We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov–Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop–Gromov volume comparison theorem and Perelman’s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these.
Moreover, we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the Green function that we prove. This is parallel to the fact that Perelman’s monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li–Yau.
In [CM4] one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross-sections of Einstein manifolds.
Finally, there are obvious parallelisms between our monotonicity and the positive mass theorem of Schoen–Yau and Witten.
"New monotonicity formulas for Ricci curvature and applications. I." Acta Math. 209 (2) 229 - 263, 2012. https://doi.org/10.1007/s11511-012-0086-2