Mabuchi and Aubin-Yau functionals over complex three-folds

In this paper we construct Mabuchi ${\mathcal{L}}_{\omega }^{\mathrm{M}}$ functional and Aubin-Yau functionals $$\mathcal{I}$^{{\rm AY}}{\omega}, $\mathcal{J}$^{{\rm AY}}{\omega}$ on any compact complex three-folds. The method presented here will be used in the forthcoming paper \cite{L1} on the construction of those functionals on any compact complex manifolds, which generalizes the previous work \cite{L}.