On the Kähler–Ricci flow near a Kähler–Einstein metric
-
Song Sun
and
Yuanqi Wang
Abstract
On a Fano manifold, we prove that the Kähler–Ricci flow starting from a Kähler metric in the anti-canonical class which is sufficiently close to a Kähler–Einstein metric must converge in a polynomial rate to a Kähler–Einstein metric. The convergence cannot happen in general if we study the flow on the level of Kähler potentials. Instead we exploit the interpretation of the Ricci flow as the gradient flow of Perelman's μ functional. This involves modifying the Ricci flow by a canonical family of gauges. In particular, the complex structure of the limit could be different in general. The main technical ingredient is a Lojasiewicz type inequality for Perelman's μ functional near a critical point.
Funding source: NSF
Award Identifier / Grant number: Research Assistantship
Both authors would like to thank Professor Xiuxiong Chen for constant support. We also thank Professor C. Arezzo and Professor X.-H. Zhu for their interest in this paper and for sending us their preprints (see [`Complexified Kähler–Ricci flow and families of Kähler–Ricci flow', preprint], [`Stability of Kähler–Ricci flow on a Fano manifold II', preprint]). The first author would also like to thank the Department of Mathematics in Stony Brook for its hospitality during the year 2009–2010.
We thank the anonymous referee for a careful reading and pointing out several typos in the first version.
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- Relatively very free curves and rational simple connectedness
- Moduli of unipotent representations II: Wide representations and the width
- Analog of selfduality in dimension nine
- A p-adic Eisenstein measure for unitary groups
- On the Kähler–Ricci flow near a Kähler–Einstein metric
- Homogeneous Ricci solitons
- Energy inequalities for cutoff functions and some applications
- Dimension of elementary amenable groups
Articles in the same Issue
- Frontmatter
- Relatively very free curves and rational simple connectedness
- Moduli of unipotent representations II: Wide representations and the width
- Analog of selfduality in dimension nine
- A p-adic Eisenstein measure for unitary groups
- On the Kähler–Ricci flow near a Kähler–Einstein metric
- Homogeneous Ricci solitons
- Energy inequalities for cutoff functions and some applications
- Dimension of elementary amenable groups
