In section 5 in "**The entropy formula for the Ricci flow and its geometric applications**" *Grisha Perelman* has written:

Fix a closed manifold $M$ with a probability measure $m$, and suppose
that our system is described by a metric $g_{ij}(\tau)$, which depends on the **temperature**
$\tau$ according to equation $\frac{\partial}{\partial \tau}g_{ij}=2(R_{ij}+\nabla_i \nabla_j f)$ where $dm = udV$, $u =(4\pi \tau)^{-\frac{n}{2}}e^{-f} $, and the partition function is given by $log Z = \int (−f + \frac{n}{2})dm$.

**Question 1:** We know $\frac{\partial}{\partial t}g_{ij}=-2(R_{ij}+\nabla_i \nabla_j f)$ is modified Ricci flow, where $t$ is time. Why the changes of the metric respect to $\tau$ is exactly backward of the changes of the metric respect to $t$?

**Question 2:** $\frac{\partial}{\partial \tau}g_{ij}=2(R_{ij}+\nabla_i \nabla_j f)$ is backward to heat equation PDE, Is there a solution to this?

**Question 3:** Why is the partition function $log Z =\int (−f + \frac{n}{2})dm$?

Thanks