Compact riemannian manifolds with boundary that have infinite volume? 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.  
 A: 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. 
