Lipschitz geometry of the image of finite mappings

2025/07/29 by Ballesteros, Juan José Nuño, Prado, Vinícius de Oliveira, Sanchis, Guillermo Peñafort +1
#14B05 (Primary) 32S05 (Secondary) #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.21405

Abstract

This paper is devoted to the study of the LNE property in complex analytic hypersurface parametrized germs, that is, the sets that are images of finite analytic map germs from (ℂⁿ,0) to (ℂⁿ⁺¹,0). We prove that if the multiplicity of f is equal to his generic degree, then the image of f is LNE at 0 if and only if it is a smooth germ. We also show that every finite corank 1 map is sattisfies the previous hypothesis. Moreover, we show that for an injective map germ f from (ℂⁿ,0) to (ℂⁿ⁺¹,0), the image of f is LNE at 0 if and only if f is an embedding.

Citations

• Universality theorem for LNE Hölder triangles
• Classification of complex algebraic curves under blow-spherical equivalence
• Global bi-Lipschitz classification of semi-algebraic surfaces
• On Lipschitz Geometry at infinity of complex analytic sets
• On characterization of smoothness of complex analytic sets
• On Lipschitz rigidity of complex analytic sets
• Tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded. Appendix by Anne Pichon and Walter D. Neumann
• Lipschitz Normal Embeddings in the Space of Matrices
• Multiple point spaces of finite holomorphic maps
• Lipschitz regular complex algebraic sets are smooth

Related

• Lipschitz geometry does not determine embedded topological type
• The infinitesimal deformations of hypersurfaces that preserve the Gauss map
• Topological invariants and Milnor fibre for A-finite germs ℂ²→ℂ³
• Good real pictures of corank one map germs from the n-space to the (n+1)-space
• Images of maps from (ℂ²,0) to (ℂⁿ,0)