Is there a natural family of finite volume hyperbolic $3$-manifolds parametrized by $n$ distinct hyperbolic points?

Question

Let $x_1,\ldots,x_n$ denote $n$ distinct points in the open $3$-ball, thought of as the Poincaré model of hyperbolic $3$-space. Let $X$ denote the complement of these $n$ points $x_i$, $1 \leq i \leq n$, and denote by $\hat{X}$ the compact $3$-manifold with boundary obtained by adding to $X$ a copy of $S^2$ at each point $x_i$, as well as a copy of $S^2$ "at infinity" (the so-called sphere at infinity).

My question is this. Is there a natural Riemannian metric on $\hat{X}$?

Of course, the question is a little vague, because I did not specify what properties I want this metric to have etc. I am hoping for some (geodesically) complete hyperbolic metric I guess, or something "along those lines". Feel free to modify a little $\hat{X}$ in order to obtain natural metrics, or to modify a little my hoped-for type of metrics.

I am sure that LeBrun's hyperbolic ansatz will pop up most probably in such a question, but I don't know the literature sufficiently well unfortunately.

Edit: the smooth manifold $\hat{X}$ I had written down unfortunately cannot have a complete hyperbolic metric, as for instance Moishe Cohen pointed out, and my own thesis advisor did too. I think it actually cannot even have a (geodesically) complete metric (indeed, how to continue a geodesic starting at a point on one of those spheres, and with direction vector which is not tangent to the sphere?).

Does anyone know however of some (non-trivial) family (or families) of finite volume hyperbolic $3$-manifolds which are parametrized by $n$ distinct points in either $\mathbb{R}^3$ or hyperbolic $3$-space $H^3$? What I am thinking of is that maybe one of those explicit constructions of hyperbolic $3$-manifolds, such as by gluing hyperbolic polyhedra (due to Poincare according to Wikipedia), may give rise to one such family.

Answer 1

1. By Mostow rigidity theorem for every $n\ge 3$, if two complete hyperbolic $n$-manifolds of finite volume have isomorphic fundamental groups, then they are isometric.

2. The fundamental group of every complete hyperbolic manifold of finite volume is finitely presented. (This is because such manifold is homotopy equivalent to a compact manifold with boundary, the complement of its "cusps".)

3. There are only countably many finitely presented groups.

4. Therefore, there are only countably many isometry classes of complete hyperbolic manifold of finite volume and dimension $\ge 3$.