Realizable geometrically finite hyperbolic 3-manifolds with prescribed conformal boundaries

Question

By Bers' simultaneous uniformization theorem, if $\Gamma$ is a Fuchsian group, then $\operatorname{QC}(\Gamma)\cong \mathcal{T}(S)\times\mathcal{T}(\overline{S})$ where $S = \Bbb H^2/\Gamma$. In particular, one can explicitly construct a hyperbolic 3-manifold $M$ whose conformal boundary has two components, each is homeomorphic to $S$ with prescribed conformal structures. The general form of the Ahlfors–Bers theory implies that if $M$ has only geometrically finite ends, then its hyperbolic structure is uniquely determined by the corresponding points in the Teichmüller spaces of each conformal boundary of $M$.
I think my question is answered but I can't find:

Q1. Let $\Sigma_1,\dotsc,\Sigma_n$ be closed surfaces of genus $g_i\geq 2$ with prescribed conformal structures. Then there exists a hyperbolic 3-manifold $M$ whose conformal boundaries are precisely $\Sigma_1,\dotsc,\Sigma_n$.

If $n=2$ with $g_1 = g_2$ then it's Bers' simultaneous uniformization theorem. The origin of this question is by questioning:

Q2. Is there a hyperbolic 3-manifold whose conformal boundary has two components which are precisely genus $2$ and genus $3$ surfaces?

I think the answer is NO but I can't come up with an explanation.
A partial answer I found: Let $M = \Bbb H^3/\Gamma$ be such manifold. Suppose its regular set $\Omega(\Gamma)$ has two $\Gamma$-invariant components $\Omega_1$ and $\Omega_2$ then each is simply connected. Each of the three types of hyperbolic isometries on $\Omega_1$ corresponds to the same type of isometries on $\Omega_2$ so that we can induce an identity isomorphism $\pi_1(R_1)\to\pi_1(R_2)$ where $R_i = \Omega_i/\Gamma$ and we're done. $\square$

(I don't know if it's possible or not but) I don't know how to exclude the case when $\Omega(\Gamma)$ has infinitely many components so that $\Gamma$ somehow identifies those $\Omega_i$s to produce hyperbolic 3-manifolds with two conformal boundaries I stated above.
Topologically, existence is easy to see but the main point is the existence of hyperbolic structure.

Edit: Consider the space $M = \Bbb S^3-\text{thicken}(X)$, i.e., we thicken $X$ a bit and see its complement of $\Bbb S^3$. This gives topologically a compact manifold with three $\Sigma_2$ boundaries. I think this space satisfies four conditions in Sam's answer. The general case can be constructed by attaching 1-handles to one of the boundary components to increase the genus of the boundary surface and attaching copies of $M$ along a single boundary surface to increase the number of boundary components.

Answer 1

The answer to question one and question two are both "yes". Here is a very special case:

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Suppose that $M$ is a connected oriented compact three-manifold with boundary. Suppose further that $M$ is:

• irreducible (all two-spheres embedded in $M$ bound balls embedded in $M$),
• boundary irreducible (the boundary of $M$ is incompressible in $M$),
• strongly atoroidal (the fundamental group does not contain a copy of $\mathbb{Z}^2$), and
• acylindrical (if two loops in $\partial M$ are freely homotopic in $M$ then they are freely homotopic in $\partial M$).

It follows that the boundary components all have genus at least two. Let $(\Sigma_i)$ be the list of boundary components.

Then the space of hyperbolic structures on $M$ (with conformal boundaries at infinity) is homeomorphic to the product of the Teichmuller spaces $\mathcal{T}(\Sigma_i)$.

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Note that this is both an existence and a uniqueness result. Namely hyperbolic structures are plentiful, and are parameterised by "data at infinity".

In this very special case (since $M$ has many side hypotheses) to obtain just one hyperbolic metric, we "simply" double $M$ across its boundary, and apply either Thurston's hyperbolisation theorem for Haken manifolds (see the first pages of the book Hyperbolic manifolds and discrete groups by Kapovich) or apply Perelman's work.

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Here is a somewhat more concrete answer to question one. Suppose that $M$ is a compact, connected, oriented manifold with the desired boundary components. (You can obtain $M$ by taking a connect sum of handlebdies.) Note that we assume that no boundary component of $M$ is a two-sphere or a two-torus. The paper Simple knots in compact, orientable 3-manifolds, by Myers, implies that there is a knot $K$ in $M$ so that (the interior of) $M - K$ is simple - that is, satisfies the four hypotheses above. In particular, $M$ admits a hyperbolic structure. (This again uses Thurston's hyperbolisation theorem.) We now apply Thurston's Dehn filling theorem; we choose a sufficiently complicated slope $r$ on the two-torus cusp and Dehn fill $M - K$ to get the manifold $M_K(r)$. This again satisfies the four hypotheses above and has the same boundary components as $M$. So, by the theorem, we can realise any desired conformal structures on the boundaries of $M_K(r)$.