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We prove that a minimal disc in a CAT(0) space is a local embedding away from a finite set of branch points. On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps. As an application, we prove Fary-Milnors theorem in the CAT(0) setting.
We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.
We construct short retractions of a CAT(1) space to its small convex subsets. This construction provides an alternative geometric description of an analytic tool introduced by Wilfrid Kendall. Our construction uses a tractrix flow which can be defi
We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate
We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space $X$ has a non-empty intersection in the visual bordification $ bar{X} = X cup partial X$. Using this fact, several results known for proper CAT(0) spaces
Given a fixed closed manifold M, we exhibit an explicit formula for the distance function of the canonical L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the manifold o