A pair of boundary circles has one dimensionless invariant under the <conformal field theory> group. If their radii are $R_1,R_2$ and their center separation is
$$
d=R_1+R_2+L,
$$
a convenient invariant is the <inversive distance>
$$
\chi=\frac{d^2-R_1^2-R_2^2}{2R_1R_2}.
$$
The choice between connected and disconnected bulk surfaces can depend only on $\chi$. Let the transition occur at the single theory-independent numerical value $\chi=X$, with $X>1$. Then
$$
(R_1+R_2+C)^2=R_1^2+R_2^2+2XR_1R_2,
$$
so
$$
\boxed{
C(R_1,R_2)
=\sqrt{R_1^2+R_2^2+2XR_1R_2}-R_1-R_2}.
$$
All dependence on the two radii is fixed by conformal symmetry; only the pure number $X$ requires the explicit minimal-surface calculation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 2 / c / Solution

{wiki}

For two disjoint Euclidean spheres of radii $R_1,R_2$ whose centers are distance $d$ apart, the inversive distance
$$
\chi=\frac{d^2-R_1^2-R_2^2}{2R_1R_2}
$$
is invariant under conformal transformations. It satisfies $\chi>1$ for externally disjoint spheres.


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