I am not a mathematician (so I apologize if this is difficult to parse) but I have some questions that I am struggling to give a satisfactory answer to. I'll begin with some underlying assumptions.
Assumptions:
Say you have have path connected (but not convex) region, $R$ as a subset of $X^2=(0,1)^2.$ Define a smooth planar homotopy that smoothly interpolates $R$ as follows: $$ \varphi_s(x):=\exp \frac{s}{\log x} $$
where any two curves $\varphi_{s_1}(x)$ and $\varphi_{s_2}$ satisfy the condition $s_1s_2=1$. In fact let, $s_1=1/2$ and $s_2=2.$
The planar homotopy has distinguished "endpoints" $p=(0,1)$ and $q=(1,0).$
You want to build a rotating homotopy. This rotating homotopy in $X^3$ must have distinguished "endpoints" $P=(0,1,1)$ and $Q=(1,0,0)$ and be in the class $C^{\infty}.$ The surface traced out by the rotating homotopy must be diffeomorphic to $S^1\times (0,1)$ and accumulate to $P,Q.$
The projection $P:X^3 \to X^2$ is given by $(x^1,x^2,x^3)\mapsto (x^1,x^2).$ For all time, the rotating homotopy must project onto the planar homotopy.
My attempt:
To solve, or rather understand this problem and its assumptions, I thought to define an infinitesimal distance between curves:
$$g(s)=\int_{(0,1)} \varphi_s(x)~dx = 2\sqrt{s}K_1(2\sqrt{s}).$$ For Bessel function $K.$
Then I did a quick calculation, $\mathrm{dist}(s_1,s_2):=|g_1-g_2|$ and verified the other axioms of a metric.
We have that $\varphi_s(x)$ obeys the diffusion like equation (not sure why):
$$s \frac{\partial^2}{\partial s^2}\varphi_s(x) = -x \frac{\partial}{\partial x}\varphi_s(x) \tag{1}$$
I also thought about the homotopy path lifting property but I don't have an algebraic representation of the surface to lift the planar homotopy onto in the first place. All I know is the homotopy type of it and that it's essentially diffeomorphic to $S^1 \times (0,1).$
I tried drawing this diagram to understand some relationships between the objects:
These "reductions" led me to believe that essentially this homotopy $\varphi_s(x)=\exp \frac{s}{\log x}$ simply behaves like a gradual linear transformation, a rotation of radial lines on the once punctured plane, which can be described with a one parameter 2d matrix, and can be made rigorous with Lie theory.
Am I correct to conclude that $\varphi_s(x)$ behaves the same as a gradual rotation? What mistakes have I made in this post?