So here is an abrupt try find connections between them. I know this is incomplete and I hope someone else adds more/edits more into this:
The Ricci flow equation
$$
\frac{dg}{dt} = - 2 Ric(g(t))
$$
Both sides are the same type of object : at each point $p \in M$, a bilinear form on $T_pM$.
In terms of local coordinates this becomes
$$
\frac{\partial g_ij}{\partial t}= - 2 R_{ij}
$$
(Hamilton, 1982).
The heat equation in 3-D is
$$
\frac{\partial f}{\partial t} = \nabla^2 f
$$
The basic differences are
- the heat flow evolves an initial function $f_0 $ towards a constant
function
- Ricci flow evolves a Riemannian metric.
More on the Ricci flow by Bennett-chow
Here is a similar intuition behind the Ricci flow
heat-type equations. The full curvature tensor
$\operatorname{Rm}$ satisfies an equation of the form $\frac{\partial
}{\partial t}\operatorname{Rm}=\Delta\operatorname{Rm}+q(\operatorname{Rm})$,
where $q$ is a quadratic polynomial. Since $\operatorname{Rm}$ is a symmetric
bilinear form on the vector space $\wedge^{2}T_{x}^{\ast}M$ at each point $x$,
we have the notion of nonnegativity of $\operatorname{Rm}$. Since
$q(\operatorname{Rm})$ satisfies a property sufficient for the maximum
principle for systems to be applied, $\operatorname{Rm}\geq0$ is preserved
under the Ricci flow. Generally, we can analyze the behavior of
$\operatorname{Rm}$ by the maximum principle under various hypotheses.
Geometric application. In particular, when $n=3$ and $\operatorname{Ric}
_{g_{0}}>0$, we have $\pi_{1}(M)=0$ and hence the universal cover $\tilde{M}$
is a homotopy $3$-sphere. Encouraged by this, Hamilton proved that the
solution to the normalized Ricci flow exists for all time and converges to a
constant positive sectional curvature metric; thus $M$ is diffeomorphic to a
spherical space form. The main gonzo estimate is $\frac{|\operatorname{Ric}%
-\frac{R}{3}g|^{2}}{R^{2}}\leq CR^{-\delta}$ for some $C$ and $\delta>0$.
Intuitively, we expect $R\rightarrow\infty$ and hence $\operatorname{Ric}
-\frac{R}{3}g\rightarrow0$.