Limits of manifolds with boundary I

TL;DR

This paper investigates the infinitesimal geometry of limit spaces of compact Riemannian manifolds with boundary under curvature and diameter bounds, focusing on boundary singularities and their dimensions.

Contribution

It characterizes the infinitesimal structure at boundary singular points and determines the Hausdorff dimensions of boundary singular sets in these limit spaces.

Findings

Limit spaces can have complex boundary singularities.

Infinitesimal structure at boundary singular points is explicitly characterized.

Hausdorff dimensions of boundary singular sets are determined.

Abstract

In this paper, we develop the infinitesimal geometry of the limit spaces of compact Riemannian manifolds with boundary, where we assume lower bounds on the sectional curvatures of manifolds and boundaries and the second fundamental forms of boundaries and an upper diameter bound. We mainly focus on the case when inradii of manifolds are uniformly bounded away from zero. In this case, many limit spaces have wild geometry, which arise as the boundary singular points of the limit spaces. We determine the infinitesimal structure at those boundary singular points. We also determine the Hausdorff dimensions of the boundary singular sets.