The Kähler-ricci Flowon Kähler Surfaces

The problem of finding Kähler-Einstein metrics on a compact Kähler manifold has been the subject of intense study over the last few decades. In his solution to Calabi’s conjecture, Yau [Ya1] proved the existence of a Kähler-Einstein metric on compact Kähler manifolds with vanishing or negative first Chern class. An alternative proof of Yau’s theorem is given by Cao [Ca] using the Kähler-Ricci flow. As is well-known, Hamilton’s Ricci flow has become one of the most powerful tools in geometric analysis with remarkable applications to the study of 3manifolds [Ha1], [Pe]. In early 90’s, Hamilton and Chow used the Ricci flow to give another proof of classical uniformization for Riemann surfaces (see [Ha2], [Ch]). Recently Perelman [Pe] has made a major breakthrough in studying the Ricci flow. The convergence of the Kähler-Ricci flow on Kähler-Einstein manifolds with positive first Chern class was claimed by Perelman and it has been generalized to any Kähler manifolds admitting a Kähler-Ricci soliton by Tian and Zhu [TiZhu]. Previously, in [ChTi], Chen and Tian proved that the Kähler-Ricci flow converges to a Kähler-Einstein metric if the initial metric is of non-negative bisectional curvature. Most algebraic varieties do not admit Kähler-Einstein metrics, for example, those with indefinite first Chern class, so it is a natural question to ask if there exist anywell-defined canonical metrics on these varieties or on their canonical models. Tsuji [Ts] applied the Kähler-Ricci flow to produce a canonical singular KählerEinstein metric on non-singular minimal algebraic varieties of general type. In [SoTi], new canonical metrics on the canonical models of projective varieties of positive Kodaira dimensionwere constructed. We also constructed such canonical metrics by the Käher-Ricci flow on Kähler surfaces. In this expository note, we present a metric classification for Kähler surfaces with non-negative Kodaira dimension or positive first Chern class by the KählerRicci flow.