On the conditions to control curvature tensors of Ricci flow

Abstract

An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (M ⁿ, g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that ${L^\frac{n+2}{2}}$ norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C ⁰ bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds. Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M ⁿ × [0, T), the curvature tensor stays uniformly bounded on M ⁿ × [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented.