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.