Limits of manifolds with boundary I
Takao Yamaguchi, Zhilang Zhang
TL;DR
This work develops a detailed infinitesimal description of limit spaces $N$ arising from sequences of $n$-dimensional manifolds with boundary under $K_M\ge\kappa$, $K_{\partial M}\ge\nu$, and $\Pi_{\partial M}\ge-\lambda$, with a diameter bound, in the non-inradius regime where ${\rm inrad}(M_i)$ stays uniformly positive. By extending manifolds via warped-cylinder gluing and analyzing the induced gluing map $\eta_0:C_0\to X_0$, the authors reveal that the limit is infinitesimally Alexandrov with rank at most one, while its boundary $N_0$ is infinitesimally sub-Alexandrov with rank zero. They introduce the boundary singular set $\mathcal{S}=\mathcal{S}^1\cup\mathcal{C}$ (singular points arising from $N_0$) and show how the differential $d\eta$ yields a quotient structure on tangent cones, leading to cusps when the associated isometric involution $f_*$ is nontrivial. The paper also establishes dimension bounds for boundary singular sets and discusses the geometry of almost parallel domains as a key tool, with sharpness illustrated by several examples. Overall, the work lays a rigorous infinitesimal framework for understanding how boundary behavior shapes limit spaces under GH-convergence and informs subsequent global convergence and stability analyses in this collapsing setting.
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.