Properties of the solutions of the conjugate heat equation

Abstract

In this paper we consider the class $\mathcal{A}$ of those solutions $u(x,t)$ to the conjugate heat equation $\frac{d}{dt}u = -\Delta u + Ru$ on compact K\"ahler manifolds $M$ with $c_1 > 0$ (where $g(t)$ changes by the unnormalized K\"ahler Ricci flow, blowing up at $T < \infty$), which satisfy Perelman's differential Harnack inequality on $[0,T)$. We show $\mathcal{A}$ is nonempty. If $|\ric(g(t))| \le \frac{C}{T-t}$, which is alaways true if we have type I singularity, we prove the solution $u(x,t)$ satisfies the elliptic type Harnack inequlity, with the constants that are uniform in time. If the flow $g(t)$ has a type I singularity at $T$, then $\mathcal{A}$ has excatly one element.