Bourguignon soliton. In addition to this, we have proved that a non-trivial gradient pseudo-Ricci—Bourguignon soliton $(M,Df,\eta,\Omega,\theta,\gamma,g)$ on real hypersurfaces with isometric Reeb flow in the complex hyperbolic two-plane Grassmannian $G^{*}_2(\mathbb{C}^{m+2})$ does not exist. In the class of contact hypersurfaces in $G^{*}_2(\mathbb{C}^{m+2})$ except a tube with certain radius $r=\coth^{-1}({\sqrt3})$ over the totally geodesic and totally real quaternionic hyperbolic space $\mathbb{H}H^n$ in $G^{*}_2(\mathbb{C}^{m+2})$, $m=2n$, it has been also proved that there does not exist a non-trivial gradient pseudo-Ricci—Bourguignon soliton in $G^{*}_2(\mathbb{C}^{m+2})$.