  We consider Brownian motion in a bounded domain $\Omega$ on a two-dimensional
Riemannian manifold $(\Sigma,g)$. We assume that the boundary $\p\Omega$ is
smooth and reflects the trajectories, except for a small absorbing arc
$\p\Omega_a\subset\p\Omega$. As $\p\Omega_a$ is shrunk to zero the expected
time to absorption in $\p\Omega_a$ becomes infinite. The narrow escape problem
consists in constructing an asymptotic expansion of the expected lifetime,
denoted $E\tau$, as $\epsilon=|\partial \Omega_a|_g/|\partial \Omega|_g\to0$.
We derive a leading order asymptotic approximation $E\tau =
\ds{\frac{|\Omega|_g}{D\pi}}[\log\ds{\frac{1}{\epsilon}}+O(1)]$. The order 1
term can be evaluated for simply connected domains on a sphere by projecting
stereographically on the complex plane and mapping conformally on a circular
disk. It can also be evaluated for domains that can be mapped conformally onto
an annulus. This term is needed in real life applications, such as trafficking
of receptors on neuronal spines, because $\log\ds{\frac{1}{\epsilon}}$ is not
necessarily large, even when $\epsilon$ is small. If the absorbing window is
located at a corner of angle $\alpha$, then $E\tau =
\ds{\frac{|\Omega|_g}{D\alpha}}[\log\ds{\frac{1}{\epsilon}}+O(1)],$ if near a
cusp, then $E\tau$ grows algebraically, rather than logarithmically. Thus, in
the domain bounded between two tangent circles, the expected lifetime is $E\tau
= \ds{\frac{|\Omega|}{(d^{-1}-1)D}}(\frac{1}{\epsilon} + O(1))$.
