We analyze weak convergence on $CAT(0)$ spaces and the existence and properties of corresponding weak topologies.
We show that closed subsets with vanishing first homology in two-dimensional spaces inherit the upper curvature bound from their ambient spaces and discuss topological applications.
We prove that iterating projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bav{c}ak.
We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold (M,g). For example, if
a convex set in (M,g) is bounded by a smooth hypersurface, then it is strictly convex.
We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.
We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a $mathcal{C}^{1,1}$-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal
curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an $alpha$-Holder continuous Lorentzian metric admit a $mathcal{C}^{1,frac{alpha}{4}}$-parametrization.
We investigate the geometric and topological structure of equidistant decompositions of Riemannian manifolds.
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
ned as a gradient flow for a family of functions of certain type. In an appendix we prove a general existence result for gradient flows of time-dependent locally Lipschitz semiconcave functions, which is of independent interest.
We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.
We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.
