Abstract

We study the prescribed scaler curvature problem on toric manifolds. We will show that the uniform stability introduced by Donaldson is a necessary condition for existing a smooth solution for any dimension n. For the case n = 2 we prove that this condition is also sufficient. More precisely, we prove the following theorem:

Theorem Let M be a compact toric surface and Delta be its Delzant polytope. Let K in C^infty (bar{Delta}) be an edge-nonvanishing function. If (M, K) is uniformly stable, then there is a smooth T²-invariant metric on M that solves the Abreu equation.

This talk is based on the joint works with Bo-hui Chen and Li Sheng