Scalar Curvatures On Group Compactifications

In this paper we introduce Delcroix’s method for computing the complex MongeAmpere of K×K-invariant functions on a reductive group.This method allows us to reduce the computations of the curvature forms of K×K-invariant metrics to the computations of Hessians of real functions on a submanifold which can be viewed as G/K × K.Then we summarize a classic construction of toric manifolds,and show how to associate to a G × G-equivariant line bundle a moment poly tope P on a corresponding toric manifold,which allows us to associate to a K ×K-invariant metric a continuous function on 2P.We also summarize the compution of scalar curvatures in Li-ZhouZhu’s paper,which generalize Abreu’s work.Finally,we give an introduction to the Guillemin metrics on toric manifolds and give an example about these topics.