It is a result of Chinburg-Friedman-Jones-Reid that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold.
There is also a result of Milley that says that if $N$ is a closed orientable hyperbolic 3-manifold with volume less than or equal to that of the Weeks manifold, then $N$ is homeomorphic to the Weeks manifold.


My question is, therefore, is it possible to have a non-arithmetic, non-closed, orientable hyperbolic 3-manifold with volume smaller than that of the Weeks manifold? I am also curious if increasing the number of cusps necessarily increases the volume, as this would be sufficient to answer my question.



Does the Weeks manifold have the smallest volume among all finite volume oriented hyperbolic 3-manifolds?
The answer given in the comment by Ryan Budney should be enough, but let me give you a couple of theorems if it is still not so clear.


Suppose you have a finite volume hyperbolic complete orientable manifold $M$. Suppose that $M$ has cusps (hence, it is not compact).


Theorem [Thick-thin decomposition]: $M$ is diffeomorphic to the interior of a compact manifold $N$ with $\partial N$ that consist of a disjoint union of tori $T\_1, \ldots, T\_c$.


Fix the notation $\mathring{N}=M$.
One operation that you can perform on $N$ is the Dehn Filling: choose a boundary component $T\_i$, choose a diffeomorphism $f\_i \colon \partial(D^2 \times S^1) \to T\_i$ and consider the new manifold
$$ N(f\_i) = N \cup\_{f\_i} D^2 \times S^1$$
obtained by gluing a solid torus to the chosen boundary component using the chosen diffeomorphism. Through this operation, you fill one boundary component with a solid torus.


The manifold that you obtain depends on the diffeomorphism $f\_i$ that you choose. Suppose that you perform Dehn Filling on every boundary component of $N$ with functions $f\_1,\ldots, f\_c$. Call the result of these Dehn Fillings $N(f\_1,\ldots, f\_c)$. It is a closed manifold. Now, there are two results that can help you. These can be found in the notes of Thurston, but I will give you another citation that may help you.


Theorem [1, Theorem 15.1.1, Hyperbolic Dehn filling, very basic version]: There are functions $f\_1,\ldots, f\_c$ such that $N(f\_1,\ldots, f\_c)$ admits a hyperbolic metric.


Theorem [1, Theorem 15.4.1]: Suppose that $N(f\_1,\ldots, f\_c)$ admits a hyperbolic metric. Then $vol(N(f\_1,\ldots, f\_c)) < vol(M)$.


Using all these results, it is easy to show that a cusped hyperbolic manifold with volume less than the smallest volume of a closed hyperbolic manifold cannot exist: in this case, you could Dehn Fill it and obtain a closed manifold with even smaller volume, contradicting the hypothesis.


[1]: Martelli Bruno, An Introduction to Geometric Topology.


