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
*T*^{2}-invariant metric on M that solves the Abreu
equation.

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