  Given a complex analytic germ $(X, 0)$ in $(\mathbb C^n, 0)$, the standard
Hermitian metric of $\mathbb C^n$ induces a natural arc-length metric on $(X,
0)$, called the inner metric. We study the inner metric structure of the germ
of an isolated complex surface singularity $(X,0)$ by means of an infinite
family of numerical analytic invariants, called inner rates. Our main result is
a formula for the Laplacian of the inner rate function on a space of
valuations, the non-archimedean link of $(X,0)$. We deduce in particular that
the global data consisting of the topology of $(X,0)$, together with the
configuration of a generic hyperplane section and of the polar curve of a
generic plane projection of $(X,0)$, completely determine all the inner rates
on $(X,0)$, and hence the local metric structure of the germ. Several other
applications of our formula are discussed in the paper.
