Harnack inequalities, Aleksandrov reflection, and ancient solution of curvature flows on the sphere

Since the seminal work of Li and Yau, Harnack inequalities have played an important role in the study of geometric PDE. Many authors have studied such inequalities and used them to obtain convergence results for curvature flows such as the Ricci flow, Mean Curvature Flow and various non-linear hyper-surface flows. I will focus on the hyper-surface flow situation, which has been studied principally in Euclidean space. There Harnack inequalities closely relate to solitons (self-similar solutions) and ancient solutions. In the sphere, the notion of self-similar is not so clear, yet a Harnack inequality holds and may be used in classifying ancient solutions of curvature flows in there sphere. I will describe the Harnack inequality (and maybe make a remark or two about conformal-solitons) and then discuss a parabolic Aleksandrov reflection argument that may be used to deduce maximal symmetry for convex, ancient solutions of curvature flows in the sphere.