J. Edson Sampaio

Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones

We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily subanalytic) between two subanalytic sets, then their tangent cones are bi-Lipschitz homeomorphic. As a consequence of this result, we show that any Lipschitz regular complex analytic set, i.e., any complex analytic set which is locally bi-Lipschitz homeomorphic to an Euclidean ball must be smooth. Finally, we give an alternative proof of S. Koike and L. Paunescu’s result about the bi-Lipschitz invariance of directional dimensions of subanalytic sets.

Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms

We give partial answers to a metric version of Zariskis multiplicity conjecture. In particular, we prove the multiplicity of complex analytic surface (not necessarily isolated) singularities in

Multiplicity and degree as bi-Lipschitz invariants for complex sets: MULTIPLICITY AND DEGREE AS BI-LIPSCHITZ INVARIANTS

\mathbb{C}^3

A proof of the differentiable invariance of the multiplicity using spherical blowing-up

is a bi-Lipschitz invariant.

Tangent cones of normally embedded sets are normally embedded

The first named author is partially supported by IAS and by ERCEA 615655 NMST Consolidator Grant, MINECO by the project reference MTM2013-45710-C2-2-P, by the Basque Government through the BERC 2014-2017 program, by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323, by Bolsa Pesquisador Visitante Especial (PVE) - Ciencias sem Fronteiras/CNPq Project number: 401947/2013-0 and by Spanish MICINN project MTM2013-45710-C2-2-P. The second named author was partially supported by CNPq-Brazil grant 302764/2014-7. The third named author was partially supported by the ERCEA 615655 NMST Consolidator Grant and also by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.

On Lipschitz rigidity of complex analytic sets.

In this paper we use some properties of spherical blowing-up to give an alternative and more geometric proof of Gau–Lipman Theorem about the differentiable invariance of the multiplicity of complex analytic sets. Moreover, we also provide a generalization of the Ephraim–Trotman Theorem.