Regularity and Long-Time Existence

In Chaps. 4 and 6 we saw that the curvature under Ricci flow obeys a parabolic equation with quadratic nonlinearity. By appealing to this view, we would expect the same kind of regularity that is seen in parabolic equa- tions to apply to the curvature. In particular we want to show that bounds on curvature automatically induce a priori bounds on all derivatives of the curvature for positive times. In the literature these are known as Bernstein– Bando–Shi derivative estimates as they follow the strategy and techniques introduced by Bernstein (done in the early twentieth century) for proving gradient bounds via the maximum principle and were derived for the Ricci flow in [Ban87] and comprehensively by Shi in [Shi89]. Here we will only need the global derivative of curvature estimates (for various local estimates see [CCG+08, Chap. 14]). In the second section we use these bounds to prove long-time existence.