Ricci Flow under Kato-type curvature lower bound

In this work, we extend the existence theory of non-collapsed Ricci flows from point-wise curvature lower bound to Kato-type lower bound. As an application, we prove that compact three dimensional non-collapsed strong Kato limit space is homeomorphic to a smooth manifold. We also use the Ricci flow smoothing to study stability problem in scalar curvature geometry.Comment: reference updated, typos fixed, 20 page

Conformal tori with almost non-negative scalar curvature

In this work, we consider sequence of metrics with almost non-negative scalar curvature on torus. We show that if the sequence is uniformly conformal to another sequence of metrics with uniformly controlled geometry, then it converges to a flat metric in the volume preserving intrinsic flat sense, $L^{p}$ sense and the measured Gromov-Hausdorff sense.Comment: 20 page