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 Oₗ, ₀. In this paper, we associate to each m-primary ideal I of Oₗ, ₀ a continuous function II defined on the set of positive (suitably normalized) semivaluations of Oₗ, ₀. We prove that the function I₌ 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 = I₌₈, where mI is the maximal ideal of Oₗ₈, 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.