A local curvature estimate for the Ricci-harmonic flow on complete Riemannian manifolds

In this paper we consider the local \(L^p\) estimate of Riemannian curvature for the Ricci-harmonic flow or List's flow introduced by List \cite{List2005} on complete noncompact manifolds. As an application, under the assumption that the flow exists on a finite time interval \([0,T)\) and the Ricci curvature is uniformly bounded, we prove that the \(L^p\) norm of Riemannian curvature is bounded, and then, applying the De Giorgi-Nash-Moser iteration method, obtain the local boundedness of Riemannian curvature and consequently the flow can be continuously extended past \(T\).