Extending the base metric to the quotient metric in $\zeta$-space

Define an analytic planar homotopy $h_t: (0,1) \to (0,1)$ for $t\in[1/3,3]$ where $h_t(x)=e^{\frac{t}{\log x}}$ s.t. $h_{1/3}=e^{\frac{1/3}{\log x}}$ and $h_3=e^{\frac{3}{\log x}}.$ I want a rotational homotopy that projects onto the planar homotopy. So, define the equivalence relation on the boundary curves of the closed subset of $(0,1)^2$, that is $h_{1/3}\sim h_3$. This yields a spherical topology $\mathcal S$ embedded in $(0,1)^3$ with the endpoints of the sphere being singularities at $p=(0,1,1)$ and $q=(1,0,0)$. 


Conceptually it's clear that a projection $\pi: (0,1)^3 \to (0,1)^2$ can take the leaves on $\mathcal S$ to $h_t(x)$ as a double cover. I would also like to put a metric on $\mathcal S$. The metric I derived on the planar space is $g_t(r)=\int_{(0,1)} x^{r-1}h_t(x)~dx = 2\sqrt{\frac{t}{r}}K_1(2\sqrt{tr})$ and it measures the distance between the $h_t(x)$ leaves. For example the distance between leaf $t=1/3$ and leaf $t=3$ would be a subtraction of integrals $d(1/3,3)= \bigg| g_{1/3}(r)-g_{3}(r) \bigg |.$ This distance changes w.r.t. to $r$ as well because it's defined to be a $1$-parameter family of metrics. 


Some questions: How do we prove that after gluing with the equivalence relation, that $g_t(r)$ has an extension to $\mathcal S$? 


Comments: It's desirable to control the geometry under the action of the quotient. We'd like to be able to put the quotient metric on $\mathcal S$. 


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