The Kähler–Ricci flow around complete bounded curvature Kähler metrics

Abstract

We produce complete bounded curvature solutions to Kähler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth Kähler metric uniformly equivalent to another complete bounded curvature Kähler metric. We obtain related flow results for nonsmooth as well as degenerate initial conditions. We also obtain a stability result for complex space forms under the flow.

Some local maximum principles along Ricci flows

“…Theorem 1.1 is known to be true if M is compact without boundary or M is noncompact and g(t) is a complete solution with uniformly bounded curvature, see [11,23] for example. Nevertheless, there are interesting results of the existence of the Ricci flows in which the initial metrics and the flows g(t) may not be complete and may have unbounded curvatures, see [1,2,3,7,10,12,14,15,16,24,26,28]. However, most of the Ricci flow solutions mentioned above satisfy the condition (1.3), which is invariant under parabolic rescaling.…”

Some local Maximum principles along Ricci Flow

“…If M is compact without boundary or M is noncompact and g(t) is complete with uniformly bounded curvature, then Theorem 1.1 is by now standard (see [26] for example). Nevertheless, there are a number of existence results in which the initial metric is complete with possibly unbounded curvature and was found to be very useful in the study on complete manifolds, see [1,2,3,8,15,16,18,11,13,27,29,31] for related works on the existence theory of Ricci flow with initial metric of possibly unbounded curvature. Moreover, most of the known examples satisfy a scaling invariant curvature upper bound Ric(g(t)) ≤ αt −1 for some α > 0 when t > 0 is small.…”