Local Solution and Extension to the Calabi Flow

The corresponding Kähler form ω = √ −1gij̄dz ∧ dz̄j is a closed (1, 1) form. So a Kähler class [ω] defines a nontrivial cohomology class. For simplicity, we use both g and ω to denote the same Kähler metric. For any Kähler metric ωφ in the same Kähler class [ω], we can write ωφ = ω + √ −1∂∂̄φ, where φ is a real valued function and it is called a Kähler potential. The Calabi flow is defined by ∂φ ∂t = Rφ −R, φ(0) = φ0, (1.1)