Inradius collapsed manifolds with a lower Ricci curvature bound

TL;DR

This paper investigates the geometric limits of inradius collapsed manifolds with boundary under Ricci curvature bounds, revealing their structure as quotients of boundary limit spaces with implications for boundary component counts.

Contribution

It establishes the structure of limit spaces of inradius collapsed manifolds with boundary, including isometric involutions and bounds on boundary components.

Findings

Limit space of boundaries admits an isometric involution.

Number of boundary components is at most two.

Limit space retains Ricci curvature bounds in a synthetic sense.

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.