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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

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you're asking for an explanation of Perelman's intuition to relate the Ricci flow to thermodynamics; many asked that question, an insightful analysis has been given here: arxiv.org/abs/1303.5193 –  Carlo Beenakker May 23 '13 at 17:59

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