Mabuchi and Aubin–Yau functionals over complex surfaces

In this note we construct Mabuchi LωM functional and Aubin–Yau functionals IωAY,JωAY on any compact complex surfaces, and establish a number of properties. Our construction coincides with the original one in the Kähler case. This method has been generalized to construct Mabuchi and Aubin–Yau functional over complex manifolds with complex dimension larger than two, see [5] and [6]. We will consider in [7] their applications to geometry.