Let $M=(0,1)^n$ and take $(M,g_n)$ where $g_n$ is the usual Euclidean metric. Let's assume that in dim. $n$ we want a smooth codimension one foliation of $M$ (can have exactly one singular leaf) whose disjoint union of $1$ dim. leaves $\bigcup_{\alpha \in A}f_{\alpha} \simeq S^{n-2} \times (0,\sqrt{n})=S$ and where each $1$-leaf limits to points $p,q$ s.t. $\mathrm{dist}_n(p,q)=\sqrt{n}.$ In other words the mutually diffeomorphic surfaces of the class $S$ accumulate to $p$ and $q.$
Say we want a sequence of projections $(x^1,x^2,...,x^n)\mapsto (x^1,x^2,...,x^{n-1})\mapsto ... \mapsto (x^1,x^2)$ which send $1$-leaves in higher dimensional space to one lower dimension at each step. In the case of $(x^1,x^2,x^3)\mapsto(x^1,x^2)$ we have $1$-leaves in $(0,1)^3$ mapped down to the plane.
Let's begin in dim. $n=2,$ with $p=(0,1)$ and $q=(1,0).$ Define a candidate foliation of $(0,1)^2$ as follows: $\varphi_s(x):=\exp\big(\frac{s}{\log x}\big).$ This foliation solves a linear parabolic diffusion equation:
$$ s \frac{\partial^2}{\partial s^2}\varphi_s(x) = -x \frac{\partial}{\partial x}\varphi_s(x) $$
Now, define a $1$-parameter family of Riemannian metrics which will be used to measure the distance between two leaves in this foliation as:
$$ g_s(r)=\int_{(0,1)} x^{r-1}\varphi_s(x)~dx = 2\sqrt{\frac{s}{r}}K_1(2\sqrt{sr}).$$
For Bessel function $K.$
For example the distance between leaf $s=1$ and leaf $s=2$ would be a subtraction of integrals:
$$d(1,2)= \bigg| g_1(r)-g_2(r) \bigg | $$
Now, we have a way of recording distances between leaves on the plane, precisely we've decided to measure "areas" between these leaves, and a differential equation that tells us a diffusion relationship between leaves. All we need to do now is to preserve these structures when lifting onto our mutually diffeomorphic class of surfaces $S.$
First, we note that $g_s(r)$ also respects a partial differential equation and we can use this geometric flow to figure out what it might look like on the curved class of surfaces, $S.$ This differential equation is third order and linear (it can be reduced to second order through the Fourier transform):
$$ s^2 \frac{\partial^3}{\partial s^3}g_s(r)=r^2 \frac{\partial}{\partial r}g_s(r) $$
I think we need a $1$-leaf to sweep out an area, $a$ on each $S$ which mutually projects onto the plane region which sweeps out an area of $a$. This corresponds to the projection map being area preserving. In other words we need the $g_s(r)$ metric to be isometric to the one on each $S$.
One more crucial thing to define are the "nested" zonoid regions, which are the area bounded by any two leaves $\varphi_{s_1}$ and $\varphi_{s_2}$ s.t. $s_1s_2=1$ and this includes the boundary leaves. This is very helpful because as $\varphi_{s_1}$ deforms into $\varphi_{s_2}$ and back into $\varphi_{s_1}$ the corresponding surface $S$, its $1$-leaf, has now swept out an area that is equal to the total surface area of $S,$ or in other words the $1$-leaf is a generator of $S.$ It also gives a diameter for each surface $S$.
How do you generalize the metric $g_s(r)$ to the class of surfaces $S$ for $n=3$?
The primary goal here is to try to map all these differential forms onto the surfaces $S.$ So hopefully one can obtain a diffusion equation on the surfaces much like is apparent on the base space. Likewise, hopefully one can obtain a higher dim. version of the varying $g_s(r)$ metric for the surfaces $S.$
I'm kind of stuck on how to achieve this. I think I'm close but need a few hints.