Limits of manifolds with boundary

In this paper, as a continuation of YZ: inrdius, we develop the geometry of the limit spaces of compact Riemannian manifolds with boundary, where we assume a lower sectional curvature bound, two sides bounds on the second fundamental forms of boundaries and an upper diameter bound. We mainly focus on the general case of non inradius collapse/convergence, where inradii of manifolds are uniformly bounded away from zero. In this case, many limit spaces have wild geometry, which arise from the boundary behavior of manifolds. Therefore the study of boundary singular points is a key to understand such limit spaces. We also present some global convergence/collapsing results.