Jiewon Park, Hessian Estimates, Monotonicity Formulae, and Applications

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 or matrix Harnack inequalities. In this talk we will focus on this connection building upon Hessian estimates for the Green function, leading to novel monotonicity formulae on manifolds with certain curvature lower bounds and Einstein manifolds. Some of the results have counterparts for Kähler manifolds.