Monotonicity and K ¨ Ahler-ricci Flow

§0 Introduction. In this paper, we shall give a geometric account of the linear trace Li-Yau-Hamilton (which will be abbreviated as LYH) inequality for the Kähler-Ricci flow proved by LuenFai Tam and the author in [NT1]. To put the result, especially the Liouville theorem for the plurisubharmonic functions, into the right perspective we would also describe some dualities existed in both linear and nonlinear analysis first. The purpose is to show the reader that the linear trace LYH for the Kähler-Ricci flow can be thought as a ‘global’ parabolic version of the classical monotonicity formula for the analytic hypersurfaces in C (or more generally for the positive (1, 1) currents). We also include some new results. For example the Harnack inequality for the nondivergent elliptic operator on complete Kähler manifolds with nonnegative holomorphic bisectional curvature was proved in Theorem 1.5. Another is the LYH inequality for the Hermitian-Einstein flow coupled with Kähler-Ricci flow (cf. Theorem 3.5). We also prove that the sufficient and necessary condition for the equality in the linear trace LYH inequality is that the solution is a Kähler-Ricci soliton. (It has no restriction on the tensor satisfying the linear Lichnerowicz-Laplacian heat equation.) See, Theorem 4.1 and Theorem 4.2. A direct consequence of these result is that type II (III) limit solutions of Ricci (Kähler-Ricci) flow are gradient (expanding) solitons (Kähler-Ricci solitons). Theorem 4.1 and Theorem 4.2 generalizes and unifies the previous theorems (cf. [H3], [Ca2], [C-Z]) of Hamilton, Cao and more recently Chen-Zhu on the limit solutions to the Ricci flow considerably. There are monotonicity formulae for the parabolic equations such as the harmonic map heat equation and mean curvature flow (cf. [Hu], [St] and [H2]). But in the author’s point of view they all are still ‘local’ in the sense that the precise monotonicity only holds for (or inside) locally symmetric manifolds. Another important distinction is that the ‘global monotonicity’ (which holds on complete Kähler manifolds with nonnegative bisectional curvature) derived from the LYH inequality here is a point-wise estimate instead of the monotonicity of an integral quantity as in the previous mentioned cases such as those in [St] and [Hu]. There have been some other geometric interpretations on LYH inequality proved by Hamilton, for example in [C-C1], as well as on the Chow-Hamilton’s linear trace