Conformal deformations of CAT(0) spaces

Abstract

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 defined 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.Proof. Note that geodesics [α(t)p] lie in Dom f t for any t.Since f t is λ-concave, we haveHence the only-if part follows. Given a point p ∈ U and t, consider a pointp ∈ [α(t)p]. Applying ➎ forp and the triangle inequality, we get

Improvements of upper curvature bounds

“…Our construction follows [LS17] and defines the metric d ′ via a conformal change of the original metric d with a sufficiently convex function. The same approach shows that any CAT(0) space (X, d) admits another CAT(0) metric which is locally negatively curved, see Theorem 5.2 below.…”

“…Auxiliary results of independent interest. It has been shown in [LS17] that conformal changes with sufficiently convex functions preserve non-positive curvature. The improvement of the curvature bound is derived from the following analog of the formula expressing the curvature of a Riemannian manifold after a conformal change.…”

“…We refer to [LS17] and Subsection 4.1 below for the definition and basic properties of conformally changed spaces and recall that a function f :…”

“…Proof. The proof of Lemma 4 of [LS17] implies e ψ ·Z is the completion of the length space e ψ · (ϕ · D). Moreover, it shows that (e ψ · ϕ) · D is isometric to e ψ · (ϕ · D).…”

Metric-minimizing surfaces revisited

“…In [8], Mikhael Gromov used them to bound the complexity of smooth maps (general metric minimizing surfaces). In [14], Alexander Lytchak and the second author used them to deform general CAT(0) spaces (minimal discs). In [25], the second author used them in the proof a CAT(0) version of the Fary-Milnor theorem to control the mapping behavior of minimal surfaces (minimal discs).…”