PROPERTIES OF THE SOLUTIONS OF THE CONJUGATE HEAT EQUATIONS

In this paper we consider the class A of those solutions u(x,t) to the conjugate heat equation partial derivative/partial derivative tu = -Delta u + Ru on compact Kahler manifolds M with c(1) > 0 (where g(t) changes by the unnormalized Kahler Ricci flow, blowing up at T < infinity), which satisfy Perelman's differential Harnack inequality (6) on [0, T]. We show A is nonempty. If vertical bar Ric (g(t))vertical bar <= C/T-1, which is always true if we have a type 1 singularity, we prove the solution u(x, t) satisfies the elliptic type Harnack inequality, with the constants that are uniform in time. If the flow g(t) has a type I singularity at T. then A has exactly one element.