Constrained and linear Harnack inequalities for parabolic equations

摘要

Differential Harnack inequalities for parabolic equations originated with the work of Li and Yau [LY] on the heat equation on a Riemannian manifold. Perhaps most importantly for problems in differential geometry is that their technique relies only on the maximum principle. Differential Harnack inequalities have now been proven for many geometric evolution equations using the maximum principle. In particular, the second author has proved Harnack inequalities for the Ricci flow [H1,H2], the mean curvature flow [H3], and a matrix harnack inequality for the heat equation [H4]. Similar Harnack inequalities have been proven by the first author for the Gauss curvature flow [C1] and the Yamabe flow [C2]. Harnack inequalities have also been proven by Cao [Ca] for the Ricci-Kahler flow and by Andrews [A] for general curvature flows of hypersurfaces.