Jiewon Park (Caltech)

Zoom: https://washington.zoom.us/j/98186259649
We will study complete Ricciflat manifolds with Euclidean volume growth. In the case when a tangent cone at infinity of the manifold has smooth cross section, the Green function for the Laplace equation can be used to define a functional which measures how fast the manifold converges to the tangent cone. Using the Łojasiewicz inequality of ColdingMinicozzi for this functional, we describe how two arbitrarily far apart scales in the manifold can be identified in a natural way. I will also discuss a matrix Harnack inequality for the Green function when there is an additional condition on sectional curvature, which is an analogue of various matrix Harnack inequalities obtained by Hamilton and LiCao in timedependent settings.