Mabuchi and Aubin-Yau functionals over complex surfaces

TL;DR

This paper constructs and analyzes Mabuchi and Aubin-Yau functionals on compact complex surfaces, extending their definitions beyond the Kähler case and establishing key properties.

Contribution

It introduces generalized Mabuchi and Aubin-Yau functionals on complex surfaces, aligning with classical definitions in the Kähler setting and expanding their applicability.

Findings

Functionals are well-defined on all compact complex surfaces.

The constructed functionals coincide with classical ones in the Kähler case.

Key properties of these functionals are established.

Abstract

In this note 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 surfaces, and establish a number of properties. Our construction coincides with the original one in the K\"ahler case.