Abstract
In this thesis, we do the Harnack estimate for the evoloved curvature R under the modified Ricci flow. The first step is to obtain a geometric quantity, called the Harnack quantity. Then we will prove that this quantity is positive under the modified Ricci flow. 2.Basic Properties of the Modified Ricci Flow...........3. 3.The Harnack Quantity..................................6. 4.The Harnack Inequality: Proof of Theorems.............8. References..............................................14.