Inradius collapsed manifolds with a lower Ricci curvature bound

Abstract

In this paper, we study a family of $n$-dimensional Riemannian manifolds with boundary having lower bounds on the Ricci curvatures of interior and boundary and on the second fundamental form of boundary. A sequence of manifolds in this family is said to be inradius collapsed if their inradii tend to zero. We prove that the limit space $C_0$ of boundaries of inradius collapsed manifolds admits an isometric involution $f$, and that the limit of the manifolds themselves is isometric to the quotient space $C_0/f$. As an application, we show that the number of boundary components of inradius collapsed manifolds is at most two. Moreover, we prove that the limit space has a lower Ricci curvature bound and an upper dimension bound in a synthetic sense if in addition their boundaries are non-collapsed.