A Property of Kähler-Ricci Solitons on Complete Complex Surfaces

where Rαβ(x, t) denotes the Ricci curvature tensor of the metric gαβ(x, t). One of the main problems in differential geometry is to find canonical structure on manifolds. The Ricci flow introduced by Hamilton [8] is an useful tool to approach such problems. For examples, Hamilton [10] and Chow [7] used the convergence of the Ricci flow to characterize the complex structures on compact Riemann surfaces, Hamilton [8] used the Ricci flow to classify compact three-manifolds with positive Ricci curvature, and the authors [4] recently used the Ricci flow to get steinness for a class of complete noncompact Kähler manifolds. By a direct computation one can see that the scalar curvature R(x, t) of gαβ(x, t) satisfies the equation ∂ ∂t R = △R + 2 ∣Rαβ ∣∣2 .