Extensions of Lipschitz maps into Hadamard spaces

Abstract.

We prove that every $\lambda$-Lipschitz map $f : S \to Y$ defined on a subset of an arbitrary metric space X possesses a $c \lambda$-Lipschitz extension $\bar{f} : X \to Y$ for some $c = c(Y) \ge 1$ provided Y is a Hadamard manifold which satisfies one of the following conditions: (i) Y has pinched negative sectional curvature, (ii) Y is homogeneous, (iii) Y is two-dimensional. In case (i) the constant c depends only on the dimension of Y and the pinching constant, in case (iii) one may take $c = 4\sqrt{2}$. We obtain similar results for large classes of Hadamard spaces Y in the sense of Alexandrov.