Various monotonicity formulae have profound applications in many different problems in geometric analysis.
Quite often these formulae can be derived from pointwise Hessian estimates, also known as Li-Yau-Hamilton estimates.
These estimates are often called differential Harnack estimates as well, since they imply Harnack estimates by integration along space
or spacetime paths. In this talk we will focus on this connection building upon Hessian estimates for the Green function under nonnegative curvature conditions, leading to a novel monotonicity formulae on Einstein manifolds.