Local curvature estimates for the Ricci-harmonic flow

In this paper we give an explicit bound of .6g(t)u(t) and the local curvature estimates for the Ricci-harmonic flow partial differential tg(t) = -2 Ricg(t) + 4du(t) & OTIMES; du(t), partial differential tu(t) = .6g(t)u(t) under the condition that the Ricci curvature is bounded along the flow. In the second part these local curvature estimates are extended to a class of generalized Ricci flow, introduced by the author Li (2019), whose stable points give Ricci flat metrics on a complete manifold, and which is very close to the (K, N)-super Ricci flow recently defined by Li and Li (0000). Next we propose a conjecture for Einstein's scalar field equations motivated by a result in the first part and the bounded L2-curvature conjecture recently solved by Klainerman et al. (2015). In the last part of this paper, we discuss the forward and backward uniqueness for the Ricci-harmonic flow. (c) 2022 Elsevier Ltd. All rights reserved.