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

Let $(X,0)$ be a normal complex surface germ embedded in $({\mathbb{C}}^n,0)$, and denote by $\mathfrak{m}$ the maximal ideal of the local ring ${\mathcal{O}}_{X,0}$. In this paper, we associate to each $\mathfrak{m}$-primary ideal $I$ of ${\mathcal{O}}_{X,0}$ a continuous function ${\mathcal{I}}_I$ defined on the set of positive (suitably normalized) semivaluations of ${\mathcal{O}}_{X,0}$. We prove that the function ${\mathcal{I}}_{\mathfrak{m}}$ is determined by the outer Lipschitz geometry of the surface $(X,0)$. We further demonstrate that for each $\mathfrak{m}$-primary ideal $I$, there exists a complex surface germ $(X_I,0)$ with an isolated singularity whose normalization is isomorphic to $(X,0)$ and ${\mathcal{I}}_I={\mathcal{I}}_{{\mathfrak{m}}_I}$, where ${\mathfrak{m}}_I$ is the maximal ideal of ${\mathcal{O}}_{X_I,0}$. 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.