A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature

Abstract

We prove that Ricci flows with bounded scalar curvature cannot develop finite time singularities in dimensions less than eight. In order to study such flows in higher dimensions, we then develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points. In particular, near any singular point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a result of Enders, Topping and the first author that relied on a global Type I assumption. We also prove analogous results for the Ricci tensor, as well as a localised version of Sesum's result, namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at least at a Type I rate. Combining these and other results, we then show that for Ricci flows with bounded scalar curvature in higher dimensions the singular set has codimension eight in a suitable sense.

Type of finite time singularities of the Ricci flow with bounded scalar curvature

“…In dimension three, this is a consequence of the Hamilton-Ivey pinching estimate(see [25]) while in higher dimensions it is known to be true for (globally) Type I Ricci flows by Ender, Müller and Topping [12] as well as in the Kähler case by Zhang [28]. In recent years, this conjecture has been the focus of many many interesting new developments, see for example [2], [5], [6], [7], [8], [9], [10], [12], [22], [23], [27], [28].…”

“…From the perspective of globally Type I or Type II, we can only analyze singularities of such Types. To analyze pointwisely singularities of Ricci flows, Buzano and Di-Matteo [7] introduced general definitions of locally Type I and locally Type II singular points (see Definition 3.1 below). Our main result in this paper is implicitly directed towards proving the same assertion in ( * ) in the local sense in dimension < 8 (exact claim is Main Theorem mentioned above).…”

“…The definition of the δ-well-behaved sequence is slightly different to one in [7]. In [7], it requires that the last estimate holds on B(p i , t i , √ δr Ric (p i , t i )). But it is sufficient that the following Theorem holds.…”

“…Motivated by Hamilton's theory [15], Buzano and Di-Matteo introduced a local analysis of singularities and curvature blow-up rates [7]. Moreover, in the same paper, they also proved that the closed Ricci flow with uniformly bounded scalar curvature has no well-behaved blow-up sequences in dimension < 8.…”

A Łojasiewicz inequality for ALE metrics

“…scalar curvature [BZ17]. Even more recently however, it has been shown that Ricci flows on a closed Riemannian manifold of dimension less than 8 with bounded scalar curvature exist for all time: [BD20]. Finally, ALE Ricci flat metrics actually appear as finite-time blow-up limits of some Ricci flows [App19].…”