Manifolds with Small Curvature Concentration

In this work, we construct distance like functions with integral Hessian bound on manifolds with small curvature concentration and use it to construct Ricci flows on manifolds with possibly unbounded curvature. As an application, we study the geometric structure of those manifolds without bounded curvature assumption. In particular, we show that manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors n-regular and small curvature concentration are topologically Euclidean.