Estimates on the Green function and applications to global geometry

Speaker: Associate Professor Jiewon Park
Affiliation: KAIST

Abstract

The Green function for the Laplace equation serves as a fundamental tool in geometric analysis. We will explore the interplay between estimates on the Green function and the curvature of Riemannian and Kähler manifolds. In particular, on manifolds with nonnegative Ricci curvature, Colding discovered a useful monotonicity formula involving the Green function. We establish a quantitative relationship between the pinching of a monotone functional defined by Colding and the distance to the nearest cone. Moreover, we demonstrate that the pinching quantitatively controls almost splitting. This talk is partly based on joint work with Christine Breiner.