Modified Kähler–Ricci flow on projective bundles

On a compact Kähler manifold, a Kähler metric ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} is called generalized quasi-Einstein (GQE) if it satisfies the equation Ric(ω)-HRic(ω)=LXω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Ric} (\omega ) - {\mathbb H}\mathrm{Ric} (\omega ) = L_X \omega $$\end{document} for some holomorphic vector field X, where HRic(ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb H}\mathrm{Ric} (\omega )$$\end{document} denotes the harmonic representative of the Ricci form Ric(ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Ric} (\omega )$$\end{document}. GQE metrics are one of the self-similar solutions of the modified Kähler–Ricci flow: ∂ωt∂t=-Ric(ωt)+HRic(ωt)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\partial \omega _t}{\partial t} = -\mathrm{Ric}(\omega _t) + {\mathbb H} \mathrm{Ric}(\omega _t)$$\end{document}. In this paper, we propose a method of studying the modified Kähler–Ricci flow on special projective bundles, called admissible bundles, from the view point of symplectic geometry. As a result, we can reduce the modified Kähler–Ricci flow to a simple PDE with one space variable. Moreover, we study the limiting behavior of the solution in some special cases.