A Modified Kähler-Ricci Flow

In this note, we study a Kähler-Ricci flow modified from the classic version. In the non-degenerate case, strong convergence at infinite time is achieved. The main focus should be on degenerate case, where some partial results are presented. 1 Set-up and Motivation Kähler-Ricci flow, which is nothing but Ricci flow with initial metric being Kähler, enjoys the same debut as Ricci flow in R. Hamilton’s original paper [5]. H. D. Cao’s paper, [1], can be taken as the first work devoted to the study of Kähler-Ricci flow and the alternative proof of Calabi Conjecture presented there has been bringing great interests to this object. Though it is essentially Ricci flow, the cohomology meaning coming with Kähler condition makes it possible to transform the metric flow to an equivalent scalar (potential) flow , which is much simpler-looking and more flexible to work with. One motivation of this note is to give a flavor of this flexibility. Let ω0 be any Kähler metric over a closed manifold X with dimCX = n > 2, and ω∞ is any smooth real closed (1, 1)-form. Set ωt = ω∞ + e(ω0 − ω∞) and consider the following flow at the level of metric potential for space-time ∂u ∂t = log (ωt + √ −1∂∂̄u) Ω , u(0, ·) = 0, (1.1) where Ω is a smooth volume form over X. This flow looks very much like the flow studied in [1], which can be considered as another motivation of this work. Let ω̃t = ωt + √ −1∂∂̄u and the corresponding flow at the level of metric is as follows ∂ω̃t ∂t = −Ric(ω̃t) + Ric(Ω)− e(ω0 − ω∞), ω̃0 = ω0, (1.2) 1This statement makes use of the uniqueness and short time existence results of Ricci flow.