Mabuchi and Aubin-Yau functionals over complex three-folds

TL;DR

This paper constructs Mabuchi and Aubin-Yau functionals on compact complex three-folds, providing a foundation for their extension to more general complex manifolds in future work.

Contribution

It introduces the construction of these functionals specifically for complex three-folds, advancing the understanding of geometric functionals in complex geometry.

Findings

Defined Mabuchi $ ext{L}^{{ m M}}_{oldsymbol{ ext{ω}}}$ functional

Constructed Aubin-Yau $ ext{I}^{{ m AY}}_{oldsymbol{ ext{ω}}}$ and $ ext{J}^{{ m AY}}_{oldsymbol{ ext{ω}}}$ functionals

Lays groundwork for generalization to all compact complex manifolds

Abstract

In this paper we construct Mabuchi $\mathcal{L}^{{\rm M}}_{\omega}$ 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}.