Bi-Lipschitz equivalence of Alexandrov surfaces and global Chebyshev coordinates.
Abstract. We study the following problem: under which conditions Riemannian 2-manifolds (or, more generally, Alexandrov surfaces) are bi-Lipschitz equivalent with a controlled Lipschitz constant? Such conditions include bounds on diameter, systolic constant, total curvature and its distribution over the manifolds. We also prove existence of global Chebyshev coordinate on a (complete simply connected) Alexandrov surface under optimal curvature bounds. Such coordinates are useful for constructing bi-Lipschitz maps between surfaces.