  For free boundary problems on Euclidean spaces, the monotonicity formulas of
Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the
regularity theory as well as the existence theory. In this article we establish
the analogs of these results for the Laplace-Beltrami operator on Riemannian
manifolds. As an application we show that our monotonicity theorems can be
employed to prove the Lipschitz continuity for the solutions of a general class
of two-phase free boundary problems on Riemannian manifolds.
