Harmonic functions on Alexandrov spaces and their applications

The main result can be stated roughly as follows: Let M M be an Alexandrov space, Ω ⊂ M M an open domain and f: Ω → R f: R a harmonic function. Then f f is Lipschitz on any compact subset of Ω. Using this result I extend proofs of some classical theorems in Riemannian geometry to Alexandrov spaces.