Extension Results for Lipschitz Mappings in Geodesic Spaces

Abstract

Geodesic metric spaces are a natural generalization of Riemannian manifolds and provide a suitable setting for the study of problems from various areas of mathematics with important applications. In this chapter we review selected properties of Lipschitz mappings in geodesic metric spaces focusing mainly on certain extension theorems which generalize corresponding ones from linear contexts. We point out that the two-volume book by Brudnyi and Brudnyi (Methods of geometric analysis in extension and trace problems, Volume 1. Monographs in mathematics, vol. 102. Birkhäuser/Springer, Basel, 2012; Methods of geometric analysis in extension and trace problems, Volume 2. Monographs in mathematics, vol. 103. Birkhäuser/Springer, Basel, 2012) vastly covers the theory of extension and trace problems ranging from classical results to recent ones and hence includes some of the aspects that we also discuss here.