Monotonicity formulas for harmonic functions on manifolds

On a complete noncompact Riemannian manifold, the existence of certain nontrivial harmonic functions can be used to discover important geometric properties of the manifold. A possible approach to monotonicity is to integrate the Bochner formula on the sublevel sets of the harmonic function. This idea can be used to establish geometric inequalities (e.g., Willmore type for hypersurfaces) or volume estimates (e.g., Bishop-Gromov type), among others. We will survey the method and present some recent applications.