Geometric & Functional Analysis GAFA

, Volume 10, Issue 6, pp 1527-1553

First online:

Extensions of Lipschitz maps into Hadamard spaces

  • U. LangAffiliated withDepartement Mathematik, ETH Zentrum, Rämistrasse 101, CH-8092 Zürich, Switzerland, e-mail:
  • , B. PavlovićAffiliated withInstitute of Mathematics, Belgrade, Yugoslavia, and School of Mathematics, Trinity College, Dublin 2, Ireland, e-mail:
  • , V. SchroederAffiliated withInstitut für Mathematik, Universität Zürich-Irchel, Winterthurer Strasse 190, CH-8057 Zürich, Switzerland, e-mail:

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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.