Ricci-DeTurck flow from rough metrics and applications to scalar curvature problems

Motivated by the recent work of Lamm and Simon, in this work we study the short-time existence theory of Ricci-DeTurck flow starting from rough metrics which are bi-Lipschitz and have small local scaling invariant gradient concentration. As applications, we use the Ricci flow smoothing to show that scalar curvature lower bound is preserved under bi-Lipschitz $W^{1,n}$ convergence. This is an counter-part of the celebrated work of Gromov and Bamler. We also use similar idea to study stability problems in scalar curvature geometry.