Bounds on volume of a Riemannian manifold given its diameter

Question

Suppose that we are given an $n$-dimensional Riemmanian manifold $M$. Then it is naturally endowed with a metric function $d$ compatible with Riemannian structure (the distance of two points is the infimum of lengths of all curves connecting the two points). Suppose that the diameter of the manifold is finite, i.e. $$ L = \mathrm{sup}_{x,y \in M} d(x,y) < \infty. $$ The question is what can be said about the volume of the manifold. In particular, is it true that there exist constants $c,C >0$ such that $$ c L^d \leq vol(M) \leq C L^d. $$ If this helps I am happy with results with additional assumptions, e.g. that $M$ is compact.

Answer 1

For the surface of a cylinder of length $L$ and diameter $D$ the surface is $\pi D L$ and since $D$ can be arbitrarily small there is no lower bound of the form you state.

For a piece of hyperbolic surface the volume grows exponentially with the diameter. So there is not such an upper bound either. See e.g. Hyperbolic Geometry

Answer 2

Considering that it's been a while since this question has been posted, this answer probably won't be very useful.

The answer given above is of the form: "There is a smooth manifold M such that one can pick a Riemannian metric for which the diameter equals $1$ while the volume can be arbitrarily small." There is in fact a stronger statement one can make, that is, for any smooth and compact manifold M with dimension at least 2, there are metrics with constant diameter whose volume can be made arbitrarily small. The proof is as follows.

Take a surjective smooth map $f: M \rightarrow [0,1]$. The interval is equipped with its standard metric. One can pull it back by $f$ to obtain a rank at most 1 semi-definite inner product $g_0$. Take any metric $g$ on $M$. The metric $ g_0 + \epsilon g$ will have diameter at least $1$, since any curve connecting $f^{-1}(0)$ with $f^{-1} (1) $ will have length at least $1$. Moreover, picking $\epsilon$ accordingly, one can make the natural volume form of $ g_0 + \epsilon g$ arbitrarily small. Neat, right?