A new Li-Yau-Hamilton estimate for the Ricci flow

Abstract

Since its discovery, Hamilton’s Li–Yau–Hamilton (LYH) estimate has proven its importance in many different works (for example [7], [9], [10], [6]), and as a result, similar inequalities have subsequently appeared in the study of numerous other geometric flows – the mean curvature flow [11], the Kahler– Ricci flow [1], and the Gauss curvature flow [2], among others. The importance of LYH-type estimates is underscored by the fact that the discovery of an LYH estimate without curvature assumption is a large step in Hamilton’s program for Geometrization. Traditionally, positivity of curvature in some form has always been needed for the existence of an LYH estimate. However, work of Hamilton [10] and Ivey [13] in three dimensions indicate that because the curvature operator becomes (in a sense) close to positive near singularities, and because there is an LYH estimate for positive curvature operator, there should be an LYH estimate without any curvature assumptions. Because of this, one approach towards finding an inequality on spaces of arbitrary curvature is to perturb the LYH estimates that are found when there is positive curvature. In fact, using this point of view, an LYH estimate was discovered on surfaces where some negative curvature was allowed [12]. This provides great motivation to find and understand the LYH estimates that do exist when positivity is assumed, and to discover the deeper reasons why such inequalities exist. One very interesting approach towards understanding LYH estimates attempts to view these inequalities in a geometric setting. This was first accomplished in the work of Chow and Chu [3], where a degenerate metric and a space-time connection is place on the flow. In this setting, Hamilton’s original LYH quantity appears very naturally and geometrically, being closely related to the curvature of this conenction. In subsequent papers, Chow and Chu [4], as well as Chow and Knopf [5] have since gained more understanding of this point of view. They have been able to refine it, clarify-