Lipschitz geometry of complex surface germs via inner rates of primary ideals

Let $(X, 0)$ be a normal complex surface germ embedded in (Cⁿ, 0), and denote by m the maximal ideal of the local ring OX,0. In this paper, we associate to each m-primary ideal I of OX,0 a continuous function II defined on the set of positive (suitably normalized) semivaluations of OX,0. We prove that the function Iₘ is determined by the outer Lipschitz geometry of the surface $(X, 0)$. We further demonstrate that for each m-primary ideal I, there exists a complex surface germ (XI, 0) with an isolated singularity whose normalization is isomorphic to $(X, 0)$ and II = ImI, where mI is the maximal ideal of OXI,0. Subsequently, we construct an infinite family of complex surface germs with isolated singularities, whose normalizations are isomorphic to $(X,0)$ (in particular, they are homeomorphic to $(X,0)$) but have distinct outer Lipschitz types.