  Using $\Gamma$-convergence, we study the Cahn-Hilliard problem with interface
width parameter $\varepsilon > 0$ for phase transitions on manifolds with
conical singularities. We prove that minimizers of the corresponding energy
functional exist and converge, as $\varepsilon \to 0$, to a function that takes
only two values with an interface along a hypersurface that has minimal area
among those satisfying a volume constraint. In a numerical example, we use
continuation and bifurcation methods to study families of critical points at
small $\varepsilon > 0$ on 2D elliptical cones, parameterized by height and
ellipticity of the base. Some of these critical points are minimizers with
interfaces crossing the cone tip. On the other hand, we prove that interfaces
which are minimizers of the perimeter functional, corresponding to $\varepsilon
= 0$, never pass through the cone tip for general cones with angle less than
$2\pi$. Thus tip minimizers for finite $\varepsilon > 0$ must become saddles as
$\varepsilon \to 0$, and we numerically identify the associated bifurcation,
finding a delicate interplay of $\varepsilon > 0$ and the cone parameters in
our example.
