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

Let ( X , 0 ) be a normal complex surface germ embedded in ( ℂ n , 0 ) , and denote by 𝔪 the maximal ideal of the local ring 𝒪 X , 0 . In this paper, we associate to each 𝔪 -primary ideal I of 𝒪 X , 0 a continuous function ℐ I defined on the set of positive (suitably normalized) semivaluations of 𝒪 X , 0 . We prove that the function ℐ 𝔪 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 ( X I , 0 ) with an isolated singularity whose normalization is isomorphic to ( X , 0 ) and ℐ I = ℐ 𝔪 I , where 𝔪 I is the maximal ideal of 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.