  In this article, we study the narrow capture problem on a Riemannian
2-manifold. This involves the derivation of the mean first passage (sojourn)
time of a surface-bound ion modelled as a Brownian particle. We use a layer
potential argument in conjunction with microlocal analysis in order to derive
the leading order singularity as well as the O(1) term of the mean first
passage time and the associated spatial average.
