[Paper Review] Collapsing Manifolds with Boundary

Jeremy Wong|ArXiv.org|Nov 21, 2007

Geometric Analysis and Curvature Flows17 references4 citations

TL;DR

This paper investigates the Gromov-Hausdorff collapse of Riemannian manifolds with boundary under curvature and second fundamental form bounds, establishing that such sequences converge to Alexandrov spaces of curvature bounded below and exhibit disc bundle structures when the limit is a closed manifold. The key contribution is a topological and geometric characterization of collapsing sequences via extrinsic extension of boundaries and convexity arguments in limit spaces.

ABSTRACT

This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result establishes a disc bundle structure for any manifold-with-boundary having two-sided bounds on sectional curvature and second fundamental form, and a lower bound on intrinsic injectivity radius, which is sufficiently close in the Gromov-Hausdorff topology to a closed manifold. The second main result identifies Gromov-Hausdorff limits of certain sequences of manifolds-with-boundary as Alexandrov spaces of curvature bounded below.

Motivation & Objective

• To understand the topological and geometric relationship between collapsing sequences of manifolds with boundary and their Gromov-Hausdorff limits.
• To characterize the structure of limits when the sequence satisfies two-sided curvature bounds, bounded second fundamental form, and lower intrinsic injectivity radius.
• To show that certain collapsing sequences of manifolds with boundary yield limits that are Alexandrov spaces of curvature bounded below.
• To establish conditions under which the limit space inherits a disc bundle structure, even when the original manifolds have boundary.
• To analyze the role of boundary geometry—particularly local convexity and second fundamental form—in controlling collapse behavior and preventing new topology.

Proposed method

• Uses an extrinsic approach by extending the boundary of a manifold-with-boundary via a non-smooth gluing procedure to study collapse independently of interior or boundary collapse.
• Applies the Gromov-Hausdorff convergence framework to analyze limits of sequences of manifolds with boundary under curvature and second fundamental form bounds.
• Employs a codimension-zero extension technique to construct a larger space where the original manifold-with-boundary is embedded, enabling analysis of limit structure.
• Utilizes convexity arguments in metric spaces: proves that limits of $(C,2,r)$-convex subsets remain convex in the limit, with a refined radius bound.
• Applies a commutation result for warped product limits: $ ext{GH-lim}(X_i imes_ ho Y) = ( ext{GH-lim} X_i) imes_ ho Y$ when $Y$ is compact and $ ho$ continuous.
• Relies on the extension procedure from [19] and homotopy finiteness theorems to characterize limit spaces topologically and geometrically.

Experimental results

Research questions

• RQ1 Under what conditions does a sequence of manifolds with boundary collapse to a closed limit space in the Gromov-Hausdorff topology?
• RQ2 How does the second fundamental form of the boundary influence the structure of the limit space during collapse?
• RQ3 Can the limit of a collapsing sequence of manifolds with boundary be identified as an Alexandrov space of curvature bounded below?
• RQ4 What topological structure (e.g., disc bundle) emerges in the limit when the original manifolds have bounded curvature and boundary geometry?
• RQ5 To what extent can the presence of geodesic terminals (e.g., boundary points) be removed or controlled in the limit via curvature and boundary control?

Key findings

• A manifold-with-boundary with two-sided curvature bounds, bounded second fundamental form, and positive lower bound on intrinsic injectivity radius admits a disc bundle structure if it is sufficiently close in Gromov-Hausdorff distance to a closed manifold.
• The Gromov-Hausdorff limit of a sequence of manifolds-with-boundary satisfying curvature and second fundamental form bounds is an Alexandrov space of curvature bounded below.
• Limits of $(C,2,r)$-convex subsets under Gromov-Hausdorff convergence remain convex in the limit with a refined radius bound: the limit is $(C,2,r/2)$-convex.
• The extension procedure from [19] combined with homotopy finiteness theorems ensures that the limit space has no new topology introduced by the collapse.
• The limit of a sequence of warped product spaces $X_i imes_ ho Y$ under Gromov-Hausdorff convergence is the warped product of the limit space with $Y$, provided $Y$ is compact and $ ho$ is continuous.
• Boundary collapse of type (3)—boundary contact—can be analyzed independently of interior or boundary collapse when curvature and second fundamental form are bounded, via the extrinsic extension method.