LOCAL BI-LIPSCHITZ CLASSIFICATION OF SEMIALGEBRAIC SURFACES

We provide bi-Lipschitz invariants for finitely determined map germs f: (Kn, 0) → (Kp, 0), where K = R or C. The aim of the paper is to provide partial answers to the following questions: Does the bi-Lipschitz type of a map germ f: (Rn, 0) → (Rp, 0) determine the bi-Lipschitz type of the link of f and the double point set of f? Reciprocally, given a map germ f: (Rn, 0) → (Rp, 0), do the bi-Lipschitz types of the link of f and the double point set of f determine the bi-Lipschitz type of the germ f: (Rn, 0) → (Rp, 0)? We provide a positive answer to the first question in the case of a finitely determined map germ f: (Rn, 0) → (Rp, 0) where p ≥ 2n − 1 (Theorem 3.3). With regard to the second question, for a finitely determined map germ f: (R2, 0) → (R3, 0), we show (Proposition 4.14) that inner bi-Lipschitz equivalence of germs is guaranteed by equivalence of the links, the images of the double point sets, and the polar curves of a generic projection onto the plane. Reciprocally, bi-Lipschitz equivalence of germs implies equivalence of the links and images of the double point sets. In particular, in the homogeneous parametrization case f: (R2, 0) → (R3, 0) of corank 1, we do not need the hypothesis on the equivalence of the image of the double point set (Theorem 5.2). Finally, we apply our results to relate the C0 − A classes of finitely determined mapgerms f of corank 1 with homogeneous parametrization and the inner bi-Lipschitz type of Xf (Proposition 5.4).