# Characterization of Lipschitz Normally Embedded complex curves

André Costa  ,  Vincent Grandjean  and  Maria Michalska  A. Costa, Centro de
Ciências e tecnologias, Universidade Estadual do Ceará, Campus do Itaperi,
60.714-903 Fortaleza - CE, Brasil<EMAIL_ADDRESS>V. Grandjean,
Departamento de Matemática, Universidade Federal de Santa Catarina, Campus
Universitário Trindade CEP 88.040-900 Florianópolis-SC, Brasil
<EMAIL_ADDRESS>M. Michalska, Wydział Matematyki i Informatyki,
Uniwersytet Łódzki, Banacha 22, 90-238 Łódź, Poland
<EMAIL_ADDRESS>
###### 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.

###### Key words and phrases:

Complex curve, inner and outer distance, Lipschitz normally embedded

###### 2010 Mathematics Subject Classification:

14H50, 32B10, 51F99

###### Contents

  1. 1 Introduction
  2. 2 Geometric Preliminaries
  3. 3 A criterion for isolated singularity germs
  4. 4 The local case
  5. 5 Preliminaries at infinity
  6. 6 Unbounded part of affine curves
  7. 7 Bounded part of affine curves
  8. 8 Main Result: how to recognize an affine LNE curve
  9. 9 Affinely LNE versus Projectively LNE
  10. 10 Lipschitz classification of LNE curves

##  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
$(S,d_{\rm inn}^{S})$ 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 $(S,d_{\rm inn}^{S})$ need not be metrically
equivalent, thus introduced the property: there exists a positive constant $L$
such that

$(P)\hfill{\bf s},{\bf s}^{\prime}\in S\;\Longrightarrow\;d_{\rm inn}^{S}({\bf
s},{\bf s}^{\prime})\;\leq\;L\cdot d_{S}({\bf s},{\bf s}^{\prime}),\hfill$

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

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 $(X,{\bf 0})$ 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 $\mathbb{K}^{n}$ are LNE, both
when $\mathbb{K}=\mathbb{C}$ and $\mathbb{K}=\mathbb{R}$. The only non-trivial
example we are aware of is from [11]: it has non-isolated singularities, but
is a $\mathbb{K}$-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 ${\mathbb{C}^{n}}$ 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 ${\mathbb{C}^{n}}$ 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 ${\bf x}_{0}$ of
$X$, the germ $(X,{\bf x}_{0})$ 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 ${\bf x}_{0}$ of $X$, the germ
$(X,{\bf x}_{0})$ 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
$\mathbb{C}_{H}^{n}:=\mathbb{CP}^{n}\setminus H$ is LNE in
$\mathbb{C}_{H}^{n}$.
(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 $\mathbb{C}_{H}^{n}$.

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
$(\mathbb{C}^{n_{1}},{\bf 0})\to(\mathbb{C}^{n_{1}},{\bf 0})$ mapping
$(X_{1},{\bf 0})$ to $(X_{2},{\bf 0})$, 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. $n_{1}=n_{2}\geq 3$.
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 $|-|$. Let $eucl$ be the
Euclidean/Hermitian metric tensor over ${\mathbb{K}^{n}}$.

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

######  Convention 1.

A Riemannian manifold $(M,g_{M})$ consists of a $C^{k}$ or
$\mathbb{K}$-analytic manifold $M$, for $k\in\mathbb{N}_{\geq 1}\cup\infty$,
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

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

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

Observe that $d_{S}\leq d_{\rm inn}^{S}$. 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

$\displaystyle{{\bf x},{\bf x}^{\prime}\in
X\;\Longrightarrow\;\frac{1}{L}\cdot d_{1}({\bf x},{\bf
x}^{\prime})\;\leq\;d_{2}({\bf x},{\bf x}^{\prime})\;\leq\;L\cdot d_{1}({\bf
x},{\bf x}^{\prime}).}$

An elementary question arising from presenting these two definitions is when
are the metric spaces $(S,d_{S})$ and $(S,d_{\rm inn}^{S})$ 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 $(S,d_{S})\to(S,d_{\rm
inn}^{S})$ is bi-Lipschitz: there exists a positive constant $L$ such that

${\bf s},{\bf s}^{\prime}\;\in\;S\;\Longrightarrow\;d_{\rm inn}^{S}({\bf
s},{\bf s}^{\prime})\;\leq\;L\cdot d_{S}({\bf s},{\bf s}^{\prime}).$

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 ${\bf 0}$ of positive radius $r$ is a LNE
representative of the germ $(\mathbb{R}^{2},{\bf 0})$, while
$\\{(x,y):y^{2}-(x-r)^{3}>0\\}\cap B_{2r}^{2}$ 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 ${\bf s}$ of ${\rm clos}(S)$, the closure of $S$ in $M$, if there exists
an open neighbourhood $\mathcal{U}$ of ${\bf s}$ in $M$ such that
$S\cap\mathcal{U}$ is LNE.

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

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 ${\bf s}$ of $S$, let $\mathcal{U}_{\bf s}$ be a neighbourhood
of ${\bf s}$ in $M$ such that $U_{\bf s}:=S\cap\mathcal{U}_{\bf s}$ is LNE.
Thus $S$ is covered by $U_{1},\ldots,U_{N},$ where $U_{i}:=U_{{\bf s}_{i}}$
for some points ${\bf s}_{1},\ldots,{\bf s}_{N}$ of $S$. Let $L_{0}$ be a LNE
constant of $U_{1},\ldots,U_{N}$. Consider the following function

$f:S\times S\setminus\Delta\to\mathbb{R},\;\;({\bf s},{\bf
s}^{\prime})\mapsto\frac{d_{\rm inn}^{S}({\bf s},{\bf s}^{\prime})}{d_{S}({\bf
s},{\bf s}^{\prime})},\;\;{\rm where}\;\;\Delta:=\\{({\bf s},{\bf s})\in
S\times S\\}.$

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 $({\bf s},{\bf s}^{\prime})$ in
$V\setminus\Delta$, thus there exists $i$ such that $({\bf s},{\bf
s}^{\prime})\in U_{i}\times U_{i}$. Therefore $f({\bf s},{\bf s}^{\prime})\leq
L_{0}$. Thus $S$ is LNE with LNE constant $\max(L_{0},L_{1})$. ∎

We recall that a subset $S$ of a $C^{k}$ manifold $M$, $k\geq 1$, 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.

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

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

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

${\bf y},{\bf y}^{\prime}\in N\cap B_{r}^{m}({\bf x})\;\Longrightarrow\;|{\bf
y}-{\bf y}^{\prime}|\leq A\cdot|\phi_{\bf x}^{\partial}({\bf y})-\phi_{\bf
x}^{\partial}({\bf y}^{\prime})|\;\leq\;A^{2}\cdot|{\bf y}-{\bf y}^{\prime}|.$

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

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

${\bf y},{\bf y}^{\prime}\in N\cap B_{r}^{m}({\bf x})\;\Longrightarrow\;|{\bf
y}-{\bf y}^{\prime}|\geq|p_{\bf x}({\bf y})-p_{\bf x}({\bf
y}^{\prime})|\;\geq\;\frac{1}{2}|{\bf y}-{\bf y}^{\prime}|.$

Let $Z:=\pi_{\bf x}(N)$. Therefore $(Z,{\bf x})$ is a $C^{1}$ sub-manifold
possibly with boundary of $T_{\bf x}$ and thus $Z$ is locally LNE at ${\bf x}$
w.r.t. the Euclidean metric of $T_{\bf x}$ by the same argument as in the
previous case. Since $T_{\bf x}$ is still an euclidean space for the
restriction of the Euclidean metric of $\mathbb{R}^{m}$, there exists a
positive radius $r^{\prime}$ such that $Z\cap B_{r^{\prime}}^{m}({\bf x})$ is
LNE since $T_{\bf x}\cap B_{r^{\prime}}^{m}({\bf x}^{\prime})$ is an Euclidean
ball of $T_{\bf x}$ of radius $r^{\prime}$. Since $p_{\bf x}$ is bi-Lipschitz
we deduce that $p_{\bf x}^{-1}(Z\cap B_{r^{\prime}}^{m}({\bf x}))$ is LNE
whenever $2r^{\prime}\leq r$.

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

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

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 $B_{1},\ldots,B_{s},$ be the connected components of ${\partial}E$. Define
the positive number

${\delta}:=\min_{1\leq i<j\leq s}{\rm dist}(B_{i},B_{j}),$

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

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

  * i)

${\bf x}={\bf x}_{1}^{0}\leq{\bf x}_{1}^{1}<{\bf x}_{2}^{0}\leq\ldots<{\bf
x}_{p}^{0}\leq{\bf x}_{p}^{1}={\bf x}^{\prime}$;

  * ii)

For each $k=1,\ldots,p$, there exists $i_{k}\geq 1$ such that ${\bf
x}_{k}^{0},{\bf x}_{k}^{1}\in B_{i_{k}}$;

  * iii)

For each $k=1,\ldots,p-1$, the interval $J^{k}$ is contained in $I\setminus
N$;

  * iv)

$i_{k}\neq i_{k+1}$ for each $k=1,\ldots,p-1$;

  * 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 $[{\bf x}_{k}^{0},{\bf x}_{k}^{1}]$, where
$k=1,\ldots,p$.

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

$d_{\rm inn}^{E}({\bf x}_{k}^{0},{\bf x}_{k}^{1})\leq d_{\rm
inn}^{B_{i_{k}}}({\bf x}_{k}^{0},{\bf x}_{k}^{1})\leq A\,|{\bf x}_{k}^{0}-{\bf
x}_{k}^{1}|.$

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

$d_{\rm inn}^{E}({\bf x}_{k}^{1},{\bf x}_{k+1}^{0})=|{\bf x}_{k}^{1}-{\bf
x}_{k+1}^{0}|\leq A\,|{\bf x}_{k}^{1}-{\bf x}_{k+1}^{0}|.$

We deduce

$d_{\rm inn}^{E}({\bf x},{\bf x}^{\prime})\leq\sum_{k=1}^{p}d_{\rm
inn}^{E}({\bf x}_{k}^{0},{\bf x}_{k}^{1})+\sum_{k=1}^{p-1}d_{\rm inn}^{E}({\bf
x}_{k}^{1},{\bf x}_{k+1}^{0})\leq A|{\bf x}-{\bf x}^{\prime}|,$

and the lemma is proved. ∎

######  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 $K_{F}$:

${\bf s},{\bf s}^{\prime}\;\in\;S\;\Longrightarrow\;d_{P}(F({\bf s}),F({\bf
s}^{\prime}))\;\leq\;K_{F}\cdot d_{S}({\bf s},{\bf s}^{\prime})=K_{F}\cdot
d_{M}({\bf s},{\bf s}^{\prime}).$

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

${\bf s},{\bf s}^{\prime}\;\in\;S\;\Longrightarrow\;d_{P}(F({\bf s}),F({\bf
s}^{\prime}))\;\leq\;K_{F}\cdot d_{S}({\bf s},{\bf s}^{\prime}).$

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

${\rm length}_{M\times P}({\gamma}_{F})\;\leq(1+K_{F})\cdot{\rm
length}_{M}({\gamma})$

where ${\rm length}_{Q}$ is the length taken w.r.t. to the Riemannian metric
$g_{Q}$, for $Q=M,\,M\times P$. ∎

######  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 ${\bf 0}$ 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
$S\setminus{\bf 0}$.

######  Lemma 3.1.

Let $(N_{1},{\bf 0}),\ldots,(N_{s},{\bf 0}),$ be germs at ${\bf 0}$ of $C^{1}$
sub-manifolds of ${\mathbb{R}^{n}}$ of positive dimensions and codimensions.
Assume that $(Y,{\bf 0}):=\cup_{j=1}^{s}(N_{j},{\bf 0})$ has an isolated
singularity at ${\bf 0}$.
The germ $(Y,{\bf 0})$ is locally LNE at ${\bf 0}$ if and only if $T_{\bf
0}N_{j}\cap T_{\bf 0}N_{k}=\\{{\bf 0}\\}$ for each $1\leq j<k\leq s$.

###### Proof.

For each $j=1,\ldots,s,$ let $T_{j}:=T_{\bf 0}N_{j}$ and let $T_{j}^{\perp}$
be the orthogonal supplement of $T_{j}$. Since each $(N_{j},{\bf 0})$ 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 $(N_{j},{\bf 0})$ is the graph $\Gamma_{j}$ of
$G_{j}$. Since $D_{\bf 0}G_{j}={\bf 0}$ and $G$ is $C^{1}$, up to shrinking
$r_{Y}$ the mapping $G_{j}$ is Lipschitz over ${\bf B}_{r}^{n}$ for any radius
$0<r\leq r_{Y}$, with Lipschitz constant tending to $0$ as $r$ goes to $0$.
For $j=1,\ldots,s,$ we define

$N_{j}^{{\leq r}}:={\Gamma}_{j}\cap{\bf B}_{r}^{n}.$

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

${\partial}N_{j}^{{\leq r}}={\Gamma}_{j}\cap{\bf S}_{r}^{n-1}.$

Each ${\partial}N_{j}^{{\leq r}}$ is diffeomorphic to ${\bf S}^{\dim
T_{j}-1}$.

Given a pair $1\leq j<k\leq s$, and $r\leq r_{Y}$, let ${\bf
x}_{l}\in(N_{l}^{{\leq r}})^{*}$ for $l=j,k$. Let $r_{l}>0$ be defined as
follows:

$r_{l}:=|{\bf x}_{l}|.$

Let $2{\alpha}\in[0,\pi]$ be the (non-oriented) angle between ${\bf x}_{j}$
and ${\bf x}_{k}$. From plane geometry classes we recall that

$\left|\frac{{\bf x}_{j}}{r_{j}}-\frac{{\bf
x}_{k}}{r_{k}}\right|=2\sin{\alpha}$

and further remember that

(3.1) $|{\bf x}_{j}-{\bf
x}_{k}|^{2}=(r_{j}-r_{k})^{2}\cos^{2}{\alpha}+(r_{j}+r_{k})^{2}\sin^{2}{\alpha}.$

By definition of ${\alpha}$, we deduce that

(3.2) $(r_{j}+r_{k})\sin{\alpha}\;\leq\;|{\bf x}_{j}-{\bf
x}_{k}|\;\leq\;r_{j}+r_{k}.$

Finally let $Y_{{\leq r}}:=Y\cap{\bf B}_{r}^{2n}$, $d_{\rm inn}^{{\leq
r}}:=d_{\rm inn}^{Y_{{\leq r}}}$ and $d_{j}^{{\leq r}}:=d_{\rm
inn}^{N_{j}^{{\leq r}}}$, for $1\leq j\leq s$.

$\bullet$ Assume that $T_{j}\cap T_{k}={\bf 0}$, for each pair $j\neq k$.

For $j=1,\ldots,s,$ we define the link of $T_{j}$ at ${\bf 0}$

$S_{j}:=T_{j}\cap{\bf S}_{1}^{n-1}.$

By hypothesis, we get

(3.3) ${\delta}:=\min_{1\leq j<k\leq w}{\rm dist}(S_{j},S_{k})>0.$

We can require that $r_{Y}$ is small enough such that for each $j=1,\ldots,s,$
and each ${\bf x}_{j}\in(N_{j}^{\leq r_{Y}})^{*}$ the followings estimates
hold true

(3.4) ${\rm dist}\left(S_{j},\frac{{\bf
x}_{j}}{r_{j}}\right)\;\leq\;\frac{{\delta}}{4}\cdot$

For each $1\leq j<k\leq s$, each ${\bf x}_{j}\in N_{j}^{*}$, ${\bf x}_{k}\in
N_{k}^{*}$, we also find the following estimates

(3.5) $\left|\frac{{\bf x}_{j}}{r_{j}}-\frac{{\bf
x}_{k}}{r_{k}}\right|\geq\frac{{\delta}}{2}\cdot$

By choice of $r_{Y}$ and $j,k$ we have

$\left|\frac{{\bf x}_{j}}{r_{j}}-\frac{{\bf
x}_{k}}{r_{k}}\right|=2\sin({\alpha})\;\geq\;\frac{{\delta}}{2}.$

From Estimate (3.2), we deduce

(3.6) $\frac{{\delta}}{4}\cdot(r_{k}+r_{j})\leq|{\bf x}_{j}-{\bf x}_{k}|.$

as well as

(3.7) $|{\bf x}_{l}|=r_{l}\leq\;\frac{4}{{\delta}}\cdot|{\bf x}_{j}-{\bf
x}_{k}|\;\;{\rm for}\;\;l=j,k.$

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

$d_{\rm inn}^{{\leq r}}({\bf x}_{j},{\bf x}_{k})\leq d_{j}^{{\leq r}}({\bf
x}_{j},{\bf 0})+d_{k}^{{\leq r}}({\bf 0},{\bf x}_{k}),$

combining Estimate (3.7) with $N_{j}^{{\leq r}}$ and $N_{k}^{{\leq r}}$ being
LNE, each with LNE constant $A$, yield an expected inequality

$d_{\rm inn}^{{\leq r}}({\bf x}_{j},{\bf
x}_{k})\;\leq\;\frac{8A}{{\delta}}\cdot|{\bf x}_{j}-{\bf x}_{k}|.$

Since each $N_{j}^{{\leq r}}$ is LNE, we obtain that $Y^{{\leq r}}$ is LNE.

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

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

$j<k\;\Longrightarrow\;(N_{j}^{\leq r_{Y}})^{*}\cap(N_{k}^{\leq
r_{Y}})^{*}=\emptyset.$

Given any pair $1\leq j<k\leq s$, observe that to connect ${\bf x}_{j}$ to
${\bf x}_{k}$ within $Y^{\leq r_{Y}}$ it is necessary to go through ${\bf 0}$.
Therefore

(3.8) $d_{\rm inn}^{Y}({\bf x}_{j},{\bf x}_{k})\geq r_{j}+r_{k}$

Let $E:=\\{\\{i_{1},\ldots,i_{t}\\}$ such that $1\leq i_{1}<\ldots<i_{t}\leq
s,$ with $t\geq 2$ and $\dim\cap_{j=1}^{t}T_{i_{j}}\geq 1\\}$.

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 ${\bf p}\in P_{J}$ and each $j\in J$, let ${\bf x}_{j}$ be the point
$({\bf p},G_{j}({\bf p}))$ of $N_{j}$. Since $P_{J}$ is tangent at ${\bf 0}$
to each $N_{l}$ and $D_{\bf 0}G_{l}$ is null, for any $l\in J$, as ${\bf p}$
goes to ${\bf 0}$ in $P_{J}$ we find that

$|{\bf x}_{j}|,|{\bf x}_{k}|=|{\bf p}|+o(|{\bf p}|)\;\;{\rm and}\;\;|{\bf
x}_{j}-{\bf x}_{k}|=o(|{\bf p}|)$

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

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 $(Y,{\bf 0})$ of Lemma 3.1. If
$(Y,{\bf 0})$ is LNE, there exists a positive radius $r_{Y}$ such that for
each radius $r\in(0,r_{Y}]$, the subsets $Y\cap{\bf B}_{r}^{n}$ and $Y\cap
B_{r}^{n}$ 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 $(Y,{\bf 0})$ be a curve germ of ${\mathbb{C}^{n}}$ at ${\bf
0}\in{\mathbb{C}^{n}}$. Let $m(Y,{\bf 0})$ be the multiplicity of $Y$ at ${\bf
0}$. We recall that $Y$ is non-singular at ${\bf y}_{0}$ if and only if
$m(Y,{\bf y}_{0})=1$.

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

$L:=\\{{\bf
x}=(x_{1},\ldots,x_{n})\in{\mathbb{C}^{n}}\,:\,x_{2}=\ldots=x_{n}=0\\}=\mathbb{C}\times{\bf
0}.$

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

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

More precisely, writing $F=(f_{2},\ldots,f_{n})$, for each $j=2,\ldots,n,$ we
know that

$f_{j}(s)=s^{a_{j}}\phi_{j}(s),\;\;{\rm with}\;\;\phi_{j}(0)\neq 0,\;\;{\rm
and}\;\;a_{j}\geq m+1$

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

Let $(X,{\bf 0})$ be a curve germ of $({\mathbb{C}^{n}},{\bf 0})$. Let
$(X_{1},{\bf 0}),\ldots,(X_{s},{\bf 0}),$ be its local irreducible components.
For each $j=1,\ldots,s,$ let $L_{j}$ be the tangent cone of $X_{j}$ at ${\bf
0}$, thus a complex line through ${\bf 0}$.

For each $j=1,\ldots,s,$ the (orthogonal) projection $\pi_{j}:(X_{j},{\bf
0})\to(L_{j},{\bf 0})$ is a complex finite mapping and it induces the germ of
a holomorphic cover $(X_{j}^{*},{\bf 0})\to(L_{j}^{*},{\bf 0})$ with
$m_{j}:=m(X_{j},{\bf 0})\geq 1$ 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 $(X,{\bf 0})$ of $({\mathbb{C}^{n}},{\bf 0})$
admits a representative which is locally LNE at ${\bf 0}$ if and only if the
following conditions are satisfied: (i) $m_{j}=1$ for each $j=1,\ldots,s$, and
(ii) $L_{j}\cap L_{k}=\\{{\bf 0}\\}$ for each pair $(j,k)$ such that $1\leq
j<k\leq s$. In other words, the curve germ $(X,{\bf 0})$ is LNE if and only if
it is a union of non-singular curve germs intersecting pairwise transversally.

###### Proof.

Let $\lambda_{1},\ldots,\lambda_{t},$ be complex lines of ${\mathbb{C}^{n}}$
through the origin. We define the following “angle”

${\delta}((\lambda_{j})_{j=1,\ldots,t}):=\min_{1\leq j<k\leq t}\left\\{|{\bf
u}_{j}-{\bf u}_{k}|:{\bf u}_{j}\in\lambda_{j}\cap{\bf S}_{1}^{2n-1}\;{\rm
and}\;{\bf u}_{k}\in\lambda_{k}\cap{\bf S}_{1}^{2n-1}\right\\}.$

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

$\bullet$ Assume that (i) and (ii) are satisfied.

Condition (i) means that each $(X_{j},{\bf 0})$ 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.

$\bullet$ Assume that $X$ admits a representative locally LNE at ${\bf 0}$.

First we show the following property.

######  Claim 4.3.

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

###### Proof.

For simplicity let $m:=m(Y,{\bf 0})$ and assume that $L=\mathbb{C}\times{\bf
0}\subset{\mathbb{C}^{n}}$. 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 $Y^{*}$

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

is a $m$-sheeted holomorphic covering. Let $x_{1}\neq 0$ and let ${\bf
y}_{1},\ldots,{\bf y}_{m}$ be the points of $\pi^{-1}(x_{1})$.

Assume that $m\geq 2$. Since ${\bf y}_{j}=(x_{1},{\upsilon}_{j})$ for each
$j=1,\ldots,m$, using Puiseux Parametrization (4.1) for each irreducible
component of $(Y,{\bf 0})$, we check there exists a positive integer $p$ such
that

$|{\upsilon}_{j}|\leq const|x_{1}|^{\frac{p+1}{p}},\;\;\forall\;1\leq j\leq m$

where $const$ is a positive constant we do not need to denote specifically.

If to connect the points ${\bf y}_{1}$ and ${\bf y}_{2}$ in $Y$ we go through
${\bf 0}$, we find.

(4.2) $d_{\rm inn}^{X}({\bf y}_{1},{\bf y}_{2})\geq|{\bf y}_{1}|+|{\bf
y}_{2}|.$

When a rectifiable path ${\gamma}:[0,1]\to Y^{*}$ connects the points ${\bf
y}_{1}$ and ${\bf y}_{2}$, 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

$2|x_{1}|\leq{\rm length}({\gamma}_{1})<{\rm length}({\gamma}).$

In any case we have obtained that $d_{\rm inn}^{Y}({\bf y}_{1},{\bf
y}_{2})\geq 2|x_{1}|$, while $|{\bf y}_{1}-{\bf y}_{2}|=o(|x_{1}|)$. Therefore
the germ $(Y,0)$ cannot admit a representative locally LNE at ${\bf 0}$. ∎

Claim 4.3 implies that $m_{j}=1$ for each $j=1,\ldots,s,$ and also that the
tangent cones $L_{j}$ are pairwise transverse. ∎

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 $U_{1}:=\psi^{-1}({\bf B}_{1}^{m})$ is a
compact sub-manifold with boundary of $\mathcal{U}$ and of $M$. Let

${\bf e}_{1}:=\psi^{*}(eucl|_{{\bf B}_{1}^{m}})$

the pull-back of the Euclidean metric restricted to ${\bf B}_{1}^{m}$. Since
$\psi_{1}:=\psi|_{U_{1}}:(U_{1},d_{U_{1}})\to({\bf B}_{1}^{m},|-|)$ is bi-
Lipschitz (it is $C^{1}$), the metric space structures obtained from the
Riemannian metrics $h_{1}:=g_{M}|_{U_{1}}$ and ${\bf e}_{1}$ 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 ${\bf
x}_{0}\in\mathcal{U}$ if and only if $\psi(\mathcal{U})$ is locally LNE at
${\bf 0}$ (w.r.t. $eucl$).

###### Proof.

We can assume that $\psi({\bf x}_{0})={\bf 0}$. The previous observation about
$h_{1}$ and ${\bf e}_{1}$, the compactness of $U_{1}$ and Lemma 2.6 altogether
imply the statement. ∎

######  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 $X_{\rm sing}$ 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 $X_{\rm sing}:=\\{{\bf
x}_{1},\ldots,{\bf x}_{s}\\}$. For each $j=1,\ldots,s,$ 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 ${\bf x}_{j}$ such that $\psi_{j}({\bf
x}_{j})={\bf 0}$. 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 $r\leq r_{j}$, the curve

$Y_{j}\cap{\bf S}_{r}^{2m-1}$

is a non-singular real analytic compact curve consisting of $d_{j}\geq 1$
connected components (see [14] for instance). Let

$r_{X}:=\min_{j=1,\ldots,s}r_{j}.$

We can assume that each intersection ${\rm clos}(\mathcal{U}_{k})\cap{\rm
clos}(\mathcal{U}_{l})$ is empty for any pair $k,l$ with $k\neq l$. From Lemma
4.4, we can also assume that for each $j=1,\ldots,s,$ and for each positive
radius $r\leq r_{X}$ the compact subset

$X_{j,r}:=X\cap\psi_{j}^{-1}({\bf B}_{r}^{2m})$

is LNE. By choice of $r_{X}$ we deduce that for each positive radius $r\leq
r_{X}$, 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

${\partial}E_{r}=\cup_{j=1}^{s}(X\cap\psi_{j}^{-1}({\bf S}_{r}^{2m-1})).$

Thus any connected component of $E_{r}$ is LNE. Observe that for any
$r\leq\frac{2r_{X}}{3}$ the subset

$T_{j,r}:=X\cap\psi_{j}^{-1}({\bf B}_{r}^{2m}\setminus B_{\frac{r}{2}}^{2m})$

is a compact real analytic sub-manifold with boundary, consisting of $d_{j}$
connected components, each diffeomorphic to the cylinder ${\bf
S}^{1}\times[\frac{r}{2},r]$. 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 $r\leq\frac{2}{3}r_{X}$, $X$ is locally
LNE at each of its points by Lemma 4.4. We conclude the proof applying again
Lemma 2.6. ∎

##  5\. Preliminaries at infinity

Let ${\mathbb{C}^{n}}$ be the complex affine space of dimension $n$ and let
${\bf x}$ 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 ${\mathbb{C}^{n}}\setminus{\bf B}_{R}^{2n}$ are LNE.

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

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

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

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

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

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

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

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

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

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) $L=\\{[1:w_{2}:0:\ldots:0]\,:\,w_{2}\in\mathbb{C}\\}.$

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

$L=\\{[1:z{\bf a}:z]\,:\,z\in\mathbb{C}\\}.$

The affine translation ${\bf x}\to(x_{1},{\bf y}-{\bf a})$ in
${\mathbb{C}^{n}}$, corresponding to the linear change of coordinates $({\bf
w},z)\to({\bf w}^{\prime},z)=({\bf v},z)=({\bf w}-z{\bf a},z)$ in
$\mathcal{A}_{1}$, shows that the line $L$ has the following equation

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

Let $(s,{\bf w}^{\prime},t)$ be either $(w_{2},{\bf w}^{\prime},z)$ or
$(z,{\bf w}^{\prime},w_{2})$ for $w_{2},z,$ as appearing in the simplified
equations (5.1) and (5.2) of $L$, so that $L=\\{t={\bf w}^{\prime}=0\\}$.

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

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

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 ${\bf H}_{\infty}$. 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

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

for a positive integer $p\geq m+1$, 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 $(x_{1},{\bf y})\in(Y^{a},\infty)$, then ${\bf y}\to{\bf 0}$ as
$x_{1}\to\infty$. Let us consider the following holomorphic map germ
$(\mathbb{C},\infty)\to(\mathbb{C}^{n-1},0)$

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

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

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

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

${\bf y}=G(x_{1}).$

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

$\bullet$ Assume $m\geq 2$. Let $z\in\mathbb{C}^{*}$ and let ${\zeta}$ be a
$m$-th root of $z$. Let ${\bf x}_{0}:=[1:F({\zeta}):z]$ and ${\bf
x}_{1}:=[1:F(\omega{\zeta}):z]$ for $\omega=e^{i\frac{2\pi}{m}}$. By
parametrization (4.1), we find $F({\zeta})\neq F(\omega{\zeta})$. Writing

${\bf
x}_{0}^{a}:=\left(\frac{1}{z},\frac{F({\zeta})}{z}\right)\in{\mathbb{C}^{n}}\;\;{\rm
and}\;\;{\bf
x}_{1}^{a}:=\left(\frac{1}{z},\frac{F(\omega{\zeta})}{z}\right)\in{\mathbb{C}^{n}}$

for $|z|$ small enough we obtain

(5.3) $|{\bf x}_{1}^{a}-{\bf
x}_{0}^{a}|=\left|\frac{F({\zeta})}{z}-\frac{F(\omega{\zeta})}{z}\right|\leq
const.|z|^{\frac{p}{m}-1}=const.\frac{1}{|x_{1}|^{q}}\;\;{\rm
for}\;\;q:=\frac{p-m}{m}>0.$

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

$\pi:Y^{a}\to\mathbb{C}\setminus{\bf B}^{2},\;\;[x_{1}:{\bf y}:1]\mapsto
x_{1}$

is a $m$-sheeted holomorphic covering, where ${\bf B}^{2}$ is an Euclidean
ball of $\mathbb{C}$ centred at ${\bf 0}$ image of ${\bf B}^{2n}$ under the
projection onto the $x_{1}$-axis.

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

${\rm length}(\pi\circ{\gamma}):=\int_{0}^{1}|x_{1}^{\prime}(t)|{\rm
d}t\,\leq\,{\rm length}({\gamma}).$

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

$|x_{1}(0)|\gg 1\;\Longrightarrow\;|x_{1}(0)|\;\leq\;{\rm
length}(\pi\circ{\gamma})\;\leq\;{\rm length}({\gamma}).$

Combining this last estimate with Estimate (5.3), which implies that $|{\bf
x}_{1}^{a}-{\bf x}_{0}^{a}|$ 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. ∎

The next result presents the case of tangency to ${\bf H}_{\infty}$. 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 $T_{\lambda}{\bf H}_{\infty}$, 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

$(Y,\lambda)=\left\\{[1:w_{2}^{m}:F(w_{2})]=[1:w_{2}^{m}:O(w_{2}^{m+1})]\,:\,w_{2}\in(\mathbb{C},0)\right\\}.$

Since the curve germ is not contained in ${\bf H}_{\infty}$, there exist a
positive integer $p\geq m+1$, a function $\varphi\in\mathbb{C}\\{w_{2}\\}$
satisfying $\varphi(0)\neq 0$ such that

$z(w_{2})=w_{2}^{p}\cdot\varphi(w_{2}),$

writing $F=({\bf w}^{\prime},z)=(w_{3},\ldots,w_{n},z)$. For each
$j=3,\ldots,n,$ there exists an exponent $q_{j}\in\mathbb{N}_{\geq
m+1}\cup\infty$ 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 $\psi_{j}(0)\neq 0$ if the function $w_{j}$ is not the null function,
equivalently if $q_{j}<\infty$. Since $\varphi(0)\neq 0$, we can re-
parametrize the affine part $(Y^{a},\lambda)$ as

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

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

$(x_{1},{\bf y})\in Y^{a}\;\Longrightarrow\;|{\bf y}|\leq
const.|x_{1}|^{r}\;\;{\rm for}\;\;r:=\frac{p-m}{p}\in]0,1[$

when $|x_{1}|$ is large enough.

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

$\pi:Y^{a}\setminus{\bf B}^{2n}\to\mathbb{C}\setminus{\bf
B}^{2},\;\;[x_{1}:{\bf y}:1]\mapsto x_{1}$

is a $p$-sheeted holomorphic covering whenever $|x_{1}|$ is large enough,
where ${\bf B}^{2}$ is Euclidean ball of $\mathbb{C}$ centred at ${\bf 0}$
image of ${\bf B}^{2n}$ under the projection onto the $x_{1}$-axis. Let ${\bf
x},{\bf x}^{\prime}$ in $Y^{a}$ such that

$\pi({\bf x})=\pi({\bf x}^{\prime})=x_{1}.$

Since $p\geq m+1\geq 2$, we assume that ${\bf x}\neq{\bf x}^{\prime}$. If
$|x_{1}|$ is large enough, we find

$|{\bf x}|,\,|{\bf x}^{\prime}|=|x_{1}|(1+o(1))$

as well as the following estimate

$|{\bf x}-{\bf x}^{\prime}|=o(|x_{1}|)\ll|x_{1}|.$

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

##  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 ${\bf
P}^{n}$ is a projective algebraic curve.

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

######  Remark 6.1.

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

We assume that $X^{\infty}$ consists of finitely many points
$\lambda_{1},\ldots,\lambda_{p},$ with $p=card(X^{\infty})\leq\deg(X)$.

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

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

Given any positive radius $R$, let

$X_{R}:=X\cap{\bf S}_{R}^{2n-1}=X^{a}\cap{\bf S}_{R}^{2n-1}.$

Using the Euclidean inversion $\iota_{2n}:{\bf x}\mapsto|{\bf
x}|^{-2}\cdot{\bf x}$, the Euclidean closure of the image of
$\iota_{2n}(X^{a}\setminus{\bf 0})$ is a real algebraic set of
$\mathbb{R}^{2n}\supset\iota_{2n}({\mathbb{C}^{n}})$, with an isolated
singularity at ${\bf 0}$. The Local Conic Structure at ${\bf 0}$ 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
$R,R^{\prime}\geq R_{X}$ the links $X_{R}$ and $X_{R^{\prime}}$ are
$C^{\infty}$ diffeomorphic. Moreover for $R\geq R_{X}$ the sub-manifold with
boundary

$X_{{\geq R}}^{a}:=X^{a}\setminus B_{R}^{2n}$

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 $X_{{\geq R}}^{a}$. Therefore the closure of $\mathcal{C}$ in
${\bf P}^{n}$ intersects the hyperplane at infinity ${\bf H}_{\infty}$ 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 $R\geq R_{X}$, let $\mathcal{C}_{1}^{R},\ldots,\mathcal{C}_{e}^{R},$ be
the connected components of $X_{{\geq R}}^{a}$. Thus $card(X^{\infty})\leq
e\leq\deg(X)$. 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 $\deg(X)=card(X^{\infty})$. Then, there exists a positive radius
$R_{X}^{\prime}$ $\geq R_{X}$ such that for each $R\geq R_{X}^{\prime}$, the
connected components $\mathcal{C}_{j}^{R}$ are LNE, $j=1,\ldots,\deg(X)$.

###### Proof.

Let $d:=\deg(X)$. Let $i(X,\lambda)$ be the local intersection multiplicity of
the germ $(X,\lambda)$ with $({\bf H}_{\infty},\lambda)$ (see [5]). Since
${\bf H}_{\infty}$ is non-singular, we deduce that $i(X,\lambda)\geq
m(X,\lambda)$ for any $\lambda\in X^{\infty}$. The hypothesis means that

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

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

Let $X^{\infty}=\\{\lambda_{1},\ldots,\lambda_{d}\\}$. Assume that
$(X^{a},\lambda_{j})=(\mathcal{C}_{j}^{R},\lambda_{j})$ for each
$j=1,\ldots,d,$ and each $R\geq R_{X}$.

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

$L_{j}=\mathbb{C}{\bf e}_{j}+{\bf a}_{j}.$

######  Claim 6.3.

$\mathcal{C}_{j}^{R}$ 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
$(\mathcal{C}_{j}^{R},\lambda_{j})$ is a Lipschitz and holomorphic graph over
$\mathbb{C}\setminus B_{r_{j}}^{2}=\mathbb{C}{\bf e}_{j}\setminus
B_{r_{j}}^{2}$ for each $j=1,\ldots,d$. Moreover the Lipschitz constant tends
to $1$ as $r_{j}$ goes to $\infty$. Let

$r_{X}:=\max_{j=1,\ldots,d}r_{j}.$

For each $R\geq r_{X}$, let

$Y_{j}^{R}:=\\{(s,G_{j}(s))=s{\bf e}_{j}+G_{j}(s)\in\mathbb{C}{\bf
e}_{j}\times H_{j}:s\in\mathbb{C}\setminus B_{R}^{2}\\}$

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

$G_{j}(s)=g_{j}\left(\frac{1}{s}\right)\;\;{\rm
where}\;\;g_{j}(t)\in\mathbb{C}\\{t\\}^{n-1}\;\;{\rm and}\;\;g_{j}(0)={\bf
a}_{j}.$

Therefore, the following estimate holds true

$\lim_{|s|\to\infty}\;G_{j}^{\prime}(s)={\bf 0}.$

Since $G_{j}$ goes to ${\bf a}_{j}$ and its derivative goes to ${\bf 0}$ as
$s$ goes to $\infty$, for every positive $\varepsilon$, there exists
$r_{\varepsilon}\geq\max(R_{X},r_{X})$ such that

$x_{1},x_{1}^{\prime}\in\mathbb{C}{\bf e}_{j}\setminus
B_{r_{\varepsilon}}^{2}\;\Longrightarrow\;|G_{j}(x_{1})-G_{j}(x_{1}^{\prime})|<\varepsilon|x_{1}-x_{1}^{\prime}|\;\;{\rm
and}\;\;|G_{j}(x_{1})|^{2}<|{\bf a}_{j}|^{2}+\varepsilon^{2}.$

Thus the mapping ${\Gamma}_{j}:s\mapsto s{\bf e}_{j}+G_{j}(s))$ 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
$\mathbb{C}{\bf e}_{j}$, is Lipschitz with constant larger than
$(1-\varepsilon)$. Natural embedding of $\mathbb{C}{\bf e}_{j}\setminus
B_{r_{\varepsilon}}^{2}$ 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. ∎

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

$Y_{j}^{R_{j,\varepsilon}}\;\subset\;\mathcal{C}_{j}^{R_{j,\varepsilon}}\;\subset\;Y_{j}^{R},\;\;{\rm
where}\;\;R_{j,\varepsilon}:=\sqrt{R^{2}+|{\bf a}_{j}|^{2}+\varepsilon^{2}}$

whenever $R\geq r_{\varepsilon}$. Under this choice of $\varepsilon$ we get

${\partial}\mathcal{C}_{j}^{R_{j,\varepsilon}}\cap
Y_{j}^{R_{j,\varepsilon}}={\partial}Y_{j}^{R}\cap\mathcal{C}_{j}^{R_{j,\varepsilon}}=\emptyset.$

where ${\partial}\mathcal{C}_{j}^{R}=\mathcal{C}_{j}^{R}\cap{\bf
S}_{R}^{2n-1}$ and ${\partial}Y_{j}^{R}={\Gamma}_{j}(L_{j}\cap{\bf
S}_{R}^{2n-1})$. We define the following smooth Jordan curve of $L_{j}$

$Z_{j}^{R}:=p_{j}({\partial}\mathcal{C}_{j}^{R}),$

which is diffeomorphic to ${\bf S}^{1}$. 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.

$\mathbb{C}{\bf e}_{j}\setminus D_{j}^{R}$ is LNE.

Since ${\bf B}_{R}^{2}\cap\mathbb{C}{\bf e}_{j}\subset
D_{j}^{R_{j,\varepsilon}}$ and the mapping ${\Gamma}_{j}$ is Lipschitz over
$\mathbb{C}{\bf e}_{j}\setminus B_{R}^{2}$, it is also Lipschitz over
$\mathbb{C}{\bf e}_{j}\setminus D_{j}^{R}$. The result follows from
Proposition 2.10. This concludes the proof of Proposition 6.2. ∎

There are two scenarii for which $card(X^{\infty})<\deg(X)$. 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 ${\bf P}^{n}$. 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 $\lambda=[1:0:\ldots:0]$. Let
$\left\\{(X_{j},\lambda)\right\\}_{j=1,\ldots,N}$ be the irreducible
components of $(X,\lambda)$. Fix $R\geq R_{X}$ so that $S:=X_{{\geq R}}^{a}$
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 $j\in\\{1,\ldots,N\\}$ 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 $({\bf x}_{l})_{l}\subset\mathcal{C}_{1}$ and $({\bf
x}_{l}^{\prime})_{l}\subset\mathcal{C}_{2}$ be two sequences converging to
$\infty$. We further require that

${\bf x}_{l}=(l,{\bf x}_{k})\;\;{\rm and}\;\;{\bf x}_{l}^{\prime}=(l,{\bf
x}_{k}^{\prime}).$

Thus we obtain the following estimate

$|{\bf x}_{l}-{\bf x}_{l}^{\prime}|=o(l).$

Let ${\gamma}_{l}$ be a rectifiable path on $X^{a}$ connecting ${\bf x}_{l}$
and ${\bf x}_{l}^{\prime}$. Its length satisfies the following inequality

${\rm length}({\gamma}_{l})\;\geq\;{\rm dist}({\bf x}_{l},{\partial}S)+{\rm
dist}({\bf x}_{l}^{\prime},{\partial}S),$

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

${\rm dist}({\bf x}_{l},{\partial}S)\geq\frac{l}{2}\;\;{\rm and}\;\;{\rm
dist}({\bf x}_{l}^{\prime},{\partial}S)\geq\frac{l}{2}.$

Since for $l$ large enough we have found that

${\rm length}({\gamma}_{l})\;\geq\;l,$

the subset $X^{a}$ cannot be LNE. ∎

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

For $R\geq R_{X}$ the following subset

$X_{{\leq R}}^{a}:=X^{a}\cap{\bf B}_{R}^{2n}$

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 $R\geq R_{X}$ the
subset $X_{{\leq R}}^{a}$ 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 $R\geq R_{X}$ the subset $X_{{\leq R}}^{a}$ is LNE if and
only if $X^{a}$ is locally LNE at each point of $X_{\rm sing}^{a}$.

###### Proof.

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

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

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

Each point ${\bf x}$ of $X^{a}\setminus X_{\rm sing}^{a}$ is one of the
following two types: the germ $(S,{\bf x})$ is either bi-holomorphic to the
germ $(\mathbb{C},0)$ if $|{\bf x}|<R$, or is $C^{\infty}$ diffeomorphic to
the germ $(\mathbb{R}\times[0,\infty),{\bf 0})$ when $|{\bf x}|=R$. In both
situations, $S$ is locally LNE at ${\bf x}$ 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. ∎

##  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 ${\bf P}^{n}$ 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 ${\bf H}_{\infty}$ 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 ${\bf H}_{\infty}$ at each $\lambda$ of $X^{\infty}$. Since
$\deg(X)$ is the intersection number of $X$ with any projective hyperplane of
${\bf P}^{n}$, the transversality of $X$ and ${\bf H}_{\infty}$ at each of
their intersection points yields $card(X^{\infty})=\deg(X)$.

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

Thus there exists $R_{X}^{\prime}$ such that $X_{{\leq R}}^{a}$ is connected
for each $R\geq R_{X}^{\prime}$. Since $X_{R}$ is a smooth compact sub-
manifold once $R\geq R_{X}^{\prime}$, hypothesis $(i)$ and Proposition 7.1
guarantee that $X_{{\leq R}}^{a}$ is LNE for each $R\geq R_{X}^{\prime}$. Let

$\mathcal{C}_{0}:=X_{{\leq R}}^{a}.$

Let $d=\deg(X)$ and let $X^{\infty}=\\{\lambda_{j}\\}_{j=1,\ldots,d}$.
Condition $(ii)$ implies that Proposition 6.2 is satisfied, that is, each
connected component of $X_{{\geq R}}^{a}$ is LNE whenever $R\geq
R_{X}^{\prime}$. Let $\mathcal{C}_{1},\ldots,\mathcal{C}_{d},$ be the
connected components of $X_{{\geq R}}^{a}$, indexed such that $\lambda_{j}$ is
the accumulation point at infinity of $\mathcal{C}_{j}$.

For each $j=0,\ldots,d,$ let $d_{j}:=d_{\rm inn}^{\mathcal{C}_{j}}$.

Since each $\mathcal{C}_{j}$ is LNE, let $A$ be a positive constant which is a
LNE constant for each $\mathcal{C}_{j}$:

${\bf x},{\bf x}^{\prime}\in\mathcal{C}_{j}\;\Longrightarrow\;d_{j}({\bf
x},{\bf x}^{\prime})\;\leq\;A|{\bf x}-{\bf x}^{\prime}|,\;\;j=0,\ldots,d.$

######  Claim 8.2.

There exists a positive constant $A^{\prime}$ such that for any $0\leq j<k\leq
d$ the following estimate holds true

${\bf x}_{j}\in\mathcal{C}_{j},{\bf
x}_{k}\in\mathcal{C}_{k}\;\Longrightarrow\;d_{\rm inn}^{X^{a}}({\bf
x}_{j},{\bf x}_{k})\leq A^{\prime}|{\bf x}_{j}-{\bf x}_{k}|.$

###### Proof of the claim.

For $j\geq 1$, let $\mathbb{C}_{j}$ be the complex line of ${\mathbb{C}^{n}}$
through the origin corresponding to complex line direction $\lambda_{j}$. Let
$S_{j}:=\mathbb{C}_{j}\cap{\bf S}_{1}^{2n-1}$ be the unit circle of
$\mathbb{C}_{j}$ centred at the origin. Whenever $j\neq k$, the complex lines
$\mathbb{C}_{j}$ and $\mathbb{C}_{k}$ only meet at ${\bf 0}$, therefore the
intersection $S_{j}\cap S_{k}$ is empty. For any $1\leq j<k\leq d$, let

${\delta}_{j,k}:={\rm dist}(S_{j},S_{k})>0,$

the Euclidean distance between $S_{j}$ and $S_{k}$, and let

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

Since $\mathcal{C}_{j}$ accumulates at $\lambda_{j}$ at infinity, we can
assume that $R$ is large enough so that

$\sup\left\\{{\rm dist}\left(S_{j},\frac{{\bf x}}{|{\bf x}|}\right):{\bf
x}\in\mathcal{C}_{j}\right\\}\;\leq\;\frac{{\delta}}{4}$

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

Let ${\bf x}_{j}\in\mathcal{C}_{j}$ and ${\bf x}_{k}\in\mathcal{C}_{k}$, for a
given pair of indices $0\leq j<k\leq d$. Let ${\bf u}_{l}\in{\bf S}^{2n-1}$
(when defined) and $r_{l}\geq 0$, for $l=j,k,$ be defined as follows:

${\bf u}_{l}:=\frac{{\bf x}_{l}}{|{\bf x}_{l}|},\;\;{\rm and}\;\;r_{l}:=|{\bf
x}_{l}|.$

Let $2{\alpha}\in[0,\pi]$ be the non-oriented angle between ${\bf u}_{j}$ and
${\bf u}_{k}$.

$\bullet$ Case 1: $1\leq j<k\leq d$.

Let ${\bf x}_{k}$ be a point of ${\partial}\mathcal{C}_{l}\subset X_{R}^{a}$
which realizes the minimum of $d_{l}({\bf x}_{l},{\partial}\mathcal{C}_{l})$
for $l=j,k$.

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

$|{\bf u}_{j}-{\bf u}_{k}|=2\sin({\alpha})\;\geq\;\frac{{\delta}}{2},$

Assume that $r_{j}\geq r_{k}$, we recall identity (3.1)

$|{\bf x}_{j}-{\bf
x}_{k}|^{2}=(r_{j}+r_{k})^{2}\sin^{2}{\alpha}+(r_{j}-r_{k})^{2}\cos^{2}{\alpha}$

from which we deduce again Estimates (3.6)

$\frac{{\delta}}{4}\cdot(r_{k}+r_{j})\leq|{\bf x}_{j}-{\bf x}_{k}|.$

Since $r_{j},r_{k}\geq R$, Estimate (3.2) yields the following estimate

(8.1) $|{\bf y}_{j}-{\bf y}_{k}|\;\leq\;2R\;\leq\;\frac{4}{{\delta}}\cdot|{\bf
x}_{j}-{\bf x}_{k}|.$

as well as

(8.2) $|{\bf x}_{l}-{\bf
x}_{k}|\;\leq\;r_{l}+R\;\leq\;\frac{4}{{\delta}}\cdot|{\bf x}_{j}-{\bf
x}_{k}|\;\;{\rm for}\;\;l=j,k.$

Since, we obviously have

$d_{\rm inn}^{X^{a}}({\bf x}_{j},{\bf x}_{k})\leq d_{j}({\bf x}_{j},{\bf
y}_{j})+d_{0}({\bf y}_{j},{\bf y}_{k})+d_{k}({\bf y}_{k},{\bf x}_{k}),$

combining Estimates (8.1) and (8.2) with
$\mathcal{C}_{0},\mathcal{C}_{j},\mathcal{C}_{k}$ being LNE with LNE constant
$A$ yield an expected inequality

$d_{\rm inn}^{X^{a}}({\bf x}_{j},{\bf
x}_{k})\;\leq\;\frac{12A}{{\delta}}\cdot|{\bf x}_{j}-{\bf x}_{k}|.$

$\bullet$ Case 2: $j=0$ and $1\leq k\leq d$.

We can assume $r_{0}<R<r_{k}$.

If $r_{k}\leq 2R$, we know that $X_{\leq 2R}^{a}$ is LNE.

Let ${\bf y}$ be a point of ${\partial}\mathcal{C}_{k}$. When $r_{k}\geq 2R$,
we find that

$\frac{1}{2}r_{k}\;\leq\;|{\bf x}_{k}-{\bf
x}_{0}|\;\leq\;\frac{3}{2}r_{k},\;\;{\rm
and}\;\;\frac{1}{2}r_{k}\;\leq\;d_{k}({\bf x}_{k},{\bf
y})\;\leq\;\frac{3A}{2}r_{k}.$

For $r_{k}\geq 2R$, we therefore deduce the following estimate

$d_{\rm inn}^{X^{a}}({\bf x}_{0},{\bf x}_{k})\;\leq\;d_{0}({\bf x}_{0},{\bf
y})+d_{k}({\bf x}_{k},{\bf y})\;\leq\;2AR+\frac{3A}{2}r_{k}\;\leq\;5A|{\bf
x}_{k}-{\bf x}_{0}|,$

proving the statement in the second case. ∎

Claim 8.2 establishes the desired LNE properties between any pair of points
belonging to different subsets $\mathcal{C}_{j}$. Since each of these subset
is LNE, the result is proved. ∎

##  9\. Affinely LNE versus Projectively LNE

The space ${\bf P}^{n}$ 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 ${\bf P}^{n}$, 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 ${\bf
P}^{n}$ as the projective metric $g_{{\rm proj}}$.

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

As a subset of the projective space ${\bf P}^{n}$, the affine space
$\mathbb{C}^{n}$ is equipped with the restriction of the projective metric
$g_{{\rm proj}}$, whereas as a complex vector space it is equipped with its
canonical Hermitian structure $eucl$. The metric space structures respectively
obtained from $({\mathbb{C}^{n}},g_{{\rm proj}})$ and
$({\mathbb{C}^{n}},eucl)$ are clearly not equivalent, since $\mathbb{C}^{n}$
is bounded for $g_{{\rm proj}}$ while it is not bounded for $eucl$. Yet the
respective restrictions of $eucl$ and $g_{{\rm proj}}$ 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 ${\bf P}^{n}$ and let $H$ be a hyperplane of ${\bf
P}^{n}$.
1) The curve $X\setminus H$ in $\mathbb{C}_{H}^{n}$ 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. $eucl$.

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

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

######  Example 9.3.

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

$P:=\\{{\bf z}:yz-x^{2}=0\\}\;\;{\rm and}\;\;H:=\\{{\bf z}:xy-z^{2}=0\\}.$

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 ${\bf
P}^{2}$ since they are non-singular and irreducible. The affine traces w.r.t.
$L$ are

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

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 ${\bf P}^{n}$. The following statements are equivalent:
(i) $X$ is LNE in ${\bf P}^{n}$.
(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 ${\bf P}^{n}$ such that the pair $X,H$ is in
general position, the variety $X\cup H$ is LNE in ${\bf P}^{n}$.
(iv) For any hyperplane $H$ of ${\bf P}^{n}$ 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 ${\bf P}^{n}$ 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 $(ii)\Longrightarrow(iii)$ follows from Lemmas 3.1 and 2.6.

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

To prove $(i)\Longrightarrow(iv)$, we observe that whenever $X$ and $H$ are in
general position, the intersection $H\cap X_{\rm sing}$ 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 $(iv)\Longrightarrow(v)$ is obvious.

To end the proof, let us show $(v)\Longrightarrow(i)$. 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
$X_{\rm sing}=(X^{a})_{\rm sing}$. 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
$X_{\rm sing}$ which is LNE in ${\mathbb{C}^{n}}$. Thus $X$ is locally LNE in
${\bf P}^{n}$ at each of its singular points. We conclude applying Proposition
4.6. ∎

##  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

$(X_{1}^{a},d_{\rm inn}^{X_{1}^{a}})\to(X_{2}^{a},d_{\rm inn}^{X_{2}^{a}}).$

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 ${\bf P}^{n}$ of degree $d\geq 1$. For each
${\bf y}\in X$, let $c({\bf y})$ be the number of irreducible components of
the curve germ $(X,{\bf y})$. 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 $X^{a}\setminus X_{\rm
sing}^{a}$ satisfies the following identity

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

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)

$DSGM(X):=\left(\deg(X);s(X);g(X);{({\bf y},c({\bf y}))}_{{\bf y}\in X_{\rm
sing}}\right),$

where $s(X):=\\#X_{\rm sing}$ 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

$DSGM(X_{i})=\left(d_{i};s_{i};g_{i};{({\bf y},c({\bf y}))}_{{\bf
y}\in(X_{i})_{\rm sing}}\right),\;\;i=1,2,$

are equivalent if $d_{1}=d_{2}$, $s_{1}=s_{2}=:s$, $g_{1}=g_{2}$, and there
exists a bijection $\varphi$ of $\\{1,\ldots,s\\}$ such that $c({\bf
y}_{j}^{1})=c({\bf y}_{\varphi(j)}^{2})$ for each $j=1,\ldots,s$, where
$(X_{i})_{\rm sing}=\\{{\bf y}_{1}^{i},\ldots,{\bf y}_{s}^{i}\\}$ for $i=1,2$.

######  Theorem 10.2.

The following statements are equivalent:

  1. (a)

Two irreducible complex affine algebraic LNE curves are Lipschitz equivalent.

  2. (b)

Two irreducible complex affine algebraic LNE curves are topologically
equivalent.

  3. (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 ${\bf
P}^{n_{i}}$. Denote

$d_{i}:=\deg(X_{i}),\;\;Y_{i}:=(X_{i})_{\rm sing}=\\{{\bf
y}_{i}^{1},\ldots,{\bf y}_{i}^{s_{i}}\\},\;\;{\rm
and}\;\;X_{i}^{\infty}=\\{\lambda_{i}^{1},\ldots,\lambda_{i}^{d_{i}}\\},$

since $X_{i}^{a}$ is LNE by hypotheses implies that $\\#X_{i}^{\infty}=d_{i}$.
For ${\bf y}_{i}\in Y_{i}$, let $T_{{\bf y}_{i}}X_{i}^{a}$ be the tangent cone
of $X_{i}^{a}$ at ${\bf y}_{i}$.
Let $R\gg 1$ be a large radius, as in the proof of Lemma 3.1, such that for
each ${\bf y}_{i}^{j}\in X_{i}^{a}$, with $i=1,2$ and $j=1,\ldots,s_{i}$, the
subset

$\mathcal{C}_{i}^{j}:=X_{i}^{a}\cap B_{i}^{j}$

is bi-Lipschitz homeomorphic to its truncated tangent cone

$T_{{\bf y}_{i}^{j}}X_{i}^{a}\cap B_{i}^{j},$

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

$\mathcal{X}_{i}:=X_{i}^{a}\setminus\mathcal{B}_{i},$

is bi-Lipschitz homeomorphic to the truncated cone

$\widehat{X_{i}^{\infty}}\setminus\mathcal{B}_{i}=\left(\cup_{j=1}^{d_{i}}\mathbb{C}\lambda_{i}^{j}\right)\setminus\mathcal{B}_{i},$

where $\mathcal{B}_{i}$ is either the open ball $B_{R}^{2n}$ or its closure
${\bf B}_{R}^{2n}$. The boundary ${\partial}\mathcal{X}_{i}$ is a finite union
of $d_{i}$ embedded ${\bf S}^{1}$.

Removing $N(\geq 1)$ 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

$N_{i}^{a}:=d_{i}+\sum_{j=1}^{s_{i}}c({\bf y}_{i}^{j})\;\;{\rm
and}\;\;\mathcal{C}_{i}^{a}:=\mathcal{X}_{i}\cup\left(\cup_{j=1}^{s_{i}}\mathcal{C}_{i}^{j}\right).$

Therefore the compact (resp. open and bounded) surface with boundary
$X_{i}^{a}\setminus\mathcal{C}_{i}^{a}$ 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).

$\bullet$ Assume (b). Let $h:X_{1}^{a}\to X_{2}^{a}$ be a homeomorphism. Since
$X_{i}^{a}$ is LNE, a point ${\bf y}_{i}$ of $X_{i}^{a}$ is singular if and
only if $c({\bf y}_{i})\geq 2$. Since $h$ maps each germ

$(X_{1}^{a}\setminus{\bf y},{\bf y})\to(X_{2}^{a}\setminus h({\bf y}),h({\bf
y}))$

we deduce that

$h(Y_{1})=Y_{2},\;\;{\rm and}\;\;c({\bf y})=c(h({\bf y}))\;\;\forall{\bf y}\in
Y_{1}.$

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.

$\bullet$ 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 $j=1,\ldots,s$ we can assume
without loss of generality that $c({\bf y}_{1}^{j})=c({\bf y}_{2}^{j})$.

We deduce $X_{1}^{a}\setminus\mathcal{C}_{1}^{a}$ and
$X_{2}^{a}\setminus\mathcal{C}_{2}^{a}$ are both diffeomorphic to
$S_{g,N^{a}}$ for $N^{a}:=N_{1}^{a}=N_{2}^{a}$.

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
$\mathcal{C}_{1}^{j}$ and $\mathcal{C}_{2}^{j}$ for each $j=1,\ldots,s$.

We can glue all these pieces together to
$S_{g,N^{a}}=X_{i}\setminus\mathcal{C}_{i}^{a}$ along the boundary
${\partial}\mathcal{C}_{i}^{a}$ 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.
∎

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 $n\geq 3$. 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 ${\bf G}(n-2,n)$ of the
$(n-2)$ planes of ${\mathbb{C}^{n}}$. Consider the Gauss mapping of $X^{a}$:

$\tau:X^{a}\setminus X_{\rm sing}\to{\bf G}(1,n),\;\;{\bf x}\to T_{\bf x}X.$

The closure $G_{0}(X)$ of $\tau(X^{a}\setminus X)$ is either a curve of ${\bf
G}(1,n)$ or a finite set of points. Let

$G(X):=G_{0}(X)\cup\left(\cup_{\lambda\in X^{\infty}}\mathbb{C}\lambda\right)$

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

$V:=\\{(\lambda,\lambda^{\prime},P)\in G(X)\times G(X)\times{\bf
G}(n-2,n):\dim_{\mathbb{C}}\mathbb{C}\lambda+\mathbb{C}\lambda^{\prime}+P\leq
n-1\\}$

We check that $\dim_{\mathbb{C}}V=2n-5$ and that $W$, the Zariski closure of
the projection of $V$ onto ${\bf G}(n-2,n)$, has also dimension $2n-5$.
Therefore

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

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 $P\in{\bf G}(n-2,n)$. 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 ${\bf z}$ be a point of $X_{P,{\rm sing}}^{a}$ and let
$\pi_{P}^{-1}({\bf z})=\\{{\bf x}_{1},\ldots,{\bf x}_{q}\\}$. For each
$i=1,\ldots,q$, the number of distinct lines in the tangent cone of $X$ at
${\bf x}_{i}$ is $c({\bf x}_{i})$. The definition of $\Omega(X)$ implies that
the tangent cone of $X_{P}^{a}$ at ${\bf z}$ will consist of
$\sum_{i=1}^{q}c({\bf x}_{i})$ distinct lines. Since $X$ is LNE and
$P\in\Omega(X)$, the germ $(X_{P}^{a},{\bf z})$ is a finite union of pairwise
transverse non-singular curve germs. Thus $X_{P}^{a}$ is locally LNE at ${\bf
z}$. Since $X^{a}$ is connected so is $X_{P}^{a}$. Theorem 8.1 shows that
$X_{P}^{a}$ is LNE. ∎

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 $n\geq 3$, 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 $n\geq 3$ and each positive integer $d$, there is a non-
negative integer $p(d,n)$, polynomial in $d$, such that for every $g$ such
that $0\leq g\leq p(d,n)$, there is an irreducible non-singular (and non-
degenerate) curve of ${\bf P}^{n}$ with degree $d$ and genus $g$ (see [10,
2]). Thus, there exist non-singular space curves with $2g\neq(d-1)(d-2)$. 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. ∎

Note that generic affine traces of non-singular irreducible projective curves
with genus zero and degree $d\geq 3$ 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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