Convergence of complete Ricci-βat manifolds

Abstract

This thesis is focused on the convergence at inαnity of complete Ricci βat manifolds. In the αrst part of this thesis, we will give a natural way to identify between two scales, potentially arbitrarily far apart, in the case when a tangent cone at inαnity has smooth cross section. The identiαcation map is given as the gradient βow of a solution to an elliptic equation. We use an estimate of Colding-Minicozzi of a functional that measures the distance to the tangent cone. In the second part of this thesis, we prove a matrix Harnack inequality for the Laplace equation on manifolds with suitable curvature and volume growth assumptions, which is a pointwise estimate for the integrand of the aforementioned functional. This result provides an elliptic analogue of matrix Harnack inequalities for the heat equation or geometric βows.