What question did this study set out to answer?

The aim is to classify the inner Lipschitz geometry of complex surface germs, particularly those with non-isolated singularities.

June 20, 2026Open Access

Inner Lipschitz Geometry of Complex Surface Germs with Non-Isolated Singularities: A Complete Classification

Key Points

The aim is to classify the inner Lipschitz geometry of complex surface germs, particularly those with non-isolated singularities.
Developed an invariant for inner Lipschitz geometry based on numerical invariants from normalization maps.
Used combinatorial techniques linked to good resolutions of complex surfaces.
Provided a complete invariant that extends previous results to non-isolated cases.
Established relationships between numerical invariants and the combinatorial structure of resolutions.

Abstract

Let (X, 0) be a germ of a reduced and irreducible complex surface embedded in (Cᵏ, 0). In this paper, we give a complete invariant of the inner Lipschitz geometry of complex surface germs, extending the result of Birbrair--Neumann--Pichon BNP to the non-isolated case. This invariant is expressed in terms of numerical invariants associated with the coordinate functions f₁, , fₖ of the normalization map n: (X, 0) (X, 0) (Cᵏ, 0), together with the combinatorics of a suitable good resolution of (X, 0).

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