An extension procedure for manifolds with boundary

Abstract

This paper introduces an isometric extension procedure for Riemannian manifolds with boundary, which preserves some lower curvature bound and produces a totally geodesic boundary. As immediate applications of this construction, one obtains in particular upper volume bounds, an upper intrinsic diameter bound for the boundary, precompactness, and a homeomorphism finiteness theorem for certain classes of manifolds with boundary, as well as a characterization up to homotopy of Gromov-Hausdorff limits of such a class.

Collapsing manifolds with boundary

“…Several results substantiate the statement that collapses of type (3) do not generate additional topology: (i) the extension procedure introduced in [19] (see §B.1) together with the attendant homeomorphism finiteness theorem and homotopy characterization of limits, and (ii) Theorems 2.1.1-2.1.3 and 2.1.5 of the present paper, which yield disc bundle structures for a sequence of manifolds collapsing in the sense of (3) under a lower sectional curvature bound and having not-too-concave boundary.…”

“…The reader is referred to [20] for the proof of Theorem 2.1.4. Theorem 2.1.5 relies on the Alexandrov extension procedure from [19] (see § B), and uses a special gluing lemma. Whereas most 'perturbation' or ǫ-versions of a theorem in Riemannian geometry are obtained from the corresponding usual proof, via an argument involving pasing to the limit, the proof here will proceed somewhat differently.…”

Volumes and limits of manifolds with Ricci curvature and mean curvature bounds

“…Kodani [11] has proven GH precompactness of families with uniform bounds on sectional curvature. Wong [22] has proven GH precompactness of families with uniform bounds for the Ricci curvature, the second fundamental form and the diameter. Neither Kodani nor Wong study the rectifiability of the GH limit spaces of manifolds in the families they study.…”

Sequences of open Riemannian manifolds with boundary

“…Compactness theorems for sequences of Riemannian manifolds with boundary, assuming curvature controls on the boundary, have been proven by Kodani [11], Anderson, Katsuda, Kurylev, Lassas, and Taylor [1], Wong [14] and Knox [10]. A survey of these results has been written by the first author [12].…”