Curvature pinching estimate and singularities of the Ricci flow

Abstract

Abstract. In this paper, we first derive a pinching estimate on the traceless Ricci curvature in term of scalar curvature and Weyl tensor under the Ricci flow. Then we apply this estimate to study finite-time singularity behavior. We show that if the scalar curvature is uniformly bounded, then the Weyl tensor has to blow up, as a consequence, the corresponding singularity model must be Ricci flat with non-vanishing Weyl tensor.

Long time existence and bounded scalar curvature in the Ricci-harmonic flow

“…As an immediate consequently of Theorem 2.2 we obtain the following theorem that is an extension of Cao's result ( [6], Corollary 3.1).…”

“…A well-known conjecture (see [6]) about the extension of RF states that (1.6) Is it true for lim sup…”

Pinching estimates for solutions of the linearized Ricci flow system in higher dimensions

“…Remark 1.3. If we let h = Rc, then Theorem 1.1 is just as a special case of Knopf 's Ricci curvature pinching estimate [19] (see also [4]). Cao and H. Tran [5] observed that the Ricci flow solution with nonnegative isotropic curvature implies a Riemannian curvature pinching result:…”

Long time existence of Ricci-harmonic flow

“…This conjecture was settled for Kähler-Ricci flow by Zhang [57] and for type-I maximal solution of RF by Enders-Müller-Topping [10]. Cao [6] proved the following:…”

“…List [27] and Müller [34] showed that, under the system of evolution equations 6) which is nonnegative. The evolution equations for other two functionals are derived in [25].…”