Local bi-Lipschitz classification of semialgebraic surfaces

We provide bi-Lipschitz invariants for finitely determined map germs $f: (\mathbb{K}^n,0) \to (\mathbb{K}^p, 0)$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. The aim of the paper is to provide partial answers to the following questions: Does the bi-Lipschitz type of a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ determine the bi-Lipschitz type of the link of $f$ and of the double point set of $f$? Reciprocally, given a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$, do the bi-Lipschitz types of the link of $f$ and of the double point set of $f$ determine the bi-Lipschitz type of the germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$? We provide a positive answer to the first question in the case of a finitely determined map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ where $2n-1 \leq p$ (Theorem 3.3). With regard to the second question, for a finitely determined map germ $f : (\mathbb{R}^2,0) \to (\mathbb{R}^3,0),$ we show that a complete set of invariants for the bi-Lipschitz classification with respect to the inner metric of $X_f=f(U)$, where $U$ is a small neighbourhood of the origin in $\mathbb R^2$, is is given by the link of $f$, the image of the double point set of $f$ and the polar curve of a generic projection into the plane (Proposition 4.13). In particular, in the homogeneous parametrization case $f: (\mathbb{R}^2, 0) \to (\mathbb{R}^3, 0)$ 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 $C^{0}- \mathcal A$ classes of finitely determined map germs $f$ of corank 1 with homogeneous parametrization and the inner bi-Lipschitz type of $X_f$ (Proposition 5.4).