Mathematics

A classical result on Riemannian manifolds satisfying a lower bound on the Ricci curvature is the monotonicity of theBishop-Gromov volume ratio. Colding and Minicozzi ('12-'14) realized that for non-negative Ricci curvature there existanalogous monotone quantities involving the Green function. Recently this has been generalized byAgostiniani, Fogagnolo and Mazzieri ('18) from the Green function to the case of an electrostatic potential and has provento be fruitful in proving geometric inequalities. We will see that the same monotonicity formulas can be proven also in thesetting of synthetic lower Ricci curvature bounds. This allows to prove some almost-rigidity results which are new also in thesmooth case. This is a joint work with professor Nicola Gigli.