Limits of manifolds with boundary II

TL;DR

The paper analyzes the Gromov-Hausdorff limits of families of $n$-dimensional compact Riemannian manifolds with boundary under lower sectional-curvature bounds and a diameter bound, focusing on the local and infinitesimal structure of the limit space $N$ and its boundary $N_0$. Using Alexandrov-geometry tools, gluing constructions, and gradient-flow arguments, it characterizes the regular and singular behavior along $N_0$, establishes almost-isometric stability near regular points, and proves that $N$ is locally Lipschitz contractible. It provides a quantitative Lipschitz-homotopy stability framework and proves volume-convergence results for boundary measures, along with criteria obstructing collapse, including a simplicial-volume obstruction. Together, these results give a detailed, stable picture of how manifolds with boundary can degenerate under GH-convergence and how their boundary components behave in the limit, including diffeomorphism-type stability in the non-collapsed regime. ${}$

Abstract

In this paper, as a continuation of [30], we consider the Gromov-Hausdorff convergence and collapsing in the family of compact Riemannian manifolds with boundary satisfying lower bounds on the sectional curvatures of interior manifolds, boundaries and the second fundamental forms of boundaries, and an upper diameter bound. We describe the local geometric structure of the limit spaces, and establish some stability results including Lipschitz homotopy stability.

Theorems & Definitions (116)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Remark 1.10