Alexandrov Spaces Of Curvature Research Articles

Barycenters in Alexandrov spaces of curvature bounded below

Barycenters in Alexandrov spaces of curvature bounded below

Collapsing manifolds with boundary

This paper 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 first group of results provide a fiber bundle structure to the manifolds-with-boundary. One of the main theorems 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. Another result is a rough version of Toponogov’s Splitting Theorem. The second group of results identify Gromov–Hausdorff limits of certain sequences of manifolds with non-convex boundaries as Alexandrov spaces of curvature bounded below.

Manifolds with lower Ricci and L 1 -Sectional curvature control

Suppose that a sequence of Riemannian manifolds with Ricci curvature ≥ - k 2 converges to a Gromov-Hausdorff limit X . We show that if the amount of sectional curvature below K of the limiting manifolds approaches 0 in a suitable L 1 -sense, then X is an Alexandrov space of curvature ≥ K . As applications we present several generalizations of classical theorems.

Cheeger Type Sobolev Spaces for Metric Space Targets

In this paper, we consider the natural generalization of Cheeger type Sobolev spaces to maps into a metric space. We solve Dirichlet problem for CAT(0)-space targets, and obtain some results about the relation between Cheeger type Sobolev spaces for maps into a Banach space and those for maps into a subset of that Banach space. We also prove the minimality of upper pointwise Lipschitz constant functions for locally Lipschitz maps into an Alexandrov space of curvature bounded above.

Totally geodesic maps into metric spaces

We prove that a totally geodesic map between a Riemannian manifold and a metric space can be represented as the composite of a totally geodesic map from a Riemannian manifold to a Finslerian manifold and a locally isometric embedding between metric spaces. As a corollary, we obtain the homotheticity of a totally geodesic map from an irreducible Riemannian manifold to an Alexandrov space of curvature bounded above. This is a generalization of the case between Riemannian manifolds.