A surface that does not admit a length nonincreasing deformation is called metric minimizing. We show that metric minimizing surfaces in CAT(0) spaces are locally CAT(0) with respect to their length metrics.

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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

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 Fáry-Milnor's theorem in the CAT(0) setting.

We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.