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I am looking for references in the literature pertaining to (essentially riemannian) metric spaces that are compact of infinite volume, such in the following example. Consider a riemannian metric on the open disk with respect to which the disk has infinite area but finite diameter, such that geodesics meet the boundary at right angles. Attach the boundary circle, and take the metric on the closed disk which completes that of the open disk. This has come up in the context of scattering in layered media (see http://arxiv.org/abs/1412.6138), and I'd like to know if such examples occur naturally in other settings.

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    $\begingroup$ What regularity assumptions do you make? The Koch snowflake is an example if you allow very irregular boundaries. If you assume enough regularity, the boundary becomes a compact Riemannian manifold with a nice metric and has thus finite volume. $\endgroup$ Jan 11, 2015 at 18:36
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    $\begingroup$ Then you should make your question clearer about your goal. The thing you describe in your comment is not really a Riemannian manifold with boundary as the term is usually understood. Moreover, I understood that you wanted the boundary to have infinite volume, not the manifold itself (since compact Riemannian manifolds have finite volume). Can you add details about what is it exactly that you want? $\endgroup$ Jan 11, 2015 at 19:23
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    $\begingroup$ It sounds like you want the Riemann metric only on the interior of the manifold. As Joonas mentions, that is not what people would generally consider to be a compact Riemann manifold. Compact Riemann manifolds have finite volume, unless you're venturing into the less-than $C^1$-manifold territory. $\endgroup$ Jan 11, 2015 at 19:40
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    $\begingroup$ Since the question has attracted some votes to close, perhaps it might help if you gave an idea of where such metric spaces/manifolds have come up in the course of your research (or related thinking) $\endgroup$
    – Yemon Choi
    Jan 11, 2015 at 20:38
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    $\begingroup$ Even with the edits I still don't understand what you are looking for: in your example the metric wrt which you have a infinite volume is different from the induced metric wrt which you exhibit the "compactness". I don't think your edits have fully addressed Joonas' or Ryan's questions; namely what exactly is the object that you are looking for? // For example, taking a look at your paper, on page 8 the construction of the scattering disk does not have anything to do with geodesics per se, so maybe you are emphasizing the wrong things in your question statement. $\endgroup$ Jan 12, 2015 at 9:04

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Looking at your paper http://arxiv.org/pdf/1412.6138.pdf and the description on page 8, there are infinitely many examples of manifolds which have "finite diameter" yet "infinite area".

Let $M$ be an arbitrary smooth compact Riemannian manifold with boundary $\partial M$. For convenience assume that $\partial M$ is compact. Let $\rho$ denote the distance function $\rho(x) = \mathrm{dist}(x,\partial M)$ defined on the $M$. For sufficiently small $\epsilon$ we have that the set $$ N_\epsilon = \{ \rho(x) < \epsilon \}$ $$ is a tubular neighbourhood of $\partial M$ diffeomorphic to $\partial M \times [0,1)$, and that $\rho$ is smooth on $N_\epsilon\setminus \partial M$.

Based on the metric $g$ on $M$ we can construct a new meric $h$ as follows. Let $\phi$ be a function on $M$ such that

  • $\phi \equiv 1$ on $M \setminus N_{\epsilon}$
  • $\phi = \epsilon^{-1} \rho$ on $N_{\epsilon / 2}$
  • $\phi$ is smooth an positive on the interior of $M$

Now consider the Riemannian metric $h = \phi^{-1} g$ on the interior $\mathring{M}$. Fix any point $y\in \mathring{M}$ and $z\in \partial M$ and a curve $\gamma$ connecting the two. The fact that $\phi$ degenerates like $1/\rho$ near the boundary means that the arc length is integrable and hence $\mathring{M}$ has finite diameter.

As long as $\dim(M) \geq 2$ however the volume of $N_{\epsilon}\setminus \partial M$ with the metric $h$ is infinite.


In this construction I chose $\phi$ to vanish like $\rho$ near the boundary. But to guarantee finite diameter you only need $\phi$ vanishing slower than $\rho^{2}$. To get infinite volume you need $\phi$ to vanish faster than or equal to $\rho^{2/d}$ where $d = \dim M$. So there is considerable amounts of freedom in the choice.


Some notes. What you are looking for, in slightly more generality, seems to be the notion of Cauchy boundary of an incomplete Riemannian manifold. It is well known that the Cauchy completion of a metric space may not admit a compatible metric extension (e.g. our examples above). I am not an expert in the state of the art, but looking on mathscinet the analysis of the Laplacian on such manifolds seems to be studied mostly by Jun Masamune and collaborators.

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