**Question 1.** Let $\epsilon > 0$ and $V > 0$. Is there always a complete connected Riemannian manifold $M$ with
$$
\operatorname{diam} M < \epsilon\quad\text{ (small diameter)} \quad \text{and} \quad \operatorname{vol}M > V\quad\text{(large volume)}?
$$
In other words, can we construct worlds with room for arbitrarily many planets, but where any two planets are arbitrarily close to each other?


Note that $M$ has to be compact by Hopf-Rinow. I'm almost certain that the answer is 'yes', but I'm having a hard time finding an explicit sequence $M\_1, M\_2, M\_3,\ldots$ such that $\operatorname{diam}(M\_n)$ goes to zero while $\operatorname{vol}(M\_n)$ goes to infinity.


**Question 2.** (Fixed dimension) Is there a sequence of complete connected Riemannian manifolds $M\_1, M\_2, M\_3, \ldots$ with
$$ \dim M\_n = m, \quad \lim\_{n \to \infty}\operatorname{diam} M\_n = 0, \quad \text{and} \quad \lim\_{n \to \infty}\operatorname{vol}M\_n = \infty
$$
for some fixed $m \in \mathbb{N}$.


For this one, I have less intuition, but I'm more inclined towards 'no'. Note that for $m = 0, 1$, this is clearly impossible. For $m = 2$ my intuition is very strongly in favour of 'no'.



Superconnected spaces
The answer to both questions is positive, even in dimension 2.
Take a round sphere of diameter $\epsilon$, and make many, say $N$ little holes in it.
Then take $N$ spheres of diameter $\epsilon$ and make one little hole in each.
Then glue these $N$ spheres to the first sphere along the boundaries of the holes. The diameter of the resulting surface is about $3\epsilon$ while the volume (area) is approximately $(N+1)\epsilon$. Now make $\epsilon$ as small as you wish, while $N>V/\epsilon$.


Here is another construction. Take a round sphere $S$ of diameter $\epsilon$, think of it as the Riemann sphere, and consider the
$N$-fold ramified covering $S\_1\to S,\;z\mapsto z^n$. Here $S\_2$ is
also a sphere, and equip it with the pullback of the metric from $S$.
Then the solume of $S\_1$ will be $N\epsilon$ while diameter is at most
$2\epsilon$. If a smooth metric is desirable, approximate it with a smooth one.


