Minimizing Weak Solutions for Calabi’s Extremal Metrics on Toric Manifolds

The existence of extremal metrics has been recently studied extensively on Kähler manifolds. The goal is to establish a sufficient and necessary condition for the existence of extremal metrics in the sense of Geometric Invariant Theory. There are many important works related to the necessary part ([Ti], [D1], [M1],[M2]). The sufficient part seems more difficult than the necessary part since the existence is related to the solvability of certain fourth-order elliptic equations. On the other hand, an extremal metric can be regarded as a critical point of some geometric energies, such as the Calabi’s energy, the modified K-energy, e.t. This gives a way to study the existence by using variational method in the sense of Nonlinear Analysis. In this paper, we focus on a class of special Kähler manifolds, namely, toric manifolds and discuss the minimizing weak solution for extremal metrics in the sense of convex functions related to a Donaldson’s version of the modified K-energy. Let (M,g) be a compact Kähler manifold of dimension n. Then Kähler form ωg of g is given by