# CHARACTERIZATION OF LIPSCHITZ NORMALLY EMBEDDED COMPLEX CURVES

ANDRÉ COSTA, VINCENT GRANDJEAN, AND MARIA MICHALSKA

**ABSTRACT.** The main result of the paper states that a connected complex affine algebraic curve is Lipschitz normally embedded (shortened to LNE afterwards) in $\mathbb{C}^n$ if and only if its germ at any singular point is a finite union of non-singular complex curve germs which are pairwise transverse, and its projective closure is in general position with the hyperplane at infinity. To this aim, we completely characterize complex analytic curves of a compact complex manifold which are LNE (regardless of the given Riemannian structure), and therefore we can relate when a projective algebraic curve is LNE in $\mathbb{CP}^n$ with the property of its general affine traces being LNE in $\mathbb{C}^n$. This allows us to deduce that Lipschitz classification of LNE curves is topological. We describe a complete invariant for the (outer and inner) Lipschitz equivalence of affine LNE curves and show that most such curves cannot be bi-Lipschitz homeomorphic to a plane one.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Geometric Preliminaries</td><td>3</td></tr><tr><td>3. A criterion for isolated singularity germs</td><td>7</td></tr><tr><td>4. The local case</td><td>9</td></tr><tr><td>5. Preliminaries at infinity</td><td>12</td></tr><tr><td>6. Unbounded part of affine curves</td><td>15</td></tr><tr><td>7. Bounded part of affine curves</td><td>18</td></tr><tr><td>8. Main Result: how to recognize an affine LNE curve</td><td>18</td></tr><tr><td>9. Affinely LNE versus Projectively LNE</td><td>20</td></tr><tr><td>10. Lipschitz classification of LNE curves</td><td>21</td></tr><tr><td>References</td><td>24</td></tr></table>

## 1. INTRODUCTION

A Riemannian manifold $(M, g_M)$ carries a natural metric space structure $(M, d_M)$. Any subset $S$ inherits the *outer metric space structure* $(S, d_S)$ where the distance $d_S$ between points of $S$ is their distance in $(M, d_M)$. When $S$ is path-connected, the infimum of the lengths of all the rectifiable curves in $S$ connecting two given points of $S$ is called the *inner distance* between these points, yielding the *inner metric space structure* $(S, d_{\text{inn}}^S)$ over $S$. While investigating $C^\infty$ functions over closed subsets of the Euclidean space, Whitney observed in [23, 24] that the metric spaces $(S, d_S)$ and $(S, d_{\text{inn}}^S)$ need not be metrically equivalent, thus introduced the property: there exists a positive constant $L$ such that

$$ (P) \qquad \mathbf{s}, \mathbf{s}' \in S \implies d_{\text{inn}}^S(\mathbf{s}, \mathbf{s}') \le L \cdot d_S(\mathbf{s}, \mathbf{s}'), $$

which guarantees that the metric spaces $(S, d_S)$ and $(S, d_{\text{inn}}^S)$ are equivalent, since $d_S \le d_{\text{inn}}^S$.

<footnote n="2010">2010 Mathematics Subject Classification. 14H50, 32B10, 51F99.</footnote>

<footnote>Key words and phrases. Complex curve, inner and outer distance, Lipschitz normally embedded.</footnote>
The equivalence of inner and outer distances relates strongly with Lipschitz Stratifications, L-regular decompositions and Pancakes of sub-analytic sets of Euclidean spaces [15, 17, 12, 18]. Property (P) is called *quasi-convexity* in the fields of Lipschitz Analysis and Metric Geometry, while to Singularity Theorists, it is known as being *Lipschitz Normally Embedded* (shortened below to LNE) since [1].

The last decade has shown growing interest in investigating analytic subset germs ($X, 0$) admitting LNE representatives. Still, outside the case of complex curve germs and isolated complex surface singularity germs, the higher dimensional situation is a terra incognita yet to be explored: the recent short survey [6] and the paper [13] present a rather exhaustive list of what is known on this topic. Similarly, asking when an algebraic set is LNE nearby infinity has been addressed in a single paper, under very restrictive hypotheses [8], which recently gave rise to a generalization to necessary condition on tangent cones in [4]. Moreover, very little is known on how to recognize which affine algebraic subsets of $K^n$ are LNE, both when $K = \mathbb{C}$ and $K = \mathbb{R}$. The only non-trivial example we are aware of is from [11]: it has non-isolated singularities, but is a $K$-cone, thus essentially a projective result. Let us mention that the new article [21] describes fully the Lipschitz outer geometry of complex plane algebraic curves by a complete finite combinatorial invariant. Last, [9] gives a complete description of quasi-convex subsets of the real plane, definable in some o-minimal structure. Still, neither of these two works characterizes LNE curves.

The goal of this paper is to start the investigation of the LNE property for subsets of the Euclidean space or of a given compact Riemannian manifold. We treat the case of complex curves in both settings: we fully characterize which complex algebraic curves of $C^n$ are LNE and which complex analytic curves of a compact complex manifold are LNE (regardless of the Riemannian metric used).

The main result of this paper is the following:

**Theorem 8.1.** Let $X$ be a complex algebraic curve of $C^n$ of positive degree $d$. It is LNE if and only if the following conditions are satisfied: (0) $X$ is connected; (i) At each singular point $x_0$ of $X$, the germ $(X, x_0)$ consists of finitely many pairwise transverse non-singular complex algebraic curve germs; (ii) The intersection of the projective closure of $X$ in $\mathbb{CP}^n$ with the hyperplane at infinity consists exactly of $d$ points.

Prior to establishing the affine result, and as a consequence of only local considerations, we obtain the following result in complex compact manifolds:

**Proposition 4.6.** Let $X$ be a complex analytic curve of a complex compact manifold. It is LNE if and only if the following conditions are satisfied: (0) $X$ is connected; (i) At each singular point $x_0$ of $X$, the germ $(X, x_0)$ consists of finitely many pairwise transverse non-singular complex analytic curve germs.

Both above results require the same condition at singular points, but the fundamental difference between the two lies of course in what happens at infinity, situation whose treatment will occupy half of the paper. These results leads to the very close interplay between some metric properties of a complex projective algebraic curve with those of its general affine traces.

**Theorem 9.4.** Let $X$ be an algebraic curve of $\mathbb{CP}^n$. The following statements are equivalent:

(1) $X$ is LNE w.r.t. to the Fubini-Study metric.

(2) There exists a hyperplane $H$ of $\mathbb{CP}^n$ intersecting $X$ only at non-singular points of $X$ at which $X$ and $H$ are transverse and such that the affine trace $X \setminus H$ of $X$ in $C_H^n := \mathbb{CP}^n \setminus H$ is LNE in $C_H^n$.

(3) For any hyperplane $H$ of $\mathbb{CP}^n$ intersecting $X$ only at non-singular points of $X$ at which $X$ and $H$ are transverse, the affine trace $X \setminus H$ is LNE in $C_H^n$.
This latter result allows us to describe a complete invariant of (outer and inner) Lipschitz equivalence which simply encodes degree, number of singular points, genus and multiplicities at singular points. From this, we deduce the following result:

**Theorem 10.2.** Let $X_1$ be an irreducible LNE complex algebraic curve of $\mathbb{C}^{n_1}$ and $X_2$ be an irreducible LNE complex algebraic curve of $\mathbb{C}^{n_2}$. Then $X_1$ and $X_2$ are bi-Lipschitz homeomorphic if and only if they are homeomorphic.

Lipschitz equivalence has been extensively studied for germs of analytic curves. In dimension $n_1 = n_2 = 2$, two curves are outer Lipschitz equivalent if and only if there exists a homeomorphism germ $(\mathbb{C}^{n_1}, \mathbf{0}) \to (\mathbb{C}^{n_1}, \mathbf{0})$ mapping $(X_1, \mathbf{0})$ to $(X_2, \mathbf{0})$, see [19]. Therefore, a complete outer Lipschitz equivalence invariant of plane curves was introduced recently in [21]. As shown in [7], the topological and outer Lipschitz classification are no longer the same for space curve germs, i.e. $n_1 = n_2 \ge 3$. Furthermore, by the result [22], a complex space curve germ is outer Lipschitz equivalent to the germ of one of its generic linear plane projections. Corollary 10.5 establishes that this is no longer true for global curves.

The paper is organized as follows. Section 2 presents some necessary background such as the definitions of LNE and locally LNE, as well as Lemma 2.6, Corollary 2.7 and Proposition 2.10, which are among the essential bricks of the paper. Lemma 3.1 of Section 3 gives a criterion for some isolated singularity set germ to be locally LNE. Corollary 3.2 about a one parameter family of LNE representatives is of interest in its own. Both results are used in Section 4 to obtain Proposition 4.2, a criterion for a complex curve germ to be LNE. It allows to deduce the aforementioned Proposition 4.6, at the end of Section 4. The next three sections develop the needed material to show the main result Theorem 8.1 in Section 8: Sections 5 and 6 deal with the *unbounded part* of the affine curve being LNE, while Section 7 focuses on its *bounded part*. Section 9 explores the relationship between the LNE properties of an affine curve of $\mathbb{C}^n$ and its projective closure. In the last Section Theorem 10.2 states that the complete invariant of the Lipschitz classification problem for irreducible complex LNE curves is actually a complete topological one for irreducible LNE curves. Last, by Corollary 10.5, the Lipschitz geometry of LNE curves cannot be reduced to that of the LNE plane curves. Since we aim at a self-contained article, proofs of the already known Lemma 2.6 and Proposition 4.2 are given.

## 2. GEOMETRIC PRELIMINARIES

Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$. The space $\mathbb{K}^n$ is equipped with the Euclidean distance denoted by $|\cdot|$. Let *eucl* be the Euclidean/Hermitian metric tensor over $\mathbb{K}^n$.

Let $\mathbf{x}$ be a point of $\mathbb{K}^n$. Let $r$ be a positive number. The open ball of $\mathbb{K}^n$ centred at $\mathbf{x}$ with radius $r$ is $B_r^{\mathbb{K}}(\mathbf{x})$, for $n_{\mathbb{K}} := \dim_{\mathbb{R}} \mathbb{K}^n$. Its closure is $\bar{B}_r^{\mathbb{K}}(\mathbf{x})$. The Euclidean sphere of $\mathbb{K}^n$ centred at $\mathbf{x}$ and of radius $r$ is $S_r^{\mathbb{K}-1}(\mathbf{x})$. When $\mathbf{x}$ is the origin $\mathbf{0}$ of $\mathbb{K}^n$, we will simply write $B_r^{\mathbb{K}}$, $\bar{B}_r^{\mathbb{K}}$ and $S_r^{\mathbb{K}-1}$.

**Convention 1.** A Riemannian manifold $(M, g_M)$ consists of a $C^k$ or $\mathbb{K}$-analytic manifold $M$, for $k \in \mathbb{N}_{\ge 1} \cup \infty$, of positive $\mathbb{K}$-dimension $m$, and of a continuous Riemannian metric $g_M$ on $M$.

Let $(M, g_M)$ be a Riemannian manifold. Thus $g_M$ induces the distance $d_M$ on $M$ defined as the infimum of the lengths of rectifiable curves connecting any given pair of points.

Any subset $S$ of $M$ admits two natural metric space structures on $S$ inherited from $(M, g_M)$:

**Definition 2.1.** 1) *The outer distance on $S$ is the distance function $d_S$ on $S \times S$ obtained by restricting $d_M$ to $S \times S$.*
2) The inner distance on S is the function

$$d_{\text{inn}}^{S} : S \times S \to [0, +\infty]$$

such that given $\mathbf{s}, \mathbf{s}' \in S$, the number $d_{\text{inn}}^{S}(\mathbf{s}, \mathbf{s}')$ is the infimum of the lengths of the rectifiable paths lying in S joining $\mathbf{s}$ and $\mathbf{s}'$. If there is no such a path, then $d_{\text{inn}}^{S}(\mathbf{s}, \mathbf{s}') = \infty$.

Observe that $d_S \le d_{\text{inn}}^S$. We recall the following notion.

**Definition 2.2.** Let $(X, d_i)$, where $i = 1, 2$, be two metric space structures over the space X. The metric space structures $(X, d_1)$ and $(X, d_2)$ are equivalent if there exists a constant $L > 1$ such that

$$\mathbf{x}, \mathbf{x}' \in X \implies \frac{1}{L} \cdot d_1(\mathbf{x}, \mathbf{x}') \le d_2(\mathbf{x}, \mathbf{x}') \le L \cdot d_1(\mathbf{x}, \mathbf{x}')$$

An elementary question arising from presenting these two definitions is when are the metric spaces $(S, d_S)$ and $(S, d_{\text{inn}}^S)$ equivalent? This motivates the next

**Definition 2.3.** A subset S of the Riemannian manifold $(M, g_M)$ is Lipschitz normally embedded (shortened to LNE) if the identity mapping $(S, d_S) \to (S, d_{\text{inn}}^S)$ is bi-Lipschitz: there exists a positive constant L such that

$$\mathbf{s}, \mathbf{s}' \in S \implies d_{\text{inn}}^{S}(\mathbf{s}, \mathbf{s}') \le L \cdot d_{S}(\mathbf{s}, \mathbf{s}')$$

Any constant L satisfying the previous inequality is called a LNE constant of S.

We will simply use the expression LNE, but it must never be forgotten that it is with respect to the metric structure given by the Riemannian metric $g_M$ on M.

**Remark 2.4.** (i) The metric space structure $(M, d_M)$ associated with the Riemannian manifold $(M, g_M)$ is LNE, with LNE constant 1. Moreover each chart is locally bi-Lipschitz.

(ii) Since any two continuous Riemannian metrics on a $C^1$ compact Riemannian manifold M yield equivalent metric space structures, the property of being LNE depends only on the $C^1$ differentiable structure. Moreover, following Sullivan's result [20], a compact topological manifold of dimension different from 4 has a unique metric space structure, up to equivalence. Therefore the property of being LNE in a compact Riemannian manifold of dimension which is not 4 depends only on the ambient topology.

Finding a local characterisation of a property is usually a first step to understand it globally, but the property being LNE does not pass to germs: the open ball $B_r^2$ centred at 0 of positive radius r is a LNE representative of the germ $(\mathbb{R}^2, 0)$, while $\{(x, y) : y^2 - (x - r)^3 > 0\} \cap B_{2r}$ is a non-LNE one.

We introduce the following notion (see also [11]), that will be central to the paper.

**Definition 2.5.** 1) A subset S of the Riemannian manifold $(M, g_M)$ is locally LNE at the point s of $\text{clos}(S)$, the closure of S in M, if there exists an open neighbourhood U of s in M such that $S \cap U$ is LNE.

2) The subset S is locally LNE if it is locally LNE at each point of $\text{clos}(S)$.

The next Lemma generalizes in our setting the Euclidean result [11, Proposition 2.4]. It presents a simple criterion for a compact subset of $(M, g_M)$ to be LNE.

**Lemma 2.6.** Let S be a compact connected subset of the Riemannian manifold $(M, g_M)$. It is LNE if and only if it is locally LNE.
Proof. We follow the proof of [11]. When $S$ is LNE, taking $\mathcal{U} = M$ yields that $S$ is locally LNE.

For each point $\mathbf{s}$ of $S$, let $\mathcal{U}_{\mathbf{s}}$ be a neighbourhood of $\mathbf{s}$ in $M$ such that $U_{\mathbf{s}} := S \cap \mathcal{U}_{\mathbf{s}}$ is LNE. Thus $S$ is covered by $U_1, \dots, U_N$, where $U_i := U_{\mathbf{s}_i}$ for some points $\mathbf{s}_1, \dots, \mathbf{s}_N$ of $S$. Let $L_0$ be a LNE constant of $U_1, \dots, U_N$. Consider the following function

$$f: S \times S \setminus \Delta \to \mathbb{R}, \quad (\mathbf{s}, \mathbf{s}') \mapsto \frac{d_{\text{inn}}^{S}(\mathbf{s}, \mathbf{s}')}{d_{S}(\mathbf{s}, \mathbf{s}')}, \quad \text{where } \Delta := \{(\mathbf{s}, \mathbf{s}) \in S \times S\}.$$

Note that $f$ is bounded below by 1. Let $V := \cup_{i=1}^N U_i \times U_i$ which is an open neighbourhood of $\Delta$ in $S \times S$. Observe that it is connected. Since $f$ is continuous and $S \times S \setminus V$ is compact it admits a maximum $L_1$. Let $(\mathbf{s}, \mathbf{s}') \in V \setminus \Delta$, thus there exists $i$ such that $(\mathbf{s}, \mathbf{s}') \in U_i \times U_i$. Therefore $f(\mathbf{s}, \mathbf{s}') \le L_0$. Thus $S$ is LNE with LNE constant $\max(L_0, L_1)$. $\square$

We recall that a subset $S$ of a $C^k$ manifold $M$, $k \ge 1$, is a $C^1$ sub-manifold (possibly with boundary) of $M$ if: i) $S$ is a $C^1$ manifold (possibly with boundary); ii) the manifold topology coincides with the subspace topology; iii) the inclusion mapping $S \hookrightarrow M$ is a $C^1$ injective immersion. As a corollary of the previous criterion we obtain the following result that we will use extensively in the paper.

**Corollary 2.7.** Let $N$ be a $C^1$ compact connected sub-manifold of the Riemannian manifold $(M, g_M)$ possibly with $C^1$ boundary. Then $N$ is LNE.

Proof. Since $N$ is connected we can assume that $M$ is also connected. Let $m, n$ be the respective dimensions of $M, N$. By Lemma 2.6, it suffices to show that $N$ is locally LNE.

• Assume that $(M, g_M) = (\mathbb{R}^m, \text{eucl})$ and $n = m$. For each $\mathbf{x}$ in $N \setminus \partial N$, there exists a positive radius $r$ such that $B_r^m(\mathbf{x})$ is contained in $N$. Since the ball is convex, it is LNE, thus $N$ is locally LNE at $\mathbf{x}$. Let $\mathbf{x} \in \partial N$. There exists a positive radius $r$ and a $C^1$ diffeomorphism

$$\phi_{\mathbf{x}}^{\partial}: N \cap B_{r}^{m}(\mathbf{x}) \to B_{2r}^{m-1} \times [0, 2),$$

mapping $\partial N \cap B_r^m(\mathbf{x})$ onto $B_{2r}^{m-1} \times 0$. Up to shrinking $r$ we can assume that $\phi_{\mathbf{x}}^{\partial}$ is bi-Lipschitz: there exists a positive constant $A$ such that

$$\mathbf{y}, \mathbf{y}' \in N \cap B_r^m(\mathbf{x}) \implies |\mathbf{y} - \mathbf{y}'| \le A \cdot |\phi_{\mathbf{x}}^{\partial}(\mathbf{y}) - \phi_{\mathbf{x}}^{\partial}(\mathbf{y}')| \le A^2 \cdot |\mathbf{y} - \mathbf{y}'|.$$

Since $B_{2r}^{m-1} \times [0, 2)$ is convex in $\mathbb{R}^m$ it is LNE, thus $N \cap B_r^m(\mathbf{x})$ is also LNE. Therefore, $N$ is locally LNE at each of its points.

• Assume that $(M, g_M) = (\mathbb{R}^m, \text{eucl})$ and $n < m$. For each $\mathbf{x}$ in $N$, let $T_{\mathbf{x}}$ be the tangent space of $N$ at $\mathbf{x}$ and let $\pi_{\mathbf{x}}: \mathbb{R}^m \to T_{\mathbf{x}}$ be the orthogonal projection onto $T_{\mathbf{x}}$. There exists a positive radius $r$ such that $p_{\mathbf{x}}$, the restriction of $\pi_{\mathbf{x}}$ to $N \cap B_r^m(\mathbf{x})$ is a $C^1$ diffeomorphism onto its image, and further we find

$$\mathbf{y}, \mathbf{y}' \in N \cap B_r^m(\mathbf{x}) \implies |\mathbf{y} - \mathbf{y}'| \ge |p_{\mathbf{x}}(\mathbf{y}) - p_{\mathbf{x}}(\mathbf{y}')| \ge \frac{1}{2}|\mathbf{y} - \mathbf{y}'|.$$

Let $Z := \pi_{\mathbf{x}}(N)$. Therefore $(Z, \mathbf{x})$ is a $C^1$ sub-manifold possibly with boundary of $T_{\mathbf{x}}$ and thus $Z$ is locally LNE at $\mathbf{x}$ w.r.t. the Euclidean metric of $T_{\mathbf{x}}$ by the same argument as in the previous case. Since $T_{\mathbf{x}}$ is still an euclidean space for the restriction of the Euclidean metric of $\mathbb{R}^m$, there exists a positive radius $r'$ such that $Z \cap B_{r'}^m(\mathbf{x})$ is LNE since $T_{\mathbf{x}} \cap B_{r'}^m(\mathbf{x}')$ is an Euclidean ball of $T_{\mathbf{x}}$ of radius $r'$. Since $p_{\mathbf{x}}$ is bi-Lipschitz we deduce that $p_{\mathbf{x}}^{-1}(Z \cap B_{r'}^m(\mathbf{x}))$ is LNE whenever $2r' \le r$.

We conclude that any sub-manifold of $\mathbb{R}^m$ is locally LNE at each of its points.

• General Case. For each $\mathbf{x}$ in $N$, let $\Phi_{\mathbf{x}}: \mathcal{V}_{\mathbf{x}} \to \mathbb{R}^m$ be a $C^k$ chart of $M$ mapping $\mathbf{x}$ onto $\mathbf{0}$. Therefore the germ $(\Phi_{\mathbf{x}}(N), \mathbf{0})$ is that of $C^1$ sub-manifold, possibly with boundary, and is of dimension $n$. Let $\delta_{\mathbf{x}}$ be the distance on $B_1^m$ obtained from the pushed-forward Riemannian metric $h_{\mathbf{x}} := (\Phi_{\mathbf{x}})_* g_M$. Let
$V_x := \Phi_x^{-1}(B_1^m)$. Since $\Phi_x$ yields a bi-Lipschitz homeomorphism $(V_x, g_M|_{V_x}) \to (B_1^m, \text{eucl}|_{B_1^m})$, the metric space structures $(B_1^m, \delta_x)$ and $(B_1^m, | - |)$ are equivalent. Since $\Phi_x(N)$ is locally LNE at $0$ in $\mathbb{R}^m$, it is also locally LNE at $0$ w.r.t. $h_x$. Therefore $N$ is locally LNE at $x$. $\square$

In the Euclidean context, the following result complements Corollary 2.7.

**Lemma 2.8.** Let $N$ be a compact $C^1$ sub-manifold with $C^1$ boundary $\partial N$ of $\mathbb{R}^n$ of dimension $n$. Then each connected component of $\mathbb{R}^n \setminus N$ and of $\mathbb{R}^n \setminus (N \setminus \partial N)$ are LNE.

*Proof.* We will treat the case $\mathbb{R}^n \setminus (N \setminus \partial N)$. The remaining one will follow from the exact same demonstration. Let $E := \mathbb{R}^n \setminus (N \setminus \partial N)$. Observe that $\partial E = \partial N$, and that $E$ has a single unbounded connected component. Since the bounded components of $E$ are compact manifolds with boundary, they are LNE. Without loss of generality we can assume that $E$ is connected.

Let $B_1, \dots, B_s$, be the connected components of $\partial E$. Define the positive number

$$ \delta := \min_{1 \le i < j \le s} \mathrm{dist}(B_i, B_j), $$

with the convention that $\delta = \infty$ if $\partial E$ is connected.

Let $\mathbf{x}, \mathbf{x}'$ be two distinct points of $E$, and let $I$ be the segment of $\mathbb{R}^n$ with end points $\mathbf{x}, \mathbf{x}'$. The segment $I$ is oriented from $\mathbf{x}$ to $\mathbf{x}'$, thus is totally ordered. When $I$ intersects with $\partial E$, we can assume without loss of generality that $\mathbf{x}, \mathbf{x}'$ are both in $\partial E$. Since $\delta$ is positive, there exists finitely many open intervals $J^k = (\mathbf{x}_k^1, \mathbf{x}_{k+1}^0)$ of $I$, for $k = 1, \dots, p$, with the following properties:

i) $\mathbf{x} = \mathbf{x}_1^0 \le \mathbf{x}_1^1 < \mathbf{x}_2^0 \le \dots < \mathbf{x}_p^0 \le \mathbf{x}_p^1 = \mathbf{x}'$;

ii) For each $k = 1, \dots, p$, there exists $i_k \ge 1$ such that $\mathbf{x}_k^0, \mathbf{x}_k^1 \in B_{i_k}$;

iii) For each $k = 1, \dots, p-1$, the interval $J^k$ is contained in $I \setminus N$;

iv) $i_k \ne i_{k+1}$ for each $k = 1, \dots, p-1$;

v) $p$ is maximal for the four previous conditions.

Observe that each connected component of the compact set $I \cap \partial E$ is contained in one of the intervals $[\mathbf{x}_k^0, \mathbf{x}_k^1]$, where $k = 1, \dots, p$.

Since each connected component of $\partial E$ is LNE by Corollary 2.7, let $A$ be a common LNE constant. For each $k = 1, \dots, p$, we find

$$ d_{\mathrm{inn}}^{E}(\mathbf{x}_{k}^{0}, \mathbf{x}_{k}^{1}) \le d_{\mathrm{inn}}^{B_{i_{k}}}(\mathbf{x}_{k}^{0}, \mathbf{x}_{k}^{1}) \le A|\mathbf{x}_{k}^{0} - \mathbf{x}_{k}^{1}|. $$

Since the segment $J^k$ is contained in $E$, we find

$$ d_{\mathrm{inn}}^{E}(\mathbf{x}_{k}^{1}, \mathbf{x}_{k+1}^{0}) = |\mathbf{x}_{k}^{1} - \mathbf{x}_{k+1}^{0}| \le A|\mathbf{x}_{k}^{1} - \mathbf{x}_{k+1}^{0}|. $$

We deduce

$$ d_{\mathrm{inn}}^{E}(\mathbf{x}, \mathbf{x}') \le \sum_{k=1}^{p} d_{\mathrm{inn}}^{E}(\mathbf{x}_{k}^{0}, \mathbf{x}_{k}^{1}) + \sum_{k=1}^{p-1} d_{\mathrm{inn}}^{E}(\mathbf{x}_{k}^{1}, \mathbf{x}_{k+1}^{0}) \le A|\mathbf{x} - \mathbf{x}'|, $$

and the lemma is proved. $\square$

**Remark 2.9.** The analogue of Lemma 2.8 is also true when the ambient metric space is a connected Riemannian manifold $(M, g_M)$. The proof of this more general case goes through similarly, using estimates of curves of length close enough to the outer/inner distance. Since we only use such a result in the Euclidean context, we presented the statement and proof in this simpler setting.

Let $(M, g_M)$ and $(P, g_P)$ be Riemannian manifolds. The manifold $M \times P$ is thus equipped with the Riemannian product metric $g_{M \times P} := g_M \otimes g_P$, whose associated distance function is $d_M + d_P$. Given
a subset $S$ of $(M, g_M)$, we can speak of a Lipschitz mapping $F : (S, d_S) \to (P, d_P)$ with Lipschitz constant $K_F$:

$$ \mathbf{s}, \mathbf{s}' \in S \implies d_P(F(\mathbf{s}), F(\mathbf{s}')) \le K_F \cdot d_S(\mathbf{s}, \mathbf{s}') = K_F \cdot d_M(\mathbf{s}, \mathbf{s}'). $$

The space $S \times P$ comes naturally equipped with the product distance $d_{S \times P} := d_S + d_P$. We conclude this section with the following very useful, though simple, result.

**Proposition 2.10.** Let $S$ be a LNE subset of $(M, g_M)$. If $F : (S, d_S) \to (P, d_P)$ is a Lipschitz mapping, then its graph is LNE in $(M \times P, d_{M \times P})$.

*Proof.* Since $F$ is Lipschitz there exists a positive constant $K_F$ such that

$$ \mathbf{s}, \mathbf{s}' \in S \implies d_P(F(\mathbf{s}), F(\mathbf{s}')) \le K_F \cdot d_S(\mathbf{s}, \mathbf{s}'). $$

Let $\gamma : [0, 1] \to S$ be a rectifiable curve and let $\gamma_F : [0, 1] \to S \times P$ be the lift of $\gamma$ to $G_F$, the graph of $F$. The result follows from the following estimate

$$ \text{length}_{M \times P}(\gamma_F) \le (1 + K_F) \cdot \text{length}_M(\gamma) $$

where $\text{length}_Q$ is the length taken w.r.t. to the Riemannian metric $g_Q$, for $Q = M, M \times P$. $\square$

**Remark 2.11.** When $M = \mathbb{R}^m, P = \mathbb{R}^p$, their product distance $d_M + d_P$ is not the Euclidean distance of $\mathbb{R}^{m+p}$, but it is equivalent to the Euclidean distance of $\mathbb{R}^{n+p}$, and this is how we will use Proposition 2.10 in the complex affine context.

### 3. A CRITERION FOR ISOLATED SINGULARITY GERMS

Let $\mathbf{0}$ be the origin of the Euclidean space $\mathbb{R}^n$. If $S$ is any subset of $\mathbb{R}^n$ containing the origin, let $S^*$ be $S \setminus \mathbf{0}$.

**Lemma 3.1.** Let $(N_1, \mathbf{0}), \dots, (N_s, \mathbf{0})$, be germs at $\mathbf{0}$ of $C^1$ sub-manifolds of $\mathbb{R}^n$ of positive dimensions and codimensions. Assume that $(Y, \mathbf{0}) := \cup_{j=1}^s (N_j, \mathbf{0})$ has an isolated singularity at $\mathbf{0}$.

The germ $(Y, \mathbf{0})$ is locally LNE at $\mathbf{0}$ if and only if $T_0 N_j \cap T_0 N_k = \{\mathbf{0}\}$ for each $1 \le j < k \le s$.

*Proof.* For each $j = 1, \dots, s$, let $T_j := T_0 N_j$ and let $T_j^{\perp}$ be the orthogonal supplement of $T_j$. Since each $(N_j, \mathbf{0})$ is a $C^1$ sub-manifold, there exists a positive radius $r_Y > 0$, and a $C^1$-mapping

$$ G_j : T_j \cap B_{2r_Y}^n \to T_j^{\perp} $$

such that a representative of $(N_j, \mathbf{0})$ is the graph $\Gamma_j$ of $G_j$. Since $D_0 G_j = \mathbf{0}$ and $G$ is $C^1$, up to shrinking $r_Y$ the mapping $G_j$ is Lipschitz over $\mathbf{B}_r^n$ for any radius $0 < r \le r_Y$, with Lipschitz constant tending to 0 as $r$ goes to 0. For $j = 1, \dots, s$, we define

$$ N_j^{\le r} := \Gamma_j \cap \mathbf{B}_r^n. $$

Whenever $r_Y$ is small enough, we can further assume that $r_Y$ is such that $N_j^{\le r}$ is LNE for any $r \le r_Y$, since they are connected compact $C^1$ sub-manifolds with boundaries

$$ \partial N_j^{\le r} = \Gamma_j \cap \mathbf{S}_r^{n-1}. $$

Each $\partial N_j^{\le r}$ is diffeomorphic to $\mathbf{S}^{\dim T_j - 1}$.

Given a pair $1 \le j < k \le s$, and $r \le r_Y$, let $\mathbf{x}_l \in (N_l^{\le r})^*$ for $l = j, k$. Let $r_l > 0$ be defined as follows:

$$ r_l := |\mathbf{x}_l|. $$
Let $2\alpha \in [0, \pi]$ be the (non-oriented) angle between $\mathbf{x}_j$ and $\mathbf{x}_k$. From plane geometry classes we recall that

$$ \left| \frac{\mathbf{x}_j}{r_j} - \frac{\mathbf{x}_k}{r_k} \right| = 2 \sin \alpha $$

and further remember that

$$ (3.1) \qquad |\mathbf{x}_j - \mathbf{x}_k|^2 = (r_j - r_k)^2 \cos^2 \alpha + (r_j + r_k)^2 \sin^2 \alpha. $$

By definition of $\alpha$, we deduce that

$$ (3.2) \qquad (r_j + r_k) \sin \alpha \le |\mathbf{x}_j - \mathbf{x}_k| \le r_j + r_k. $$

Finally let $Y_{\le r} := Y \cap \mathbf{B}_r^{2n}$, $d_{\text{inn}}^{\le r} := d_{\text{inn}}^{Y_{\le r}}$ and $d_j^{\le r} := d_{\text{inn}}^{N_j^{\le r}}$, for $1 \le j \le s$.

• Assume that $T_j \cap T_k = \mathbf{0}$, for each pair $j \ne k$.

For $j = 1, \dots, s$, we define the link of $T_j$ at $\mathbf{0}$

$$ S_j := T_j \cap \mathbf{S}_1^{n-1}. $$

By hypothesis, we get

$$ (3.3) \qquad \delta := \min_{1 \le j < k \le w} \operatorname{dist}(S_j, S_k) > 0. $$

We can require that $r_Y$ is small enough such that for each $j = 1, \dots, s$, and each $\mathbf{x}_j \in (N_j^{\le r_Y})^*$ the followings estimates hold true

$$ (3.4) \qquad \operatorname{dist} \left( S_j, \frac{\mathbf{x}_j}{r_j} \right) \le \frac{\delta}{4}. $$

For each $1 \le j < k \le s$, each $\mathbf{x}_j \in N_j^*$, $\mathbf{x}_k \in N_k^*$, we also find the following estimates

$$ (3.5) \qquad \left| \frac{\mathbf{x}_j}{r_j} - \frac{\mathbf{x}_k}{r_k} \right| \ge \frac{\delta}{2}. $$

By choice of $r_Y$ and $j, k$ we have

$$ \left| \frac{\mathbf{x}_j}{r_j} - \frac{\mathbf{x}_k}{r_k} \right| = 2 \sin(\alpha) \ge \frac{\delta}{2}. $$

From Estimate (3.2), we deduce

$$ (3.6) \qquad \frac{\delta}{4} \cdot (r_k + r_j) \le |\mathbf{x}_j - \mathbf{x}_k|. $$

as well as

$$ (3.7) \qquad |\mathbf{x}_l| = r_l \le \frac{4}{\delta} \cdot |\mathbf{x}_j - \mathbf{x}_k| \quad \text{for } l = j, k. $$

Since $Y$ has an isolated singularity at $\mathbf{0}$, we obviously have

$$ d_{\text{inn}}^{\le r}(\mathbf{x}_j, \mathbf{x}_k) \le d_j^{\le r}(\mathbf{x}_j, \mathbf{0}) + d_k^{\le r}(\mathbf{0}, \mathbf{x}_k), $$

combining Estimate (3.7) with $N_j^{\le r}$ and $N_k^{\le r}$ being LNE, each with LNE constant $A$, yield an expected inequality

$$ d_{\text{inn}}^{\le r}(\mathbf{x}_j, \mathbf{x}_k) \le \frac{8A}{\delta} \cdot |\mathbf{x}_j - \mathbf{x}_k|. $$

Since each $N_j^{\le r}$ is LNE, we obtain that $Y^{\le r}$ is LNE.

• Assume that $(Y, \mathbf{0})$ is locally LNE at $\mathbf{0}$.

Since $(Y, \mathbf{0})$ has an isolated singularity at $\mathbf{0}$, we can assume that $r_Y$ is small enough so that the following identity holds true

$$ j < k \implies (N_j^{\le r_Y})^* \cap (N_k^{\le r_Y})^* = \emptyset. $$
Given any pair $1 \le j < k \le s$, observe that to connect $\mathbf{x}_j$ to $\mathbf{x}_k$ within $Y^{\le r_Y}$ it is necessary to go through **0**. Therefore

$$ (3.8) \qquad d_{\text{inn}}^{Y}(\mathbf{x}_j, \mathbf{x}_k) \ge r_j + r_k $$

Let $E := \{\{i_1, \dots, i_t\} \text{ such that } 1 \le i_1 < \dots < i_t \le s, \text{ with } t \ge 2 \text{ and } \dim \cap_{j=1}^t T_{i_j} \ge 1\}.$

Assume that $E$ is not empty. Let $J$ be an element of $E$ and let

$$ P_J := \cap_{j \in J} T_j. $$

For any $\mathbf{p} \in P_J$ and each $j \in J$, let $\mathbf{x}_j$ be the point $(\mathbf{p}, G_j(\mathbf{p}))$ of $N_j$. Since $P_J$ is tangent at **0** to each $N_l$ and $D_0 G_l$ is null, for any $l \in J$, as $\mathbf{p}$ goes to **0** in $P_J$ we find that

$$ |\mathbf{x}_j|, |\mathbf{x}_k| = |\mathbf{p}| + o(|\mathbf{p}|) \quad \text{and} \quad |\mathbf{x}_j - \mathbf{x}_k| = o(|\mathbf{p}|) $$

for any pair $j, k$ of $J$ with $j \neq k$. Comparing the last estimate with Estimate (3.8) shows that the germ $(Y, \mathbf{0})$ cannot admit a LNE representative in any neighbourhood of **0**. Necessarily $E$ is empty. $\square$

A careful reading of the proof of the only if part of Lemma 3.1 yields the following result about LNE representatives

**Corollary 3.2.** Let $Y$ be a representative of the germ $(Y, \mathbf{0})$ of Lemma 3.1. If $(Y, \mathbf{0})$ is LNE, there exists a positive radius $r_Y$ such that for each radius $r \in (0, r_Y]$, the subsets $Y \cap \mathbf{B}_r^n$ and $Y \cap \mathbf{B}_r^n$ are LNE.

## 4. THE LOCAL CASE

We work in the local complex analytic category.

We present here Proposition 4.2, a known local criterion for a complex analytic curve germ to be LNE (see for instance [16, 3]), with a complete proof. This result is used to obtain Proposition 4.6 which fully characterizes complex curves which are LNE in a given compact complex manifold (independently of any prescribed continuous Riemannian metric).

**Convention 2.** A curve germ is a complex analytic curve germ at a prescribed point.

Let $(Y, \mathbf{0})$ be a curve germ of $\mathbb{C}^n$ at $\mathbf{0} \in \mathbb{C}^n$. Let $m(Y, \mathbf{0})$ be the multiplicity of $Y$ at $\mathbf{0}$. We recall that $Y$ is non-singular at $\mathbf{y}_0$ if and only if $m(Y, \mathbf{y}_0) = 1$.

Assume that $Y$ is irreducible at $\mathbf{0}$ and write $m$ for $m(Y, \mathbf{0})$. Let $L$ be the tangent cone of $Y$ at $\mathbf{0}$, which is a complex line through $\mathbf{0}$. The restriction to $(Y, \mathbf{0})$ of the (orthogonal) projection $\mathbb{C}^n \to L$ is a complex analytic finite mapping $p_L : (Y, \mathbf{0}) \to (L, \mathbf{0})$ inducing a holomorphic $m$-sheeted covering $(Y^*, \mathbf{0}) \to (L^*, \mathbf{0})$. After an orthonormal change of complex coordinates we can assume that

$$ L := \{\mathbf{x} = (x_1, \dots, x_n) \in \mathbb{C}^n : x_2 = \dots = x_n = 0\} = \mathbb{C} \times \mathbf{0}. $$

From Puiseux Theorem, there exists a holomorphic map germ $F : (\mathbb{C}, 0) \to (\mathbb{C}^{n-1}, \mathbf{0})$ such that

$$ (4.1) \qquad (Y, \mathbf{0}) = \{(s^m, F(s)) : s \in (\mathbb{C}, 0)\}. $$

More precisely, writing $F = (f_2, \dots, f_n)$, for each $j = 2, \dots, n$, we know that

$$ f_j(s) = s^{a_j} \phi_j(s), \text{ with } \phi_j(0) \neq 0, \text{ and } a_j \ge m+1 $$

with the convention $a_j = \infty$ if and only if $f_j \equiv 0$. Moreover there exists no holomorphic map germ $H : (\mathbb{C}, 0) \to (\mathbb{C}^{n-1}, \mathbf{0})$ such that $F(s) = H(s^N)$, for a positive integer $N \ge 2$ dividing $m$.

Let $(X, \mathbf{0})$ be a curve germ of $(\mathbb{C}^n, \mathbf{0})$. Let $(X_1, \mathbf{0}), \dots, (X_s, \mathbf{0})$, be its local irreducible components. For each $j = 1, \dots, s$, let $L_j$ be the tangent cone of $X_j$ at $\mathbf{0}$, thus a complex line through $\mathbf{0}$.
For each $j = 1, \dots, s$, the (orthogonal) projection $\pi_j : (X_j, \mathbf{0}) \to (L_j, \mathbf{0})$ is a complex finite mapping and it induces the germ of a holomorphic cover $(X_j^*, \mathbf{0}) \to (L_j^*, \mathbf{0})$ with $m_j := m(X_j, \mathbf{0}) \ge 1$ sheets.

**Remark 4.1.** The length of any rectifiable curve of an Euclidean space is taken w.r.t. the Euclidean metric.

The first principal result of this section is the following

**Proposition 4.2.** The germ of analytic curve $(X, \mathbf{0})$ of $(\mathbb{C}^n, \mathbf{0})$ admits a representative which is locally LNE at $\mathbf{0}$ if and only if the following conditions are satisfied: (i) $m_j = 1$ for each $j = 1, \dots, s$, and (ii) $L_j \cap L_k = \{\mathbf{0}\}$ for each pair $(j, k)$ such that $1 \le j < k \le s$. In other words, the curve germ $(X, \mathbf{0})$ is LNE if and only if it is a union of non-singular curve germs intersecting pairwise transversally.

*Proof.* Let $\lambda_1, \dots, \lambda_t$, be complex lines of $\mathbb{C}^n$ through the origin. We define the following “angle”

$$ \delta((\lambda_j)_{j=1,\dots,t}) := \min_{1 \le j < k \le t} \{ |\mathbf{u}_j - \mathbf{u}_k| : \mathbf{u}_j \in \lambda_j \cap \mathbf{S}_1^{2n-1} \text{ and } \mathbf{u}_k \in \lambda_k \cap \mathbf{S}_1^{2n-1} \}. $$

Condition (ii) holds true if and only if $\delta((L_j)_{j=1,\dots,s}) > 0$.

* Assume that (i) and (ii) are satisfied.

Condition (i) means that each $(X_j, \mathbf{0})$ is a non-singular germ. Condition (ii) means that the lines $L_j$s are pairwise transverse. This is the setting of Lemma 3.1.

* Assume that $X$ admits a representative locally LNE at $\mathbf{0}$.

First we show the following property.

**Claim 4.3.** Let $(Y, \mathbf{0})$ be a curve germ whose tangent cone at $\mathbf{0}$ consists of the single line $L$. If $m(Y, \mathbf{0}) \ge 2$, then the germ $(Y, \mathbf{0})$ cannot admit any representative locally LNE at $\mathbf{0}$.

*Proof.* For simplicity let $m := m(Y, \mathbf{0})$ and assume that $L = \mathbb{C} \times \mathbf{0} \subset \mathbb{C}^n$. Let $\pi_L : \mathbb{C}^n \to L$ be the projection onto $L$. Therefore the map germ induced by the restriction of $\pi_L$ to $Y^*$

$$ \pi : (Y^*, \mathbf{0}) \to (L^*, \mathbf{0}) $$

is a $m$-sheeted holomorphic covering. Let $x_1 \ne 0$ and let $\mathbf{y}_1, \dots, \mathbf{y}_m$ be the points of $\pi^{-1}(x_1)$.

Assume that $m \ge 2$. Since $\mathbf{y}_j = (x_1, v_j)$ for each $j = 1, \dots, m$, using Puiseux Parametrization (4.1) for each irreducible component of $(Y, \mathbf{0})$, we check there exists a positive integer $p$ such that

$$ |v_j| \le \mathrm{const}|x_1|^{\frac{p+1}{p}}, \quad \forall 1 \le j \le m $$

where $\mathrm{const}$ is a positive constant we do not need to denote specifically.

If to connect the points $\mathbf{y}_1$ and $\mathbf{y}_2$ in $Y$ we go through $\mathbf{0}$, we find.

$$ (4.2) \qquad d_{\mathrm{inn}}^{X}(\mathbf{y}_1, \mathbf{y}_2) \ge |\mathbf{y}_1| + |\mathbf{y}_2|. $$

When a rectifiable path $\gamma : [0, 1] \to Y^*$ connects the points $\mathbf{y}_1$ and $\mathbf{y}_2$, its projection $\gamma_1$ onto $L$ is contained in $L^*$ and goes from $x_1$ to $x_1$. Since $\gamma_1$ is rectifiable and not contractible in $L^*$, we deduce

$$ 2|x_1| \le \mathrm{length}(\gamma_1) < \mathrm{length}(\gamma). $$

In any case we have obtained that $d_{\mathrm{inn}}^Y(\mathbf{y}_1, \mathbf{y}_2) \ge 2|x_1|$, while $|\mathbf{y}_1 - \mathbf{y}_2| = o(|x_1|)$. Therefore the germ $(Y, \mathbf{0})$ cannot admit a representative locally LNE at $\mathbf{0}$. $\square$

Claim 4.3 implies that $m_j = 1$ for each $j = 1, \dots, s$, and also that the tangent cones $L_j$ are pairwise transverse. $\square$
A by-product of Proposition 4.2 is a criterion for a complex curve of a compact complex manifold to be LNE presented in Proposition 4.6 below.

Let $(M, g_M)$ be a Riemannian manifold. Let $\psi: \mathcal{U} \to B_2^m$ be a $C^1$ chart. The subset $U_1 := \psi^{-1}(B_1^m)$ is a compact sub-manifold with boundary of $\mathcal{U}$ and of $M$. Let

$$ \mathbf{e}_1 := \psi^*(\mathrm{eucl}|_{B_1^m}) $$

the pull-back of the Euclidean metric restricted to $B_1^m$. Since $\psi_1 := \psi|_{U_1} : (U_1, d_{U_1}) \to (B_1^m, | - |)$ is bi-Lipschitz (it is $C^1$), the metric space structures obtained from the Riemannian metrics $h_1 := g_M|_{U_1}$ and $\mathbf{e}_1$ are equivalent. Thus $U_1$ is LNE in $(M, g_M)$.

**Lemma 4.4.** The manifold $M$ is locally LNE (w.r.t. $g_M$) at $\mathbf{x}_0 \in \mathcal{U}$ if and only if $\psi(\mathcal{U})$ is locally LNE at $\mathbf{0}$ (w.r.t. eucl).

*Proof.* We can assume that $\psi(\mathbf{x}_0) = \mathbf{0}$. The previous observation about $h_1$ and $\mathbf{e}_1$, the compactness of $U_1$ and Lemma 2.6 altogether imply the statement. $\square$

**Definition 4.5.** Let $M$ be a complex manifold. A complex curve of $M$ is a complex analytic subset of $M$, thus closed, and which is of local (complex) dimension 1 at each of its points.

When the complex manifold $M$ is compact, any complex curve $X$ of $M$ is compact. Therefore the set $X_{\mathrm{sing}}$ of singular points of $X$ is either empty or consists of finitely many points.

Following point (ii) of Remark 2.4, since any compact complex manifold can be equipped with a $C^\infty$ Hermitian structure, we will not refer to the Riemannian structure in the next statement, the other main result of this section.

**Proposition 4.6.** A complex curve $X$ of a compact complex manifold $M$ is LNE if and only if $X$ is connected and is locally LNE at each point of its singular locus.

*Proof.* If $X$ is LNE, it is connected and it is locally LNE at each of its points by Definition 2.5.

Let $m$ be the complex dimension of $M$. Assume that $X$ is connected and is locally LNE at each of its singular points. Let $X_{\mathrm{sing}} := \{\mathbf{x}_1, \dots, \mathbf{x}_s\}$. For each $j = 1, \dots, s$, let

$$ \psi_j: \mathcal{U}_j \to \mathcal{V}_m := B_2^{2m} \subset \mathbb{C}^m $$

be a complex chart of $M$ centred at $\mathbf{x}_j$ such that $\psi_j(\mathbf{x}_j) = \mathbf{0}$. The image $Y_j := \psi(X \cap \mathcal{U}_j)$ is a complex analytic set of $\mathcal{V}_m$ of local complex dimension 1 at each of its points. There exists a positive radius $r_j < 2$ such that for every positive radius $r \le r_j$, the curve

$$ Y_j \cap \mathbf{S}_r^{2m-1} $$

is a non-singular real analytic compact curve consisting of $d_j \ge 1$ connected components (see [14] for instance). Let

$$ r_X := \min_{j=1,\dots,s} r_j. $$

We can assume that each intersection $\mathrm{clos}(\mathcal{U}_k) \cap \mathrm{clos}(\mathcal{U}_l)$ is empty for any pair $k, l$ with $k \ne l$. From Lemma 4.4, we can also assume that for each $j = 1, \dots, s$, and for each positive radius $r \le r_X$ the compact subset

$$ X_{j,r} := X \cap \psi_j^{-1}(B_r^{2m}) $$

is LNE. By choice of $r_X$ we deduce that for each positive radius $r \le r_X$, the subset

$$ E_r := X \setminus \cup_{j=1}^s \psi_j^{-1}(B_r^{2m}) $$
is a compact real analytic sub-manifold with boundary

$$ \partial E_r = \cup_{j=1}^{s} (X \cap \psi_j^{-1}(\mathbf{S}_r^{2m-1})). $$

Thus any connected component of $E_r$ is LNE. Observe that for any $r \le \frac{2r_X}{3}$ the subset

$$ T_{j,r} := X \cap \psi_j^{-1}(\mathbf{B}_r^{2m} \setminus B_{\frac{2}{3}}^{2m}) $$

is a compact real analytic sub-manifold with boundary, consisting of $d_j$ connected components, each diffeomorphic to the cylinder $\mathbf{S}^1 \times [\frac{r}{2}, r]$. In particular each connected component of $T_{j,r}$ is LNE. Since any point of $X$ is an interior point of $X_{j,r}$, $T_{j,r}$ or $E_r$, for some radius $r \le \frac{2}{3}r_X$, $X$ is locally LNE at each of its points by Lemma 4.4. We conclude the proof applying again Lemma 2.6. $\square$

## 5. PRELIMINARIES AT INFINITY

Let $\mathbb{C}^n$ be the complex affine space of dimension $n$ and let $\mathbf{x}$ be some affine complex coordinates.

**Convention 3.** Unless mentioned otherwise, being LNE is to be understood as being LNE as a subset of $\mathbb{C}^n$ w.r.t. the Euclidean metric.

We will need the following result, consequence of Lemma 2.8

**Claim 5.1.** For any positive radius $R$, the subsets $\mathbb{C}^n \setminus B_R^{2n}$ and $\mathbb{C}^n \setminus \mathbf{B}_R^{2n}$ are LNE.

Let $\mathbf{P}^n := \mathbb{CP}^n$ be the complex projective space with projective coordinates $[z_1 : \dots : z_{n+1}]$. Let $\mathbf{H}_\infty$ be the projective hyperplane $z_{n+1} = 0$.

The affine space $\mathbb{C}^n$ embeds in $\mathbf{P}^n$ as $\mathbf{x} \mapsto [\mathbf{x} : 1]$, that is $\mathbf{P}^n = \mathbb{C}^n \cup \mathbf{H}_\infty$. The hyperplane $\mathbf{H}_\infty$ is called the hyperplane at infinity and any point of $\mathbf{H}_\infty$ is called a point at infinity.

If $S$ is any subset of $\mathbf{P}^n$, the affine part $S^a$ of $S$ is defined as

$$ S^a := S \cap \mathbb{C}^n = S \setminus \mathbf{H}_\infty. $$

The subset $S$ is affine if $S = S^a$.

When $S$ is affine, the germ $(S, \infty) := (S, \mathbf{H}_\infty)$ is well defined as a germ in $\mathbb{C}^n$, and so is the germ $(S, \lambda)$ for $\lambda \in \mathbf{H}_\infty$.

Let $\lambda$ be a point of $\mathbf{H}_\infty$. If $(Y, \lambda)$ is a (complex analytic) curve germ at $\lambda$, we will denote by $Y$ any representative of the germ $(Y, \lambda)$ in $\mathbf{P}^n$. We are only interested in representatives of $(Y, \lambda)$ whose intersection with $\mathbf{H}_\infty$ is just $\lambda$, property which we will always implicitly assume true.

Assume that $(Y, \lambda)$ is irreducible at $\lambda$ and is not contained in $\mathbf{H}_\infty$. Thus the tangent cone to $Y$ at $\lambda$ is a line $L$ of the vector space $T_\lambda \mathbf{P}^n$. The line $L$ is either transverse to $\mathbf{H}_\infty$ or contained in $T_\lambda \mathbf{H}^\infty$. After a linear change of coordinates in $\mathbb{C}^n$, we can assume that $\lambda := [1 : 0 : \dots : 0]$. Let $\mathcal{A}_1$ be the affine chart of $\mathbf{P}^n$ defined as $\{z_1 \neq 0\}$. Let $[1 : \mathbf{w} : z]$ be affine coordinates in $\mathcal{A}_1$ so that in $\mathbf{H}_\infty \cap \mathcal{A}_1 = \{z = 0\}$. We further write $\mathbf{w} = (w_2, \mathbf{w}') \in \mathbb{C} \times \mathbb{C}^{n-2}$. We identify $T_\lambda \mathbf{P}^n$ with $\mathcal{A}_1$ as well.

Since $\mathbf{x}$ are the affine coordinates of $\mathbb{C}^n$, in $\mathbb{C}^n \cap \mathcal{A}_1$ we write $[x_1 : \mathbf{y} : 1] = [\mathbf{x} : 1] = [1 : \mathbf{w} : z]$.

If $L$ is contained in $\{z = 0\}$, after a linear transformation of $\{z = 0\}$ in $\mathcal{A}_1$, corresponding to a linear transformation of $\{x_1 = 0\}$ in $\mathbb{C}^n$, we can further require that

$$ (5.1) \qquad L = \{[1 : w_2 : 0 : \dots : 0] : w_2 \in \mathbb{C}\}. $$

The line $L$ is transverse to $\mathbf{H}_\infty$ at $\lambda$ if and only if there exists $\mathbf{a} \in \mathbb{C}^{n-1}$ such that

$$ L = \{[1 : z\mathbf{a} : z] : z \in \mathbb{C}\}. $$
The affine translation $\mathbf{x} \to (x_1, \mathbf{y} - \mathbf{a})$ in $\mathbb{C}^n$, corresponding to the linear change of coordinates
$(\mathbf{w}, z) \to (\mathbf{w}', z) = (\mathbf{v}, z) = (\mathbf{w} - z\mathbf{a}, z)$ in $\mathcal{A}_1$, shows that the line $L$ has the following equation

$$
(5.2) \qquad L = \{[1:0:\dots:0:z] : z \in \mathbb{C}\}.
$$

Let $(s, \mathbf{w}', t)$ be either $(w_2, \mathbf{w}', z)$ or $(z, \mathbf{w}', w_2)$ for $w_2, z$, as appearing in the simplified equations (5.1) and (5.2) of $L$, so that $L = \{t = \mathbf{w}' = 0\}$.

The multiplicity $m := m(Y, \lambda)$ is preserved by this change of coordinates to obtain the simpler equation of $L$. Let

$$
Y^* := Y \setminus \lambda \text{ and } L^* := L \setminus \lambda.
$$

The projection $p_L : (Y^*, \lambda) \to (L^*, \lambda)$ is a holomorphic $m$-sheeted covering.

We introduce the following avatar at infinity of being locally LNE at a point (see also [8]).

**Definition 5.2.** A subset $S$ of $\mathbb{C}^n$ is locally LNE at infinity (w.r.t. the euclidean metric of $\mathbb{C}^n$) if there exists a compact subset $K$ of $\mathbb{C}^n$ such that $S \setminus K$ is LNE (w.r.t. the euclidean metric of $\mathbb{C}^n$).

The second non-compact result about being LNE is the following

**Lemma 5.3.** *Assume $L$ is transverse to $H_\infty$. The germ $(Y^a, \lambda)$ is locally LNE at infinity if and only if $m=1$.*

*Proof.* In this case we have $(s,t) = (z, w_2)$, and using parametrization (4.1) we find

$$
(Y, \lambda) = \{[1 : F(z) : z^m] = [1 : O(z^p) : z^m] : z \in (\mathbb{C}, 0)\}
$$

for a positive integer $p \ge m + 1$, maximal for this property. In this case $F \equiv 0$ would mean $p = \infty$,
which corresponds to the case $Y^a = L$. Observe also that from the affine point of view we check that

$$
(Y^a, \lambda) = \left\{ \left(x_1^m, x_1^m F(x_1^{-1})\right) = \left(x_1^m, O\left(x_1^{-(p-m)}\right)\right) : x_1 \in (\mathbb{C}, \infty) \right\}
$$

so that if $(x_1, \mathbf{y}) \in (Y^a, \infty)$, then $\mathbf{y} \to \mathbf{0}$ as $x_1 \to \infty$. Let us consider the following holomorphic map
germ $(\mathbb{C}, \infty) \to (\mathbb{C}^{n-1}, 0)$

$$
x_1 \to G(x_1) := x_1^m F(x_1^{-1}) = O(x_1^{-(p-m)}).
$$

• Assume $m = 1$. The germ $(Y, \lambda)$ is non-singular at $\lambda$,

$$
\mathbf{w} = z^p \alpha(z)
$$

where either $p \ge 2$ and $\alpha(z) \in \mathbb{C}\{z\}^{n-1}$ with $\alpha(0) \ne \mathbf{0}$, or $(Y, \lambda) = (L, \lambda)$.
This presentation is
equivalent to the following affine one

$$
\mathbf{y} = G(x_1).
$$

Since the components of $G$ are power series in $x_1^{-1}$ converging for $|x_1| \ge R_1 > 0$, $G$ goes to $\mathbf{0}$ as $x_1$ goes to infinity. Thus its derivative goes to $\mathbf{0}$ as $x_1$ goes to infinity, therefore it is a Lipschitz mapping $\mathbb{C} \setminus \mathbf{B}_R^2 \to \mathbb{C}^{n-1}$ for any radius $R \ge R_1$. Proposition 2.10 and Claim 5.1 imply that the subset $\{(x_1, G(x_1)) : |x_1| \ge R\}$ is LNE since it is a Lipschitz graph over a LNE subset. Therefore the germ $(Y^a, \lambda)$ is locally LNE at infinity.

• Assume $m \ge 2$. Let $z \in \mathbb{C}^*$ and let $\zeta$ be a $m$-th root of $z$. Let $\mathbf{x}_0 := [1 : F(\zeta) : z]$ and $\mathbf{x}_1 := [1 : F(\omega\zeta) : z]$ for $\omega = e^{i\frac{2\pi}{m}}$. By parametrization (4.1), we find $F(\zeta) \ne F(\omega\zeta)$. Writing

$$
\mathbf{x}_0^a := \left( \frac{1}{z}, \frac{F(\zeta)}{z} \right) \in \mathbb{C}^n \quad \text{and} \quad \mathbf{x}_1^a := \left( \frac{1}{z}, \frac{F(\omega\zeta)}{z} \right) \in \mathbb{C}^n
$$
for $|z|$ small enough we obtain

$$ (5.3) \qquad |\mathbf{x}_1^a - \mathbf{x}_0^a| = \left| \frac{F(\zeta)}{z} - \frac{F(\omega\zeta)}{z} \right| \le \text{const.} |z|^{\frac{p}{m}-1} = \text{const.} \frac{1}{|x_1|^q} \quad \text{for } q := \frac{p-m}{m} > 0. $$

We can take a representative of $Y^a$ outside a large Euclidean closed ball $\mathbf{B}^{2n}$ centred at $0$ such that the projection mapping

$$ \pi : Y^a \to \mathbb{C} \setminus \mathbf{B}^2, \quad [x_1 : \mathbf{y} : 1] \mapsto x_1 $$

is a $m$-sheeted holomorphic covering, where $\mathbf{B}^2$ is an Euclidean ball of $\mathbb{C}$ centred at $0$ image of $\mathbf{B}^{2n}$ under the projection onto the $x_1$-axis.

Let $\gamma : [0, 1] \to Y^a$ be a rectifiable path from $\mathbf{x}_0 := \gamma(0)$ to $\mathbf{x}_1 := \gamma(1)$. We write $\gamma(t) = (x_1(t), \mathbf{y}(t))$. The path $\pi \circ \gamma : t \mapsto x_1(t)$ is well defined over $[0, 1]$, and is a loop since $\pi(\mathbf{x}_0) = \pi(\mathbf{x}_1) \in \mathbb{C}^*$. Therefore it is not contractible in $\mathbb{C} \setminus \mathbf{B}^2$. We obtain

$$ \text{length}(\pi \circ \gamma) := \int_{0}^{1} |x'_{1}(t)| dt \le \text{length}(\gamma). $$

Since $\pi \circ \gamma$ is not contractible in $\mathbb{C} \setminus \mathbf{B}$ the following estimate is satisfied

$$ |x_1(0)| \gg 1 \implies |x_1(0)| \le \text{length}(\pi \circ \gamma) \le \text{length}(\gamma). $$

Combining this last estimate with Estimate (5.3), which implies that $|\mathbf{x}_1^a - \mathbf{x}_0^a|$ goes to 0 as $|x_1(0)|$ goes to $\infty$, shows that no representative of the germ $(Y^a, \lambda)$ can be locally LNE at infinity. $\square$

The next result presents the case of tangency to $\mathbf{H}_\infty$. Its proof will follow from arguments similar to those used for the transverse case (see also [8] for a similar situation).

**Lemma 5.4.** If $L$ is contained in $T_\lambda \mathbf{H}_\infty$, then the germ $(Y^a, \lambda)$ is not locally LNE at infinity.

*Proof.* In this case we have $(s, t) = (w_2, z)$. Using again parametrization (4.1) we find

$$ (Y, \lambda) = \{[1 : w_2^m : F(w_2)] = [1 : w_2^m : O(w_2^{m+1})] : w_2 \in (\mathbb{C}, 0)\}. $$

Since the curve germ is not contained in $\mathbf{H}_\infty$, there exist a positive integer $p \ge m + 1$, a function $\varphi \in \mathbb{C}\{w_2\}$ satisfying $\varphi(0) \ne 0$ such that

$$ z(w_2) = w_2^p \cdot \varphi(w_2), $$

writing $F = (\mathbf{w}', z) = (w_3, \dots, w_n, z)$. For each $j = 3, \dots, n$, there exists an exponent $q_j \in \mathbb{N}_{\ge m+1} \cup \infty$ and a function $\psi_j \in \mathbb{C}\{w_2\}$ such that

$$ w_j(w_2) = w_2^{q_j} \cdot \psi_j(w_2) $$

where $\psi_j(0) \ne 0$ if the function $w_j$ is not the null function, equivalently if $q_j < \infty$. Since $\varphi(0) \ne 0$, we can re-parametrize the affine part $(Y^a, \lambda)$ as

$$ (Y^a, \lambda) = \{[s^p : O(|s|^{p-m}) : 1] : s \in (\mathbb{C}, \infty)\}. $$

From the affine point of view, since $p - m > p - q_j$ for all $j$, we get

$$ (x_1, \mathbf{y}) \in Y^a \implies |\mathbf{y}| \le \text{const.} |x_1|^r \quad \text{for } r := \frac{p-m}{p} \in ]0, 1[ $$

when $|x_1|$ is large enough.

We can take a representative of $Y^a$ outside a large Euclidean closed ball $\mathbf{B}^{2n}$ centred at $0$ so that, by parametrization (4.1), the projection mapping

$$ \pi : Y^a \setminus \mathbf{B}^{2n} \to \mathbb{C} \setminus \mathbf{B}^2, \quad [x_1 : \mathbf{y} : 1] \mapsto x_1 $$
is a p-sheeted holomorphic covering whenever $|x_1|$ is large enough, where $\mathbf{B}^2$ is Euclidean ball of $\mathbb{C}$ centred at $\mathbf{0}$ image of $\mathbf{B}^{2n}$ under the projection onto the $x_1$-axis. Let $\mathbf{x}, \mathbf{x}'$ in $Y^a$ such that

$$\pi(\mathbf{x}) = \pi(\mathbf{x}') = x_1.$$

Since $p \ge m + 1 \ge 2$, we assume that $\mathbf{x} \ne \mathbf{x}'$. If $|x_1|$ is large enough, we find

$$|\mathbf{x}|, |\mathbf{x}'| = |x_1|(1 + o(1))$$

as well as the following estimate

$$|\mathbf{x} - \mathbf{x}'| = o(|x_1|) \ll |x_1|.$$

From here on, we conclude as in the proof of the case $m \ge 2$ of Lemma 5.3. $\square$

## 6. UNBOUNDED PART OF AFFINE CURVES

This section continues what was initiated in Section 5, but working with explicit representatives of germs of infinity.

**Convention 4.** A curve in $\mathbb{C}^n$ is an affine algebraic curve. A curve in $\mathbf{P}^n$ is a projective algebraic curve.

Let $X$ be a curve of $\mathbf{P}^n$ and let $X^\infty$ be the intersection $X \cap \mathbf{H}^\infty$. Let $\deg(X)$ be the degree of $X$.

**Remark 6.1.** 1) If $X$ has no irreducible component contained in $\mathbf{H}_\infty$, then $\deg(X) = \deg(X^a)$. 2) As a consequence of Bézout Theorem, if $\mathrm{card}(X^\infty) = \deg(X)$, then for each $\lambda \in X^\infty$, the germ $(X, \lambda)$ is non-singular at $\lambda$ and is transverse to $\mathbf{H}_\infty$ at $\lambda$.

We assume that $X^\infty$ consists of finitely many points $\lambda_1, \dots, \lambda_p$, with $p = \mathrm{card}(X^\infty) \le \deg(X)$.

The germ of the affine part $X^a$ at infinity is

$$(X^a, \infty) = \bigsqcup_{j=1}^p (X^a, \lambda_j).$$

Given any positive radius $R$, let

$$X_R := X \cap \mathbf{S}_R^{2n-1} = X^a \cap \mathbf{S}_R^{2n-1}.$$

Using the Euclidean inversion $\iota_{2n} : \mathbf{x} \mapsto |\mathbf{x}|^{-2} \cdot \mathbf{x}$, the Euclidean closure of the image of $\iota_{2n}(X^a \setminus \mathbf{0})$ is a real algebraic set of $\mathbb{R}^{2n} \supset \iota_{2n}(\mathbb{C}^n)$, with an isolated singularity at $\mathbf{0}$. The Local Conic Structure at $\mathbf{0}$ when combined with the inversion yields a Conical Structure Theorem at infinity, which states that there exists a positive radius $R_X$ such that for $R, R' \ge R_X$ the links $X_R$ and $X_{R'}$ are $C^\infty$ diffeomorphic. Moreover for $R \ge R_X$ the sub-manifold with boundary

$$X_{\ge R}^a := X^a \setminus B_R^{2n}$$

is $C^\infty$ diffeomorphic to the cylinder $X_R \times [R, +\infty[$, mapping $X_r$ onto $X_R \times r$. Let $\mathcal{C}$ be a connected component of $X_{\ge R}^a$. Therefore the closure of $\mathcal{C}$ in $\mathbf{P}^n$ intersects the hyperplane at infinity $\mathbf{H}_\infty$ in a single point $\lambda_\mathcal{C}$. Two such connected components $\mathcal{C}_1, \mathcal{C}_2$ may accumulate at the same point at infinity: $\lambda_{\mathcal{C}_1} = \lambda_{\mathcal{C}_2}$.

For $R \ge R_X$, let $\mathcal{C}_1^R, \dots, \mathcal{C}_e^R$, be the connected components of $X_{\ge R}^a$. Thus $\mathrm{card}(X^\infty) \le e \le \deg(X)$. A first key piece to the main result is the following

**Proposition 6.2.** Let $X^a$ be an affine curve and let $X$ be its projective closure. Assume that $\deg(X) = \mathrm{card}(X^\infty)$. Then, there exists a positive radius $R'_X \ge R_X$ such that for each $R \ge R'_X$, the connected components $\mathcal{C}_j^R$ are LNE, $j = 1, \dots, \deg(X)$.
**Proof.** Let $d := \deg(X)$. Let $i(X, \lambda)$ be the local intersection multiplicity of the germ $(X, \lambda)$ with $(\mathbf{H}_\infty, \lambda)$ (see [5]). Since $\mathbf{H}_\infty$ is non-singular, we deduce that $i(X, \lambda) \ge m(X, \lambda)$ for any $\lambda \in X^\infty$. The hypothesis means that

$$d = \sum_{\lambda \in X^\infty} i(X, \lambda),$$

and thus $1 = i(X, \lambda) \ge m(X, \lambda) \ge 1$. Therefore the curve germ $(X, \lambda)$ is non-singular and is in general position with $\mathbf{H}_\infty$ at $\lambda$ for each $\lambda \in X^\infty$, that is transverse to $\mathbf{H}_\infty$. Thus we are in the hypotheses of Lemma 5.3.

Let $X^\infty = \{\lambda_1, \dots, \lambda_d\}$. Assume that $(X^a, \lambda_j) = (C_j^R, \lambda_j)$ for each $j = 1, \dots, d$, and each $R \ge R_X$.

Let $L_j$ be the line tangent to $X$ at $\lambda_j$. Since it is transverse to $\mathbf{H}_\infty$ we can consider $L_j$ as an affine line of $\mathbb{C}^n$. Let $\mathbf{e}_j$ be a unit vector of $\mathbb{C}^n$ such that the point $\lambda_j \in \mathbf{H}_\infty$ corresponds to the complex line $\mathbb{C}\mathbf{e}_j$. Let $H_j$ be the complex hyperplane orthogonal to $\mathbb{C}\mathbf{e}_j$. Since $L_j$ is transverse to $\mathbf{H}_\infty$ there exists $\mathbf{a}_j \in H_j$ such that

$$L_j = \mathbb{C}\mathbf{e}_j + \mathbf{a}_j.$$

**Claim 6.3.** $C_j^R$ is bi-Lipschitz homeomorphic to $L_j \setminus B_R^{2n}$ with respect to the outer distance.

**Proof of the claim.** Similarly to the proof of Lemma 5.3, we deduce that $(C_j^R, \lambda_j)$ is a Lipschitz and holomorphic graph over $\mathbb{C} \setminus B_{r_j}^2 = \mathbb{C}\mathbf{e}_j \setminus B_{r_j}^2$ for each $j = 1, \dots, d$. Moreover the Lipschitz constant tends to 1 as $r_j$ goes to $\infty$. Let

$$r_X := \max_{j=1,\dots,d} r_j.$$

For each $R \ge r_X$, let

$$Y_j^R := \{(s, G_j(s)) = s\mathbf{e}_j + G_j(s) \in \mathbb{C}\mathbf{e}_j \times H_j : s \in \mathbb{C} \setminus B_R^2\}$$

where $G_j$ is the holomorphic mapping taking values in $\mathbb{C}^{n-1} = H_j$ we mentioned. We recall that

$$G_j(s) = g_j\left(\frac{1}{s}\right) \quad \text{where} \quad g_j(t) \in \mathbb{C}\{t\}^{n-1} \quad \text{and} \quad g_j(0) = \mathbf{a}_j.$$

Therefore, the following estimate holds true

$$\lim_{|s| \to \infty} G_j'(s) = \mathbf{0}.$$

Since $G_j$ goes to $\mathbf{a}_j$ and its derivative goes to $\mathbf{0}$ as $s$ goes to $\infty$, for every positive $\varepsilon$, there exists $r_\varepsilon \ge \max(R_X, r_X)$ such that

$$x_1, x'_1 \in \mathbb{C}\mathbf{e}_j \setminus B_{r_\varepsilon}^2 \implies |G_j(x_1) - G_j(x'_1)| < \varepsilon|x_1 - x'_1| \quad \text{and} \quad |G_j(x_1)|^2 < |\mathbf{a}_j|^2 + \varepsilon^2.$$

Thus the mapping $\Gamma_j : s \mapsto s\mathbf{e}_j + G_j(s)$ is holomorphic over $\mathbb{C} \setminus B_{r_\varepsilon}^2$ and is Lipschitz, with Lipschitz constant smaller than $(1 + \varepsilon)$.

Since $\Gamma_j$ is bi-holomorphic, up to taking a smaller $\varepsilon$ we find that $p_j := \Gamma_j^{-1}$, that is the projection onto $\mathbb{C}\mathbf{e}_j$, is Lipschitz with constant larger than $(1 - \varepsilon)$. Natural embedding of $\mathbb{C}\mathbf{e}_j \setminus B_{r_\varepsilon}^2$ into $\mathbb{C}^n$ yields the first part of the proof. Last, there is just to recall that the orthogonal projection onto an affine space is Lipschitz. $\square$

In the notation of the proof above, if $\varepsilon$ and $r_\varepsilon$ are given, we find that for each $j = 1, \dots, d$, the following inclusions are satisfied

$$Y_j^{R_{j,\varepsilon}} \subset C_j^{R_{j,\varepsilon}} \subset Y_j^R, \quad \text{where} \quad R_{j,\varepsilon} := \sqrt{R^2 + |\mathbf{a}_j|^2 + \varepsilon^2}$$

whenever $R \ge r_\varepsilon$. Under this choice of $\varepsilon$ we get

$$\partial C_j^{R_{j,\varepsilon}} \cap Y_j^{R_{j,\varepsilon}} = \partial Y_j^R \cap C_j^{R_{j,\varepsilon}} = \emptyset.$$
where $\partial C_j^R = C_j^R \cap S_R^{2n-1}$ and $\partial Y_j^R = \Gamma_j(L_j \cap S_R^{2n-1})$. We define the following smooth Jordan curve of $L_j$

$$Z_j^R := p_j(\partial C_j^R),$$

which is diffeomorphic to $S^1$. Let $D_j^R$ be the open disk it bounds. We cannot expect $D_j^R$ to be convex.

The following claim follows from Lemma 2.8.

**Claim 6.4.** $\mathbb{C}e_j \setminus D_j^R$ is LNE.

Since $B_R^2 \cap \mathbb{C}e_j \subset D_j^{R_{j,\varepsilon}}$ and the mapping $\Gamma_j$ is Lipschitz over $\mathbb{C}e_j \setminus B_R^2$, it is also Lipschitz over $\mathbb{C}e_j \setminus D_j^R$. The result follows from Proposition 2.10. This concludes the proof of Proposition 6.2. $\square$

There are two scenarios for which $\mathrm{card}(X^\infty) < \mathrm{deg}(X)$. The first one has already been mentioned in Lemma 5.4, when the germ $(X, \lambda)$ is irreducible and with tangent cone $L$ at $\lambda$ contained in the hyperplane at infinity. The second is when $(X, \lambda)$ is not irreducible.

**Lemma 6.5.** Let $X$ be a curve of $\mathbf{P}^n$. Let $\lambda$ be an isolated point of $X^\infty$. If the germ $(X, \lambda)$ is not irreducible, then $X^a$ is not LNE.

*Proof.* We can assume that $\lambda = [1:0:\dots:0]$. Let $\{(X_j, \lambda)\}_{j=1,\dots,N}$ be the irreducible components of $(X, \lambda)$. Fix $R \ge R_X$ so that $S := X^a \ge R$ is not connected. Note that

$$(X^a, \lambda) = (S, \lambda).$$

For each connected components $\mathcal{C}$ of $S$ accumulating at $\lambda$, there exists a unique $j \in \{1, \dots, N\}$ such that

$$(\mathcal{C}, \infty) = (X_j^a, \lambda).$$

Let $\mathcal{C}_1, \mathcal{C}_2$ be two connected components of $S$ whose germ at $\infty$ are $(X_1^a, \lambda), (X_2^a, \lambda)$, respectively. Let $(\mathbf{x}_l)_l \subset \mathcal{C}_1$ and $(\mathbf{x}'_l)_l \subset \mathcal{C}_2$ be two sequences converging to $\infty$. We further require that

$$\mathbf{x}_l = (l, \mathbf{x}_k) \quad \text{and} \quad \mathbf{x}'_l = (l, \mathbf{x}'_k).$$

Thus we obtain the following estimate

$$|\mathbf{x}_l - \mathbf{x}'_l| = o(l).$$

Let $\gamma_l$ be a rectifiable path on $X^a$ connecting $\mathbf{x}_l$ and $\mathbf{x}'_l$. Its length satisfies the following inequality

$$\mathrm{length}(\gamma_l) \ge \mathrm{dist}(\mathbf{x}_l, \partial S) + \mathrm{dist}(\mathbf{x}'_l, \partial S),$$

where $\partial S$ is the boundary of $S$, which is compact. For $l$ large enough we check that

$$\mathrm{dist}(\mathbf{x}_l, \partial S) \ge \frac{l}{2} \quad \text{and} \quad \mathrm{dist}(\mathbf{x}'_l, \partial S) \ge \frac{l}{2}.$$

Since for $l$ large enough we have found that

$$\mathrm{length}(\gamma_l) \ge l,$$

the subset $X^a$ cannot be LNE. $\square$

**Remark 6.6.** The demonstration of Lemma 6.5 still holds whenever a germ $(S, \infty)$ of a closed connected subset $S$ of $\mathbb{R}^q$ has two connected components at infinity, whose respective accumulation loci at infinity contain a common point.
## 7. BOUNDED PART OF AFFINE CURVES

We continue the investigations of Section 6, using the exact same objects, notations, and hypotheses to complete what was done there in looking more closely this time at the “bounded part” of $X^a$.

Let $X_{\text{sing}}$ be the singular locus of $X$, consisting at most of finitely many points. Observe that the affine part $X_{\text{sing}}^a$ of $X_{\text{sing}}$ is exactly $(X^a)_{\text{sing}}$ the set of singular points of the affine part $X^a$.

For $R \ge R_X$ the following subset

$$X_{\le R}^a := X^a \cap \mathbf{B}_R^{2n}$$

is a semi-algebraic subset of $\mathbb{C}^n$ with $C^\infty$ boundary $X_R$. If the affine part $X^a$ is connected, then for $R \ge R_X$ the subset $X_{\le R}^a$ is also connected.

The next result is another key piece of our main result and is similar to Proposition 4.6.

**Proposition 7.1.** Let $X$ be a projective curve such that its affine part $X^a$ is connected and not empty. For $R \ge R_X$ the subset $X_{\le R}^a$ is LNE if and only if $X^a$ is locally LNE at each point of $X_{\text{sing}}^a$.

*Proof.* Let $S := X_{\le R}^a$, which is semi-algebraically-path-connected. Since $S \times S$ is connected, compact and semi-algebraic (thus rectifiable), the function $d_{\text{inn}}^S$ admits a maximum $\ell_S$.

* The condition of $S$ being locally LNE at each point of $X_{\text{sing}}^a$ is certainly necessary to have $S$ LNE.

* Assume that $X^a$ is locally LNE at each point of $X_{\text{sing}}^a$.

Each point $\mathbf{x}$ of $X^a \setminus X_{\text{sing}}^a$ is one of the following two types: the germ $(S, \mathbf{x})$ is either bi-holomorphic to the germ $(\mathbb{C}, 0)$ if $|\mathbf{x}| < R$, or is $C^\infty$ diffeomorphic to the germ $(\mathbb{R} \times [0, \infty), 0)$ when $|\mathbf{x}| = R$. In both situations, $S$ is locally LNE at $\mathbf{x}$ by Corollary 2.7. Therefore $S$ is locally LNE at each of its points. Since it is compact, Lemma 2.6 yields that $S$ is LNE. $\square$

## 8. MAIN RESULT: HOW TO RECOGNIZE AN AFFINE LNE CURVE

We have gathered in the previous sections all the ingredients to prove the main result of the paper:

**Theorem 8.1.** Let $X$ be a connected projective curve of $\mathbf{P}^n$ of degree $\deg(X)$ such that $X^\infty$ is finite. The affine curve $X^a$ is LNE if and only if the following conditions are satisfied: (0) $X^a$ is connected; (i) $X^a$ is locally LNE at each of its singular points; (ii) The curve $X$ intersects $\mathbf{H}_\infty$ in exactly $\deg(X)$ points.

*Proof.* Assume $X^a$ is LNE. It is connected and it is locally LNE at each of its points, thus we obtained (0) and (i). Since Lemma 6.5 cannot happen, by Lemma 5.4 we deduce that at each $\lambda \in X^\infty$ the germ $(X, \lambda)$ is non-singular. Lemma 5.3 further implies that $(X, \lambda)$ is transverse to $\mathbf{H}_\infty$ at each $\lambda$ of $X^\infty$. Since $\deg(X)$ is the intersection number of $X$ with any projective hyperplane of $\mathbf{P}^n$, the transversality of $X$ and $\mathbf{H}_\infty$ at each of their intersection points yields $\mathrm{card}(X^\infty) = \deg(X)$.

Assume that conditions (0), (i), (ii) are satisfied.

Thus there exists $R'_X$ such that $X_{\le R}^a$ is connected for each $R \ge R'_X$. Since $X_R$ is a smooth compact sub-manifold once $R \ge R'_X$, hypothesis (i) and Proposition 7.1 guarantee that $X_{\le R}^a$ is LNE for each $R \ge R'_X$. Let

$$C_0 := X_{\le R}^a.$$

Let $d = \deg(X)$ and let $X^\infty = \{\lambda_j\}_{j=1,...,d}$. Condition (ii) implies that Proposition 6.2 is satisfied, that is, each connected component of $X_{\le R}^a$ is LNE whenever $R \ge R'_X$. Let $C_1, ..., C_d$, be the connected components of $X_{\le R}^a$, indexed such that $\lambda_j$ is the accumulation point at infinity of $C_j$.

For each $j = 0, ..., d$, let $d_j := d_{\text{inn}}^j$.
Since each $C_j$ is LNE, let $A$ be a positive constant which is a LNE constant for each $C_j$:

$$ \mathbf{x}, \mathbf{x}' \in C_j \implies d_j(\mathbf{x}, \mathbf{x}') \le A|\mathbf{x} - \mathbf{x}'|, \quad j = 0, \dots, d. $$

**Claim 8.2.** There exists a positive constant $A'$ such that for any $0 \le j < k \le d$ the following estimate holds true

$$ \mathbf{x}_j \in C_j, \mathbf{x}_k \in C_k \implies d_{\text{inn}}^{X^a}(\mathbf{x}_j, \mathbf{x}_k) \le A'|\mathbf{x}_j - \mathbf{x}_k|. $$

*Proof of the claim.* For $j \ge 1$, let $C_j$ be the complex line of $\mathbb{C}^n$ through the origin corresponding to complex line direction $\lambda_j$. Let $S_j := C_j \cap \mathbf{S}_1^{2n-1}$ be the unit circle of $C_j$ centred at the origin. Whenever $j \ne k$, the complex lines $C_j$ and $C_k$ only meet at $0$, therefore the intersection $S_j \cap S_k$ is empty. For any $1 \le j < k \le d$, let

$$ \delta_{j,k} := \operatorname{dist}(S_j, S_k) > 0, $$

the Euclidean distance between $S_j$ and $S_k$, and let

$$ \delta := \min_{1 \le j < k \le d} \delta_{j,k} > 0. $$

Since $C_j$ accumulates at $\lambda_j$ at infinity, we can assume that $R$ is large enough so that

$$ \sup \left\{ \operatorname{dist} \left( S_j, \frac{\mathbf{x}}{|\mathbf{x}|} \right) : \mathbf{x} \in C_j \right\} \le \frac{\delta}{4} $$

for each $j = 1, \dots, d$.

Let $\mathbf{x}_j \in C_j$ and $\mathbf{x}_k \in C_k$, for a given pair of indices $0 \le j < k \le d$. Let $\mathbf{u}_l \in \mathbf{S}^{2n-1}$ (when defined) and $r_l \ge 0$, for $l = j, k$, be defined as follows:

$$ \mathbf{u}_l := \frac{\mathbf{x}_l}{|\mathbf{x}_l|}, \quad \text{and} \quad r_l := |\mathbf{x}_l|. $$

Let $2\alpha \in [0, \pi]$ be the non-oriented angle between $\mathbf{u}_j$ and $\mathbf{u}_k$.

*Case 1:* $1 \le j < k \le d$.

Let $\mathbf{x}_k$ be a point of $\partial C_l \subset X_R^a$ which realizes the minimum of $d_l(\mathbf{x}_l, \partial C_l)$ for $l = j, k$.

By choice of $R$ and $j, k$ we have

$$ |\mathbf{u}_j - \mathbf{u}_k| = 2 \sin(\alpha) \ge \frac{\delta}{2}, $$

Assume that $r_j \ge r_k$, we recall identity (3.1)

$$ |\mathbf{x}_j - \mathbf{x}_k|^2 = (r_j + r_k)^2 \sin^2 \alpha + (r_j - r_k)^2 \cos^2 \alpha $$

from which we deduce again Estimates (3.6)

$$ \frac{\delta}{4} \cdot (r_k + r_j) \le |\mathbf{x}_j - \mathbf{x}_k|. $$

Since $r_j, r_k \ge R$, Estimate (3.2) yields the following estimate

$$ (8.1) \qquad |\mathbf{y}_j - \mathbf{y}_k| \le 2R \le \frac{4}{\delta} \cdot |\mathbf{x}_j - \mathbf{x}_k|. $$

as well as

$$ (8.2) \qquad |\mathbf{x}_l - \mathbf{x}_k| \le r_l + R \le \frac{4}{\delta} \cdot |\mathbf{x}_j - \mathbf{x}_k| \quad \text{for } l = j, k. $$

Since, we obviously have

$$ d_{\text{inn}}^{X^a}(\mathbf{x}_j, \mathbf{x}_k) \le d_j(\mathbf{x}_j, \mathbf{y}_j) + d_0(\mathbf{y}_j, \mathbf{y}_k) + d_k(\mathbf{y}_k, \mathbf{x}_k), $$

combining Estimates (8.1) and (8.2) with $C_0, C_j, C_k$ being LNE with LNE constant $A$ yield an expected inequality

$$ d_{\text{inn}}^{X^a}(\mathbf{x}_j, \mathbf{x}_k) \le \frac{12A}{\delta} \cdot |\mathbf{x}_j - \mathbf{x}_k|. $$
• Case 2: $j = 0$ and $1 \le k \le d$.

We can assume $r_0 < R < r_k$.

If $r_k \le 2R$, we know that $X_{\le 2R}^a$ is LNE.

Let $\mathbf{y}$ be a point of $\partial C_k$. When $r_k \ge 2R$, we find that

$$ \frac{1}{2}r_k \le |\mathbf{x}_k - \mathbf{x}_0| \le \frac{3}{2}r_k, \quad \text{and} \quad \frac{1}{2}r_k \le d_k(\mathbf{x}_k, \mathbf{y}) \le \frac{3A}{2}r_k. $$

For $r_k \ge 2R$, we therefore deduce the following estimate

$$ d_{\text{inn}}^{X^a}(\mathbf{x}_0, \mathbf{x}_k) \le d_0(\mathbf{x}_0, \mathbf{y}) + d_k(\mathbf{x}_k, \mathbf{y}) \le 2AR + \frac{3A}{2}r_k \le 5A|\mathbf{x}_k - \mathbf{x}_0|, $$

proving the statement in the second case.

Claim 8.2 establishes the desired LNE properties between any pair of points belonging to different subsets $C_j$. Since each of these subset is LNE, the result is proved. □

## 9. AFFINELY LNE VERSUS PROJECTIVELY LNE

The space $\mathbf{P}^n$ is naturally equipped with a Khäler metric, the Fubini-Study metric. As mentioned in point (ii) of Remark 2.4, any continuous Riemannian structure equipping $\mathbf{P}^n$, Hermitian or not, will yield a metric space structure equivalent to the one induced from the Fubini-Study metric. We will refer to any such continuous Riemannian metric on $\mathbf{P}^n$ as the projective metric $g_{\text{proj}}$.

Let $H$ be a hyperplane of $\mathbf{P}^n$, and let $C_H^n$ be the affine space $\mathbf{P}^n \setminus H$. We can embed $\mathbb{C}^n$ in $\mathbf{P}^n$ as $C_H^n$ for any a priori given hyperplane $H$ of $\mathbf{P}^n$ such that the embedding provides a unitary isomorphism between $\mathbb{C}^n$ and $C_H^n$.

As a subset of the projective space $\mathbf{P}^n$, the affine space $\mathbb{C}^n$ is equipped with the restriction of the projective metric $g_{\text{proj}}$, whereas as a complex vector space it is equipped with its canonical Hermitian structure $\text{eucl}$. The metric space structures respectively obtained from $(\mathbb{C}^n, g_{\text{proj}})$ and $(\mathbb{C}^n, \text{eucl})$ are clearly not equivalent, since $\mathbb{C}^n$ is bounded for $g_{\text{proj}}$ while it is not bounded for $\text{eucl}$. Yet the respective restrictions of $\text{eucl}$ and $g_{\text{proj}}$ to any given compact subset of $\mathbb{C}^n$ produce equivalent metric space structures over the compact subset, for the outer metric and thus also for the inner metric.

In this section, Theorem 9.4 establishes the relation between LNE property of affine curve w.r.t. the projective and euclidean metrics. We present some conventions first.

**Definition 9.1.** Let $X$ be a curve of $\mathbf{P}^n$ and let $H$ be a hyperplane of $\mathbf{P}^n$.

1) The curve $X \setminus H$ in $C_H^n$ is the affine trace of $X$ w.r.t. $H$.

2) The pair $X, H$ is in general position if the intersection $X \cap H$ consists only of non-singular points of $X$ at each of which $X$ and $H$ are transverse.

Here we forget about Convention 3, introducing in its stead the following

**Definition 9.2.** (i) A subset $S$ of $\mathbb{C}^n$ is LNE in $\mathbb{C}^n$ if it is LNE in $\mathbb{C}^n$ w.r.t. $\text{eucl}$.

(ii) A subset $S$ of $\mathbf{P}^n$ is LNE in $\mathbf{P}^n$ if it is LNE in $\mathbf{P}^n$ w.r.t. $g_{\text{proj}}$.

Observe that when the curve $X$ is LNE in $\mathbf{P}^n$, its affine trace $X \setminus H$ is not necessarily LNE in $C_H^n$ for any hyperplane $H$, as already suggested by Lemma 5.4 and Lemma 6.5.

**Example 9.3.** Let $\mathbf{P}^2$ be equipped with projective coordinates $\mathbf{z} = [x : y : z]$. Let $L$ be the hyperplane $\{z = 0\}$ and $\mathbb{C}^2 := \mathbb{C}_L^2$. Consider the non-singular quadrics

$$ P := \{\mathbf{z} : yz - x^2 = 0\} \quad \text{and} \quad H := \{\mathbf{z} : xy - z^2 = 0\}. $$
The parabola P and the hyperbola H are isomorphic via the unitary linear change of coordinates $[x : y : z] \to [z : y : x]$. Both curves are LNE in $\mathbb{P}^2$ since they are non-singular and irreducible. The affine traces w.r.t. L are

$$P \setminus L := \{(x, y) : y - x^2 = 0\} \text{ and } H \setminus L := \{(x, y) : xy - 1 = 0\}.$$

By Theorem 8.1, the hyperbola $H \setminus L$ is LNE in $\mathbb{C}^2$, while the parabola $P \setminus L$ is not LNE in $\mathbb{C}^2$. The difference being that $H, L$ are in general position while $P, L$ are not.

The next and penultimate result points at the very close interplay between a projective curve X and its general affine traces while investigating the LNE natures of X.

**Theorem 9.4.** Let X be a curve of $\mathbb{P}^n$. The following statements are equivalent:

(i) X is LNE in $\mathbb{P}^n$.

(ii) X is connected and, at each singular point $\lambda$ of X the germ $(X, \lambda)$ is a finite union of pairwise transverse non-singular curve germs.

(iii) For any hyperplane H of $\mathbb{P}^n$ such that the pair X, H is in general position, the variety $X \cup H$ is LNE in $\mathbb{P}^n$.

(iv) For any hyperplane H of $\mathbb{P}^n$ such that the pair X, H is in general position, the affine curve $X \setminus H$ is LNE in $\mathbb{C}_H^n$.

(v) There exists a hyperplane H of $\mathbb{P}^n$ in general position with X such that the affine curve $X \setminus H$ is LNE in $\mathbb{C}_H^n$.

*Proof.* The equivalence (i) and (ii) is simply Proposition 4.6.

Implication (ii) $\implies$ (iii) follows from Lemmas 3.1 and 2.6.

To obtain (iii) $\implies$ (i), note that if the union $X \cup H$ is LNE in $\mathbb{P}^n$ then it is locally LNE in $\mathbb{P}^n$ at each singular point of X. Thus Proposition 4.6 yields the implication.

To prove (i) $\implies$ (iv), we observe that whenever X and H are in general position, the intersection $H \cap X_{\text{sing}}$ is empty, and thus the affine trace $X \setminus H$ is connected. Therefore the affine trace $X \setminus H$ is LNE in $\mathbb{C}_H^n$ by point (ii) and Theorem 8.1.

Implication (iv) $\implies$ (v) is obvious.

To end the proof, let us show (v) $\implies$ (i). Let $\mathbb{C}^n := \mathbb{C}_H^n$ and let $X^a := X \setminus H$. By hypothesis any point of $X \cap H$ is a non-singular point of X, therefore $X_{\text{sing}} = (X^a)_{\text{sing}}$. Since $X^a$ is LNE in $\mathbb{C}^n$, it is locally LNE in $\mathbb{C}^n$ at each of its singular points, and so the affine curve $X^a$ contains a compact neighbourhood of each point of $X_{\text{sing}}$ which is LNE in $\mathbb{C}^n$. Thus X is locally LNE in $\mathbb{P}^n$ at each of its singular points. We conclude applying Proposition 4.6. $\square$

# 10. LIPSCHITZ CLASSIFICATION OF LNE CURVES

In this section we present a Lipschitz complete invariant of LNE affine curves.

**Definition 10.1.** Let $X_1^a$ in $\mathbb{C}^{n_1}$ and $X_2^a$ in $\mathbb{C}^{n_2}$ be two affine curves.

1) They are topologically equivalent if there exists a homeomorphism $X_1^a \to X_2^a$.

2) They are outer Lipschitz equivalent if there exist a bi-Lipschitz homeomorphism

$$(X_1^a, d_{X_1^a}) \to (X_2^a, d_{X_2^a}).$$

3) They are inner Lipschitz equivalent if there exist a bi-Lipschitz homeomorphism

$$(X_1^a, d_{\text{inn}}^{X_1^a}) \to (X_2^a, d_{\text{inn}}^{X_2^a}).$$
Obviously Lipschitz equivalence implies topological equivalence. Moreover, for LNE curves outer Lipschitz equivalence is the same as inner Lipschitz equivalence, thus in case of LNE curves we will speak simply of Lipschitz equivalence. We will show that in fact for LNE curves all three classifications coincide.

Let $X$ be an irreducible curve of $\mathbf{P}^n$ of degree $d \ge 1$. For each $\mathbf{y} \in X$, let $c(\mathbf{y})$ be the number of irreducible components of the curve germ $(X, \mathbf{y})$. The curve $X$ is bi-rationally equivalent to a compact Riemann surface $S_g$, which is a compact connected oriented surface of genus $g$. Therefore the Euler Characteristic of $X^a \setminus X_{\text{sing}}^a$ satisfies the following identity

$$ (10.1) \qquad \chi(X^a \setminus X_{\text{sing}}^a) = 2 - 2g - \sum_{\mathbf{y} \in X_{\text{sing}}^a} c(\mathbf{y}) - \sum_{\lambda \in X^{\infty}} c(\lambda). $$

Given the irreducible LNE curve $X^a$ of $\mathbb{C}^n$ with projective closure $X$, we define the following ordered object (encoding degree, number of singular points, genus, and multiplicities at singular points)

$$ DSGM(X) := (\deg(X); s(X); g(X); (\mathbf{y}, c(\mathbf{y}))_{\mathbf{y} \in X_{\text{sing}}}), $$

where $s(X) := \#X_{\text{sing}}$ and $g(X)$ is the genus of the surface $S_g$ resolved from $X$. When $s(X) = 0$, we suppress the last entry, that is $DSGM(X) = (\deg(X), 0, g(X))$. Two objects

$$ DSGM(X_i) = (d_i; s_i; g_i; (\mathbf{y}, c(\mathbf{y})))_{\mathbf{y} \in (X_i)_{\text{sing}}}, \quad i = 1, 2, $$

are equivalent if $d_1 = d_2$, $s_1 = s_2 =: s$, $g_1 = g_2$, and there exists a bijection $\varphi$ of $\{1, \dots, s\}$ such that $c(\mathbf{y}_j^1) = c(\mathbf{y}_{\varphi(j)}^2)$ for each $j = 1, \dots, s$, where $(X_i)_{\text{sing}} = \{\mathbf{y}_1^i, \dots, \mathbf{y}_s^i\}$ for $i = 1, 2$.

**Theorem 10.2.** The following statements are equivalent:

(a) Two irreducible complex affine algebraic LNE curves are Lipschitz equivalent.

(b) Two irreducible complex affine algebraic LNE curves are topologically equivalent.

(c) The DSGM invariants of the two irreducible complex affine algebraic LNE curves are equivalent.

*Proof.* Let $X_i^a$ be an irreducible LNE curve of $\mathbb{C}^{n_i}$, for $i = 1, 2$. Let $X_i$ be the projective closure of $X_i^a$ in $\mathbf{P}^{n_i}$. Denote

$$ d_i := \deg(X_i), \quad Y_i := (X_i)_{\text{sing}} = \{\mathbf{y}_i^1, \dots, \mathbf{y}_i^{s_i}\}, \quad \text{and} \quad X_i^{\infty} = \{\lambda_i^1, \dots, \lambda_i^{d_i}\}, $$

since $X_i^a$ is LNE by hypotheses implies that $\#X_i^\infty = d_i$.

For $\mathbf{y}_i \in Y_i$, let $T_{\mathbf{y}_i} X_i^a$ be the tangent cone of $X_i^a$ at $\mathbf{y}_i$.

Let $R \gg 1$ be a large radius, as in the proof of Lemma 3.1, such that for each $\mathbf{y}_i^j \in X_i^a$, with $i = 1, 2$ and $j = 1, \dots, s_i$, the subset

$$ C_i^j := X_i^a \cap B_i^j $$

is bi-Lipschitz homeomorphic to its truncated tangent cone

$$ T_{\mathbf{y}_i^j} X_i^a \cap B_i^j, $$

where $B_i^j$ is either the open ball $B^{2n}(\mathbf{y}_i^j, R^{-1})$ or its closure $\mathbf{B}^{2n}(\mathbf{y}_i^j, R^{-1})$. In particular the boundary $\partial C_i^j$ is a finite union of $c(\mathbf{y}_i^j)$ embedded $\mathbf{S}^1$. Similarly, for $i = 1, 2$ and by Claim 6.3, once $R$ is large enough, the subset

$$ \chi_i := X_i^a \setminus B_i, $$

is bi-Lipschitz homeomorphic to the truncated cone

$$ \widehat{X_i^\infty} \setminus B_i = (\bigcup_{j=1}^{d_i} \mathbb{C}\lambda_i^j) \setminus B_i, $$
where $\mathcal{B}_i$ is either the open ball $B_R^{2n}$ or its closure $\mathbf{B}_R^{2n}$. The boundary $\partial\mathcal{X}_i$ is a finite union of $d_i$ embedded $\mathbf{S}^1$.

Removing $N(\ge 1)$ open (resp. closed) disks from $S_g$, whose closures do not intersect, yields the compact (resp. open and bounded) connected surface $S_{g,N}$ with $N$ boundary components.

For $i = 1, 2$, let

$$N_i^a := d_i + \sum_{j=1}^{s_i} c(\mathbf{y}_i^j) \quad \text{and} \quad C_i^a := \mathcal{X}_i \cup (\cup_{j=1}^{s_i} C_i^j).$$

Therefore the compact (resp. open and bounded) surface with boundary $X_i^a \setminus C_i^a$ is smoothly diffeomorphic to the compact (resp. open and bounded) surface with boundary $S_{g_i,N_i^a}$.

Obviously, (a) implies (b). We will show that (b) implies (c) implies (a).

• Assume (b). Let $h: X_1^a \to X_2^a$ be a homeomorphism. Since $X_i^a$ is LNE, a point $\mathbf{y}_i$ of $X_i^a$ is singular if and only if $c(\mathbf{y}_i) \ge 2$. Since $h$ maps each germ

$$(X_1^a \setminus \mathbf{y}, \mathbf{y}) \to (X_2^a \setminus h(\mathbf{y}), h(\mathbf{y}))$$

we deduce that

$$h(Y_1) = Y_2, \text{ and } c(\mathbf{y}) = c(h(\mathbf{y})) \quad \forall \mathbf{y} \in Y_1.$$

For $R$ large enough, the number of connected components $X_i \setminus B_R^{2n}$ are equal for $i = 1, 2$, that is $d_1 = d_2$. Thus we deduce that $g_1 = g_2$ by (10.1). Condition (c) is then satisfied.

• Assume (c). Let $d := d_1 = d_2$ and $g := g_1 = g_2$, $s := s_1 = s_2$. For each $i = 1, 2$ and each $j = 1, \dots, s$ we can assume without loss of generality that $c(\mathbf{y}_1^j) = c(\mathbf{y}_2^j)$.

We deduce $X_1^a \setminus C_1^a$ and $X_2^a \setminus C_2^a$ are both diffeomorphic to $S_{g,N_i^a}$ for $N_i^a := N_1^a = N_2^a$.

Since $X_1^a$ and $X_2^a$ are both LNE, we further deduce that $\mathcal{X}_1$ and $\mathcal{X}_2$ are bi-Lipschitz homeomorphic. So are $C_1^j$ and $C_2^j$ for each $j = 1, \dots, s$.

We can glue all these pieces together to $S_{g,N_i^a} = X_i \setminus C_i^a$ along the boundary $\partial C_i^a$ to produce a bi-Lipschitz homeomorphism $(X_1^a, d_{X_1^a}) \to (X_2^a, d_{X_2^a})$. Thus (a) is verified. $\square$

As a straightforward consequence of Theorem 10.2 we obtain the full Lipschitz classification of LNE curves:

**Corollary 10.3.** Let $X_1$ be a LNE complex algebraic curve of $\mathbb{C}^{n_1}$ and $X_2$ be a LNE complex algebraic curve of $\mathbb{C}^{n_2}$. Then $X_1$ and $X_2$ are bi-Lipschitz homeomorphic if and only if they are homeomorphic.

We obtain in our LNE setting the following

**Corollary 10.4.** Let $X^a$ be a LNE curve of $\mathbb{C}^n$ with $n \ge 3$. There exists a Zariski dense open subset of linear projections $\mathbb{C}^n \to \mathbb{C}^2$ mapping $X^a$ onto a LNE curve of $\mathbb{C}^2$.

*Proof.* The linear projections $\mathbb{C}^n \to \mathbb{C}^2$, identified with their kernels, consist of the Grassmann manifold $\mathbf{G}(n-2, n)$ of the $(n-2)$ planes of $\mathbb{C}^n$. Consider the Gauss mapping of $X^a$:

$$\tau: X^a \setminus X_{\text{sing}} \to \mathbf{G}(1, n), \quad \mathbf{x} \to T_{\mathbf{x}}X.$$

The closure $G_0(X)$ of $\tau(X^a \setminus X)$ is either a curve of $\mathbf{G}(1, n)$ or a finite set of points. Let

$$G(X) := G_0(X) \cup (\cup_{\lambda \in X^\infty} \mathbb{C}\lambda)$$

Thus $G(X)$ is a sub-variety of $\mathbf{G}(1, n)$ of dimension 0 or 1. Consider the following sub-variety

$$V := \{(\lambda, \lambda', P) \in G(X) \times G(X) \times \mathbf{G}(n-2, n) : \dim_{\mathbb{C}} \mathbb{C}\lambda + \mathbb{C}\lambda' + P \le n-1\}$$
We check that $\dim_{\mathbb{C}} V = 2n-5$ and that $W$, the Zariski closure of the projection of $V$ onto $\mathbf{G}(n-2, n)$, has also dimension $2n-5$. Therefore

$$ \Omega(X) := \mathbf{G}(n-2, n) \setminus W $$

is Zariski open and dense. Let $\pi_P : \mathbb{C}^n \to \mathbb{C}^2 = P^{\perp}$ be the orthogonal projection onto the orthogonal $P^{\perp}$ of $P \in \mathbf{G}(n-2, n)$. By definition of $\Omega(X)$, we check that

$$ X_P^a := \pi_P(X^a) $$

has exactly $d = \deg(X^a)$ points at infinity at each of which $X_P^a$ is transverse with the line at infinity. Therefore the degree of $X_P^a$ is $d$. Let $\mathbf{z}$ be a point of $X_P^a_{\text{sing}}$ and let $\pi_P^{-1}(\mathbf{z}) = \{\mathbf{x}_1, \dots, \mathbf{x}_q\}$. For each $i=1, \dots, q$, the number of distinct lines in the tangent cone of $X$ at $\mathbf{x}_i$ is $c(\mathbf{x}_i)$. The definition of $\Omega(X)$ implies that the tangent cone of $X_P^a$ at $\mathbf{z}$ will consist of $\sum_{i=1}^q c(\mathbf{x}_i)$ distinct lines. Since $X$ is LNE and $P \in \Omega(X)$, the germ $(X_P^a, \mathbf{z})$ is a finite union of pairwise transverse non-singular curve germs. Thus $X_P^a$ is locally LNE at $\mathbf{z}$. Since $X^a$ is connected so is $X_P^a$. Theorem 8.1 shows that $X_P^a$ is LNE. $\square$

To end the section let us remark that the above classification allows more types than represented by plane algebraic curves.

**Corollary 10.5.** There exist LNE curves of $\mathbb{C}^n$, with $n \ge 3$, which are not Lipschitz equivalent to a plane curve. In particular, there are LNE curves of $\mathbb{C}^n$ not Lipschitz equivalent to any of their plane projections.

*Proof.* Indeed, for each $n \ge 3$ and each positive integer $d$, there is a non-negative integer $p(d, n)$, polynomial in $d$, such that for every $g$ such that $0 \le g \le p(d, n)$, there is an irreducible non-singular (and non-degenerate) curve of $\mathbf{P}^n$ with degree $d$ and genus $g$ (see [10, 2]). Thus, there exist non-singular space curves with $2g \ne (d-1)(d-2)$. The general affine trace of such a projective curve is LNE by Theorem 9.4. By Theorem 10.2, such an affine trace cannot be Lipschitz equivalent to a non-singular plane curve, which must satisfy the equality $2g = (d-1)(d-2)$. It can neither be equivalent to any of its plane projections since the previous genus argument implies that these must have singular points. The claim follows from Theorem 10.2. $\square$

Note that generic affine traces of non-singular irreducible projective curves with genus zero and degree $d \ge 3$ are the simplest examples of LNE curves which are not Lipschitz equivalent to any plane curve, since $DSGM = (d, 0, 0)$ cannot be realized by a plane curve, which is opposite to the results of [22, 7] in the local case.

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A. COSTA, CENTRO DE CIÊNCIAS E TECNOLOGIAS, UNIVERSIDADE ESTADUAL DO CEARÁ, CAMPUS DO ITAPERI, 60.714-903 FORTALEZA - CE, BRASIL

Email address: andrecosta.math@gmail.com

V. GRANDJEAN, DEPARTAMENTO DE MATEMÁTICA, UNIVERSIDADE FEDERAL DE SANTA CATARINA, CAMPUS UNIVERSITÁRIO TRINDADE CEP 88.040-900 FLORIANÓPOLIS-SC, BRASIL

Email address: vincent.grandjean@ufsc.br

M. MICHALSKA, WYDZIAL MATEMATYKI I INFORMATYKI, UNIWERSYTET ŁÓDZKI, BANACHA 22, 90-238 ŁÓDŹ, POLAND

Email address: maria.michalska@wmii.uni.lodz.pl