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Extensions of Lipschitz maps into Hadamard spaces. (English) Zbl 0990.53070

Main result: Let \(Y\) be an \(n\)-dimensional Hadamard manifold with sectional curvature \(-b^2\leq K\leq-(pb)^2\) for some \(b>0\) and \(0<p\leq 1\). There exists a constant \(c=c(n,p)\) such that every \(\lambda\)-Lipschitz map \(f:S \to Y\) defined on a subset \(S\subset X\) of a metric space \(X\) possesses a \(c \lambda\)-Lipschitz extension \(g:X\to Y\).
In fact the article provides a nice invitation into the general theory of Hadamard spaces, i.e., into the theory of simply connected and complete geodesic metric spaces of nonpositive curvature in the sense of Alexandrov. All fundamental concepts (e.g., geodesics, horospheres, angles, tangent cones) are recalled and rather powerful tolls (e.g., Hausdorff measure, center of mass construction, generalized Weyl chambers and Coxeter complexes) are employed in order to prove several modifications of the above mentioned result.

MSC:

53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
26B35 Special properties of functions of several variables, Hölder conditions, etc.
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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