Curvature bounds via Ricci smoothing

Abstract

We give a proof of the fact that the upper and the lower sectional curvature bounds of a complete manifold vary at a bounded rate under the Ricci flow. Let (Mn, g) be a complete Riemannian manifold with | sec(M)| ≤ 1. Consider the Ricci flow of g given by (0.1) ∂ ∂t g = −2Ric(g) It is known ( see [Ham82, BMOR84, Shi89]) that (0.1) has a solution on [0, T ] for some T > 0 which smoothes out the metric. Namely, gt satisfies (0.2) e−c(n)tg ≤ gt ≤ eg |∇ −∇t| ≤ c(n)t |∇Rijkl(t)| ≤ c(n,m, T ) tm In particular, the sectional curvature of g(t) satisfies (0.3) |Kgt | ≤ C(n, T ) This result proved to be a very useful technical tool in many situations and in particular in the theory of convergence with two-sided curvature bounds ( see [CFG92, Ron96, PT99] etc). However, it turns out that in applications to convergence with two-sided curvature bounds in addition to the above properties, it is often convenient to know that supKgt and infKgt also vary at the bounded rate and in particular, the upper and the lower curvature bounds for gt are almost the same as for g for sufficiently small t . For example, it is very useful to know that if g0 has pinched positive [Ron96] or negative [Kan89, BK] curvature, then gt has almost the same pinching. This fact has apparently been known to some experts and it was used without a proof by various people (see e.g [Kan89, Fuk90, FJ98]). A careful proof was given in [Ron96] in case of a compact M . To the best of our knowledge, no proof exists in the literature in case of a noncompact M . The purpose of this note is to rectify this situation. To this end we prove 2000 Mathematics Subject classification. Primary 53C20.