Canonical identification between scales on Ricci-flat manifolds

Geometry & Topology
Event time: 
Tuesday, September 14, 2021 - 4:15pm
LOM 214
Jiewon Park
Speaker affiliation: 
Yale University
Event description: 

Uniqueness of tangent cone has been a central theme in many topics in geometric analysis. For complete Ricci-flat manifolds with Euclidean volume growth, 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. If a tangent cone at infinity of the manifold has smooth cross section, Colding-Minicozzi proved that the tangent cone is unique, by showing a Ɓojasiewicz-Simon inequality for this functional. As an application of this inequality, we will describe how one can identify two arbitrarily far apart scales in the manifold in a natural way. We will also discuss a matrix Harnack inequality when there is an additional condition on sectional curvature, which is an elliptic analogue of matrix Harnack inequalities obtained by Hamilton and Li-Cao for geometric flows.