Abstract
In this paper we give an explicit bound of Δg(t)u(t) and the local curvature estimates for the Ricci-harmonic flow ∂tg(t)=−2Ricg(t)+4du(t)⊗du(t),∂tu(t)=Δg(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.
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