Monday

On a complete non-compact Riemannian manifold, the existence of certain nontrivial harmonic functions provides a useful tool for obtaining key geometric properties through monotonic quantities. One approach to establishing monotonicity is by integrating the Bochner formula, which involves Ricci curvature, over the sub-level sets of a harmonic function. This method has been employed to derive various geometric inequalities, such as Minkowskitype inequalities for hypersurfaces, as well as volume estimates of Bishop- Gromov type. In this talk, we will survey this technique and highlight some recent applications.