Space of Ricci Flows I

Abstract

In this paper, we study the moduli spaces of m‐dimensional, κ‐noncollapsed Ricci flow solutions with bounded $\int |Rm|^{{m}/{2}}$ and bounded scalar curvature. We show a weak compactness theorem for such moduli spaces and apply it to study the estimates of isoperimetric constants, the Kähler‐Ricci flows, and the moduli spaces of gradient shrinking solitons. © 2012 Wiley Periodicals, Inc.

The Kähler–Ricci flow on Hirzebruch surfaces

“…By a conjecture of Yau [40], a necessary and su‰cient condition for M to admit a Kähler-Einstein metric is that M be 'stable in the sense of geometric invariant theory'. Indeed, the problem of using stabil-ity conditions to prove convergence properties of the Kähler-Ricci flow is an area of considerable current interest and we refer the reader to [21], [19], [22], [23], [25], [31], [24], [36] and [5] for some recent advances (however, this list of references is far from complete). One might expect that the su‰ciency part of the Yau-Tian-Donaldson conjecture can be proved via the flow (1.1).…”

Notes on Ricci flows with collapsinginitial data (I): Distance distortion

“…In order to obtain uniform estimates of the change of volume ratio along the Ricci flow, in the Kähler case Chen-Wang [10] studied the Bergman kernel, while in the Riemannian case, Bamler-Zhang [2] relies on Qi S. Zhang's heat kernel estimates in [33].…”

Asymptotic stability for Kähler–Ricci solitons

“…Kähler-Ricci solitons arise from the geometric analysis, such as Kähler-Ricci flow, and have been studied extensively for recent years. For instance, Chen and Wang [6] solved the Hamilton-Tian conjecture, which says that Kähler-Ricci flow on a Fano manifold always converges to a singular Kähler-Ricci soliton defined on a singular normal Fano variety in the sense of Cheeger-Gromov.…”