Constrained Matrix Li-Yau-Hamilton Estimates On Khler Manifolds

In 1986, Li and Yau first proved a gradient estimate for any positive solutions of heat equation on a Riemann manifold and derived the classical Harnack inequality by integrat-ing the gradient estimate along the paths in space-time. Later on, similar technique was employed by Hamilton in the study of Harnack inequalities for Ricci flow, mean curva-ture flow and matrix Harnack inequalities for heat equation. Besides, Chow and Hamilton extended Li-Yau’s gradient estimate of heat equation to the constrained case. Cao and Ni get a Li-Yau-Hamilton estimate for heat equation on a Kahler manifold with fixed Kahler metric. Chow and Ni get the estimate on Kahler manifolds with time dependent Kahler metrics evolving under the Kahler-Ricci flow.In this thesis, we firstly review the research background of Li-Yau-Hamilton esti-mates, we also review the importance and introduce the research status at home and abroad. Then, we introduce the basic knowledge of the Kahler geometry and the Kahler-Ricci flow. In the main part, we give the main results of this thesis. We mainly derive a family of con-strained matrix Li-Yau-Hamilton estimates on Kahler manifolds. As a result, we firstly get a constrained matrix Li-Yau-Hamilton estimate for heat equation on a Kahler manifold with fixed Kahler metric. Secondly, we get such estimate for heat equation on Kahler manifolds with time dependent Kahler metrics evolving under the Kahler-Ricci flow.