Inner geometry of complex surfaces: a valuative approach

Given a complex analytic germ (X, 0) in (C-n, 0), the standard Hermitian metric of 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 nonarchimedean 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.