On stability and the convergence of the Kähler-Ricci flow

Abstract

The normalized Kähler-Ricci flow exists for all times, and converges when the first Chern class is negative or zero [4,35]. However, when the first Chern class is positive, there are very few known cases of convergence. In one complex dimension, Hamilton [19] used entropy estimates to show convergence under the assumption of an initial metric of everywhere positive scalar curvature. This last assumption was removed later by Chow [12]. In higher dimensions, convergence was established only in the case of positive biholomorphic sectional curvature, first by X.X. Chen and Tian [9, 10] using Liouville energy functionals, and then by Cao, B.L. Chen, and Zhu [5] using the recent injectivity radius bound of Perelman [25].The convergence of the Kähler-Ricci flow for a Kähler manifold X of positive Chern class can be expected to be a difficult issue, since the limit would give a Kähler-Einstein metric, and not all Fano manifolds admit such metrics. According to a well-known conjecture of S.T. Yau [36], the existence of a Kähler-Einstein metric should be equivalent to the stability of X in the sense of geometric invariant theory. It is then an important problem to relate stability to the convergence of the Kähler-Ricci flow.In this paper, we take a first step in this direction. More specifically, we consider the convergence of the Kähler-Ricci flow on a compact Kähler manifold (X, ω 0 ) under two assumptions which are both a form of stability:(A) The Mabuchi energy ν ω 0 (φ) is bounded from below; (B) Let J be the complex structure of X, viewed as a tensor. Then the C ∞ closure of the orbit of J under the diffeomorphism group of X does not contain any complex structure J ∞ with the property that the space of holomorphic vector fields with respect to J ∞ has

Stability of Kähler-Ricci Flow

“…Now we need the following convergence result of a sequence of Kähler metrics, which is well-known in literature (cf. [26], [31]). …”

Energy functionals and canonical Kähler metrics

“…The functionals E k were used by Chen and Tian [ChTi1,ChTi2] to obtain convergence of the normalized Kähler-Ricci flow on Kähler-Einstein manifolds with positive bisectional curvature (see [PhSt4] for a related result). The Mabuchi energy is decreasing along the flow.…”

Kähler-Ricci flow on stable Fano manifolds

“…So we have shown that there is a sequence of times t i and diffeomorphisms F i such that the metrics F * i ω t i converge smoothly to a Kähler-Einstein metric on (M, J). Then by a theorem of Bando-Mabuchi [BM] the Mabuchi energy M ω has a lower bound, and the arguments in section 2 of [PS2] show that…”