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.

Abstract. We prove that a smooth compact submanifold of codimension 2 immersed in R n , n ≥ 3, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman.

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-withboundary as Alexandrov spaces of curvature bounded below.