## ANNALES

### DE LA FACULTÉ DES SCIENCES

### Mathématiques

MOHAMED JLELI

End-to-end gluing of constant mean curvature hypersurfaces

Tome XVIII, n^{o}4 (2009), p. 717-737.

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pp. 717–737

**End-to-end gluing of constant mean curvature** **hypersurfaces**

Mohamed Jleli^{(1)}

**R**^{´}**ESUM´****E.**—Il a ´et´e observ´e par R. Kusner et prouv´e par J. Ratzkin
qu’on peut recoller ensemble deux surfaces `a courbure moyenne constante
ayant deux bouts de mˆeme param`etre de Delaunay. Cette proc´edure de
recollement est connu comme«somme connexe bout-`a-bout». Dans ce
papier, nous donnons une g´en´eralisation de cette construction en dimen-
sion quelconque dans le but de construire des nouvelles hypersurfaces `a
courbure moyenne constante `a partir des hypersurfaces connues.

**A****BSTRACT.**—It was observed by R. Kusner and proved by J. Ratzkin
that one can connect together two constant mean curvature surfaces hav-
ing two ends with the same Delaunay parameter. This gluing procedure
is known as a “end-to-end connected sum”. In this paper we generalize,
in any dimension, this gluing procedure to construct new constant mean
curvature hypersurfaces starting from some known hypersurfaces.

**1. Introduction and statement of results**

Surfaces of revolution whose mean curvature is constant equal to 1 are classiﬁed in 1841 by C. Delaunay [1]. The proﬁle curves of these surfaces (known as Delaunay surfaces) are conic roulettes [2].

In higher dimension, a generalization of the Delaunay classiﬁcation for
hypersurfaces with some symmetries is given in [4]. In particular, for*n*3,
there exist a constant mean curvature hypersurface of revolution in R* ^{n+1}*
denoted by

*n-Delaunay hypersurfaces. These hypersurfaces construct a one*

(*∗*) Re¸cu le 29/02/08, accept´e le 21/11/08

(1) D´epartement de math´ematiques. Ecole sup´erieure des Sciences et Techniques de Tunis, 5 Avenue Taha Hussein 1008, Tunisia.

jlelimohamed@gmail.com

parameter family*D**τ* for

*τ∈*(−∞,0)*∪*(0, τ* _{∗}*].

Let us denote by *M**k*, for *k* *∈* N* ^{∗}*, the set of complete noncompact
constant mean curvature hypersurfaces in R

*with*

^{n+1}*k*ends asymptotic to

*n-Delaunay hypersurface. Then, any element Σ of*

*M*

*k*is said to be nondegenerate if the linearized operator (the Jacobi operator)

*L*Σ:*L*^{2}(Σ)*−→L*^{2}(Σ)

is injective. In general, it is diﬃcult to prove that a given hypersurface is
nondegenerate. But, we will prove in the beginning of the paper that any
*n-Delaunay hypersurface is nondegenerate.*

The structure of the set *M**k* is now fairly well understood. In particular,
it is proved in [5] that if Σ is a nondegenerate element of *M**k* and if the
Delaunay parameters *τ* of the ends of Σ satisfy *τ* *∈* [τ^{∗}*,*0)*∪*(0, τ* _{∗}*], for
some

*τ*

*depends only in*

^{∗}*n, there exists an open setU ⊂ M*

*k*containing Σ which is a smooth manifold of dimension

*k*(n+ 1). This result generalizes in any dimension and for constant mean curvature hypersurfaces with ends asymptotic to

*n*Delaunay with negative parameter the result of R. Kusner, R. Mazzeo and D. Pollack [14].

In 1990, a method of construction of constant mean curvature surfaces is
elaborated by N. Kapouleas to construct a compact surfaces [9], [10] and a
noncompact completes surfaces ofR^{3}[8]. This method is based in analogous
method used by R. Schoen to construct a constant scalar curvature in a
ﬁnitely punctured sphere. Later, this technique has been adopted in [15],
[16], [3] and generalized in [6]. More precisely, starting from a nondegenerate
element of *M**k*1 we can add a ﬁnite number (k2 *∈* N* ^{∗}*) of constant mean
curvature hypersurfaces asymptotic to

*n-Delaunay to this hypersurface to*construct a new nondegenerate element of

*M*

*k*1+k2.

Recently, a new method of construction of constant mean curvature sur- faces which is known as a “end-to-end connected sum” in introduced by J.

Ratzkin [19]. This gluing method is crucial in the construction of many ex- amples of compact constant mean curvature surfaces with nontrivial topol- ogy [7].

In this paper, we give a generalization of this technique in any dimension in
the aim to construct new constant mean curvature hypersurfaces starting
from some known hypersurfaces. We start in Section 2 by giving a parame-
terization of a one parameter family of hypersurfaces of revolution inR* ^{n+1}*,
which have constant mean curvature normalized to be equal to 1. These
hypersurfaces, which were originally studied in [12], generalize the classical

constant mean curvature surfaces inR^{3}which were discovered by Delaunay
in [1] in the middle of the 19-th century. Next, we deﬁne the Jacobi operator
about a*n-Delaunay hypersurface and we give the expression of the geomet-*
ric Jacobi ﬁelds (some solutions of the homogenous problem *L**D**τ**w* = 0 ).

In Section 3, we recall some important results of the moduli space concern-
ing the Fredholm properties of the Jacobi operator acting in some weighted
functional space. Next, we describe the construction by considering two non-
degenerate hypersurfaces Σ1*∈ M**k*1and Σ2*∈ M**k*2 and we assume that the
ends *E*1 *⊂* Σ1 and *E*2 *⊂* Σ2 are asymptotic to the same *n-Delaunay* *D**τ*

with parameter*τ* *∈*[τ^{∗}*,*0)*∪(0, τ** _{∗}*). Then, we can align Σ1and Σ2such that
the axis of

*E*1 and of

*E*2 coincide (with opposite directions ). Finally, we assume that

*E*1or

*E*2 is a ”regular end” then we can translate one of these hypersurface along this axis and we prove

Theorem 1.1*. — Let*Σ_{1}*∈ M**k*1*and*Σ_{2}*∈ M**k*2 *two nondegenerate con-*
*stant mean curvature hypersurfaces described as above. There exists a family*
*of hypersurfaces which is a connected sum of* Σ_{1} *and* Σ_{1}*. These hypersur-*
*faces can be perturbed into a constant mean curvature hypersurface which is*
*element ofM**k*1+k2*−*2*.*

The principal advantage of this construction (in addition to the fact that it can be used to construct new examples of compact or complete noncompact constant mean curvature hypersurfaces) is the simplicity of its proof.

**2. Delaunay hypersurfaces**
**2.1. Parameterization:**

It will be more interesting to consider an isothermal type parameteriza- tion for which will be more convenient for analytical purposes. Hence, we looking for hypersurfaces of revolution which can be parameterized by

*X*(s, θ) = (*|τ|e*^{σ(s)}*θ, κ(s)),* (2.1)
for (s, θ) *∈* R*×S*^{n}^{−}^{1}. The constant *τ* being ﬁxed, the functions *σ* and *κ*
are determined by asking that the hypersurface parameterized by *X* has
constant mean curvature equal to *H* and also by asking that the metric
associated to the parameterization is conformal to the product metric on
R*×S** ^{n−1}*, namely

(∂_{s}*κ)*^{2}=*τ*^{2}*e*^{2σ}

1*−*(∂_{s}*σ)*^{2}

*.* (2.2)

We choose the orientation of the hypersurface parameterized by*X* so that,
the unit normal vector ﬁeld is given by

*N* :=

*−* *∂**s**κ*

*|τ|e*^{σ}*θ, ∂*_{s}*σ*

*.* (2.3)

This time, using (2.2) the ﬁrst fundamental form *g* of the hypersurface
parameterized by*X* is given by

*g*=*τ*^{2}*e*^{2σ}(ds*⊗ds*+*dθ**i**⊗dθ**j*),
and its second fundamental form*b*is given by

*b*=

*∂*_{s}^{2}*κ ∂**s**σ−∂**s**κ*(∂_{s}^{2}*σ*+ (∂*s**σ)*^{2})

*ds⊗ds*+*∂**s**κ dθ**i**⊗dθ**j**.*
Therefore, the mean curvature*H* of the hypersurface parameterized by *X*
is given by

*H* = 1

*nτ*^{2}*e*^{2σ}

(n*−*1)*∂**s**κ−∂**s**κ*(∂_{s}^{2}*σ*+ (∂*s**σ)*^{2}) +*∂*^{2}_{s}*κ ∂**s**σ*
*.*

This is a rather intricate second order ordinary diﬀerential equation in the
functions *σ*and*τ* which has to be complimented by the equation (2.2). In
order to simplify our analysis, we use of (2.2) to get rid of the factor*τ*^{2}*e*^{2σ}
in the above equation. This yields

*∂**s**σ ∂*_{s}^{2}*κ*=*∂**s**κ*

1*−n*+*∂*_{s}^{2}*σ*+ (∂*s**σ)*^{2}+*n H ∂**s**κ*

1*−*(∂*s**σ)*^{2}_{−1}*.*
Now, we can diﬀerentiate (2.2) with respect to*s, and we obtain*

*∂*_{s}*κ ∂*_{s}^{2}*κ*=*τ*^{2}*e*^{2σ}*∂*_{s}*σ*

1*−∂*_{s}^{2}*σ−*(∂_{s}*σ)*^{2}
*.*

The diﬀerence between the last equation, multiplied by*∂**s**σ, and the former*
equation, multiplied by*∂**s**κ, yields*

*∂*_{s}^{2}*σ*+ (1*−n)(1−*(∂*s**σ)*^{2}) +*n H ∂**s**κ*= 0. (2.4)
Hence, in order to ﬁnd constant mean curvature hypersurfaces of revolution,
we have to solve (2.2) together with (2.4).

Let us deﬁne

*τ** _{∗}*:= 1

*n*(1*−n)*^{n−1}^{n}*.*

For all*τ* *∈*(−∞,0)*∪*(0, τ* _{∗}*], we deﬁne

*σ*

*τ*to be the unique smooth noncon- stant solution of

(∂*s**σ)*^{2}+*τ*^{2}

*e** ^{σ}*+

*ι e*

^{(1−n)σ}2

= 1, (2.5)

with initial condition*∂**s**σ(0) = 0 andσ(0)<*0. Next, we deﬁne the function
*κ**τ* to be the unique solution of

*∂**s**κ*=*τ*^{2}

*e*^{2σ}+ *ι e*^{(2−n)σ}

*,* with *κ(0) = 0.* (2.6)

Here, *ι*is the sign of*τ.*

In particular, the hypersurface parameterized by
*X**τ*(s, θ) := (|τ|e^{σ}^{τ}^{(s)}*θ, κ**τ*(s)),

for (s, θ)*∈*R*×S** ^{n−1}*, is an embedded constant mean curvature hypersur-
face of revolution when

*τ*belongs (0, τ

*], this hypersurface will be referred to as the “n-unduloid” of parameter*

_{∗}*τ*. In the other case, if

*τ <*0, this hypersurface is only immersed and will be referred to as the “n-nodoid” of parameter

*τ.*

*Remark 2.1. —* Thanks to the Hamiltonian structure of (2.5), the func-
tion*s→σ(s) is periodic. Let denote by* *s** _{τ}* this period. Then, it is proved
in [6]

*s**τ*=*−* *n*

*n−*1 log*τ*^{2}+*O(1)* (2.7)

as *τ* tends to 0.

**2.2. The Jacobi operator about a** *n-Delaunay*

It is well known [20] that the linearized mean curvature operator about
*D**τ*, which is usually referred to as the Jacobi operator, is given by

*L**τ* := ∆*τ*+*|A**τ**|*^{2}*,*

where ∆*τ* is the Laplace-Beltrami operator and *|A**τ**|*^{2} is the square of the
norm of the shape operator*A**τ* on*D**τ*.

Let us deﬁne the function*ϕ** _{τ}* :=

*|τ|e*

^{σ}*. We ﬁnd the expression of the Jacobi operator in term of the function*

^{τ}*ϕ*

_{τ}*L**τ*:=*ϕ*^{−n}_{τ}*∂**s*(ϕ^{n−2}_{τ}*∂**s*) +*ϕ*^{−2}* _{τ}* ∆

_{S}*n−1*+

*n*+

*n*(n

*−*1)

*τ*

^{2n}

*ϕ*

^{−2n}

_{τ}*.*(2.8) It will be convenient to deﬁne the conjugate operator

*L**τ*:=*ϕ*

*n+2*

*τ*2 *L**τ**ϕ**τ*^{2−n}^{2} *,* (2.9)
which is explicitly given in terms of the function*ϕ**τ* by

*L** _{τ}*=

*∂*

_{s}^{2}+ ∆

_{S}

^{n}*−*1

*−*

*n−*2

2 2

+*n(n*+ 2)

4 *ϕ*^{2}* _{τ}*+

*n(3n−*2)

4 *τ*^{2n}*ϕ*^{2}_{τ}^{−}^{2n}*.* (2.10)
Since the operators *L**τ* and *L**τ* are conjugate, the mapping properties of
one of them will easily translate for the other one. With slight abuse of
terminology, we shall refer to any of them as the Jacobi operator about*D**τ*.

Now, we give the following deﬁnition.

Definition 2.2*. — Let us denote by* *θ* *→* *e** _{j}*(θ), for

*j*

*∈*N

*, the eigen-*

*functions of the Laplace-Beltrami operator on*

*S*

^{n}

^{−}^{1}

*, which will be normal-*

*ized to have*

*L*

^{2}

*norm equal to*1

*and correspond to the eigenvalueλ*

_{j}*. That*

*is*

*−*∆_{S}^{n}*−*1*e** _{j}* =

*λ*

_{j}*e*

_{j}*,*

*and*

*λ*0= 0, *λ*1=*. . .*=*λ**n* =*n−*1, *λ**n+1*= 2n, . . . *and* *λ**j**λ**j+1**.*
We ended this section by giving only the expression of some Jacobi ﬁelds,
i.e., solution of the homogeneous problem

*L**τ**ω*= 0,

since these Jacobi ﬁelds follow from a rigid motion or by changing the De-
launay parameter*τ. More details are given in [5].*

*•* For*τ∈*(*−∞,*0)*∪*(0, τ* _{∗}*), we deﬁne Φ

^{0,+}

*to be the Jacobi ﬁeld corre- sponding to the translation of*

_{τ}*D*

*τ*along the

*x*

*axis*

_{n+1}Φ^{0,+}* _{τ}* :=

*ϕ*

^{n−4}^{2}

*∂*

*s*

*ϕ.*

It is easy to check that Φ^{0,+}* _{τ}* only depends on

*s*and is periodic. Then, this Jacobi ﬁeld is bounded.

*•* Since we have *n*directions orthogonal to *x**n+1*, there are *n*linearly
independent Jacobi ﬁelds which are obtained by translating*D**τ* in a
direction orthogonal to its axis. We get for*j*= 1, . . . , n

Φ^{j,+}* _{τ}* :=

*ϕ*^{n}^{2} *± |τ|*^{n}*ϕ*^{−}^{n}^{2}
*e**j**.*

Again, we see that Φ^{j,+}* _{τ}* is periodic (hence bounded) for all

*j*= 1, . . . , n.

*•* For*j* = 1, . . . , n, we deﬁne
Φ^{j,−}* _{τ}* (s, θ) :=

*ϕ*

^{n−4}^{2}

*ϕ ∂**s**ϕ*+*κ ∂**s**κ*

*e**j*

to be the Jacobi ﬁeld corresponding to the rotation of the axis of*D**τ*.
Observe that Φ^{j,−}* _{τ}* is not bounded, but grows linearly.

*•* Finally, The Jacobi ﬁeld corresponding to a change of parameter*τ∈*
(*−∞,*0)*∪*(0, τ* _{∗}*) is given by

Φ^{0,}_{τ}* ^{−}* :=

*ϕ*

^{n−4}^{2}(∂

_{τ}*κ ∂*

_{s}*ϕ−∂*

_{τ}*ϕ ∂*

_{s}*κ).*

Because of the rotational invariance of the operator*L** _{τ}*, we can introduce
the eigenfunction decomposition with respect to the cross-sectional Laplace-
Beltrami operator ∆

_{S}*n−1*. In this way, we obtain the sequence of operators

*L** _{τ,j}*=

*∂*

^{2}

_{s}*−λ*

_{j}*−*

*n−*2

2 2

+*n(n*+ 2)

4 *ϕ*^{2}+*n(3n−*2)

4 *τ*^{2n}*ϕ*^{2}^{−}^{2n} (2.11)
for*j∈*N. By deﬁnition, the*indicial roots*of the operator*L** _{τ,j}*characterize
the rate of growth (or rate of decay) of the solutions of the homogeneous
equation

*L**τ,j**ω*= 0

at inﬁnity (see [14]). Observe that the explicit knowledge of some Jacobi
ﬁelds yields some information about the indicial roots of the operator*L**τ*.
Indeed, since the Jacobi ﬁelds Φ^{j,±}* _{τ}* , described below, are at most linearly
growing, the associated indicial roots are all equal to 0. Hence, we conclude
that for all

*τ*

*∈*(−∞,0)

*∪*(0, τ

*]*

_{∗}*γ**j*(τ) = 0, for *j*= 0, . . . , n.

The situation is completely diﬀerent when*j* *n*+ 1. Indeed, its proved in
[5]:

Proposition 2.3*. — There exists* *τ*^{∗}*<* 0, depending only on *n, such*
*that for all*

*τ* *∈*[τ^{∗}*,*0)*∪*(0, τ* _{∗}*],

*γ** _{j}*(τ)

*>*0,

*for all*

*jn*+ 1.

*Remark 2.4. —* Thanks to the last analysis, it is easy to see that the
*n-Delaunay is a nondegenerate element ofM**k*.

**3. End-to-End connected sum**
**3.1. Moduli space theory**

Assume that we are given Σ a complete noncompact, constant mean
curvature hypersurfaces in R* ^{n+1}*, with

*k*ends which are modeled after

*n-*Delaunay hypersurfaces.

We recall that, for all*τ* *∈*(−∞,0)*∪*(0, τ* _{∗}*], the Delaunay hypersurface

*D*

*τ*is parameterized by

*X**τ* = (ϕ*τ**θ, κ**τ*)*,*

where*κ**τ* and*ϕ**τ* are deﬁned respectively in subsection 2.1 and section 2.2.

Now, we denote by*E*1*, . . . , E**k*the ends of the hypersurface Σ. We require
that these ends are asymptotic to half *n-unduloids or a half* *n-nodoids.*

More precisely, we require that, up to some rigid motion, each end can be
parameterized as the normal graph of some exponentially decaying function
over some Delaunay hypersurface, i.e. up to some rigid motion, the end*E** _{}*
is parameterized by

*Y** _{}*:=

*X*

_{τ}*+*

_{}*ω*

_{}*N*

_{τ}

_{}*,*(3.12)

where *Y** _{}* is deﬁned in (0,+

*∞*)

*×S*

^{n}

^{−}^{1}and where the function

*ω*

*is expo- nentially decaying as well as all its derivatives. Hence, for all*

_{}*k*

*∈*N, there exists

*c*

_{k}*>*0 such that

*|∇*^{k}*ω**|c**k**e*^{−γ}^{n+1}^{(τ}^{}^{)}* ^{s}* (3.13)
on (0,+∞)

*×S*

^{n}

^{−}^{1}.

The Jacobi operator about the end *E** _{}* is close to the Jacobi opera-
tor about the

*n-Delaunay hypersurface*

*D*

*τ*. The content of the following Lemma is to make this result quantitatively precise.

Lemma 3.1*. — The Jacobi operator about* Σ, restricted to the end*E**is*
*given by*

*L*Σ:= ∆_{Σ}+*|A*Σ*|*^{2}=*L**τ*+*L*_{}*,*

*where the* *L*_{}*is a second order linear operator whose coeﬃcients and their*
*derivatives are bounded by a constant timese*^{−γ}^{n+1}^{(τ}^{}^{)}^{s}*.*

*Proof. — This follows at once from the fact that the coeﬃcients of the*
ﬁrst and second fundamental forms associated to the end *E* are equal to
the coeﬃcients of the ﬁrst and second fundamental form of*D**τ* up to some
functions which are exponentially decaying like*e*^{−γ}^{n+1}^{(τ}^{}^{)}* ^{s}*.

Now, we decompose Σ into slightly overlapping pieces which are a com-
pact piece *K* and the ends *E** _{}*. Then, we deﬁne the following functional
space

Definition 3.2*. — For allr∈*N,*δ∈*R*and allα∈*(0,1), the function
*spaceC*_{δ}* ^{r,α}*(Σ)

*is deﬁned to be the space of functions*

*w∈ C*

_{loc}*(Σ)*

^{r,α}*for which*

*the following norm is ﬁnite*

*w*_{C}^{r,α}* _{δ}* (Σ):=

*k*

*=1*

*w◦Y*_{}_{E}_{δ}^{r,α}_{((0,+∞)×S}* ^{n−1}*)+

*w*

_{C}*(K)*

^{r,α}*,*

*where the space*

*E*_{δ}* ^{,α}*([s0

*,*+∞)

*×S*

*)*

^{n−1}*is the set of functions* *C*_{loc}^{,α}*which are deﬁned on* [s0*,*+∞)*×S*^{n}^{−}^{1} *and for*
*which the following norm is ﬁnite :*

*ω*_{E}^{,α}

*δ* (R*×S** ^{n−1}*):= sup

*s**s*0

*|e*^{−}^{δs}*ω|**C** ^{,α}*([s,s+1]×S

*)*

^{n−1}*.*

*Here,|* *.* *|**C** ^{,α}*([s,s+1]

*×*

*S*

*)*

^{n−1}*denotes the usual H¨older norm in*[s, s+ 1]

*×*

*S*

^{n−1}*.*

Let*χ** _{}* be a cutoﬀ function which is equal to 0 on

*E*

_{}*∩K*and equal to 1 on

*Y*

*((c*

_{}

_{}*,*+

*∞*)

*×S*

*) for some*

^{n−1}*c*

_{}*>*0 chosen large enough. We deﬁne the deﬁciency space

*W*(Σ) by

*W(Σ) :=⊕*^{k}* _{=1}*Span

*{χ*Φ

^{j,±}

_{}*|j*= 0, . . . , n}.

The analogue of the following result for *n* = 2 is usually known as the

“Linear Decomposition Lemma” (see [18] and [15]).

Proposition 3.3 *[5]. — We assume that* *τ*_{}*∈* [τ^{∗}*,*0) *∪* (0, τ* _{∗}*],

*δ*

*∈*(

*−*inf

_{}*γ*

*(τ*

_{n+1}*),0),*

_{}*α*

*∈*(0,1)

*and*Σ

*is nondegenerate. Let*

*N*(Σ)

*the*

*trace of the kernel of*

*L*Σ

*over the deﬁciency space*

*W*Σ

*, then*

*N*(Σ)

*is a*

*k*(n+ 1)

*dimensional subspace ofW(Σ)*

*which satisﬁes*

*Ker(L*Σ)*⊂ C*_{−δ}^{2,α}(Σ)*⊕ N*(Σ).

*If* *K(Σ)* *is ak*(n+ 1) *dimensional subspace ofW(Σ)* *such that*
*W(Σ) =K(Σ)⊕ N*(Σ),

*we have*

*L*Σ:*C*_{δ}^{2,α}(Σ)*⊕ K*(Σ)*−→ C*_{δ}^{0,α}(Σ)
*is an isomorphism.*

**3.2. Regular ends of constant mean curvature hypersurfaces**
We need to introduce the following

Definition 3.4*. — LetM* *be a constant mean curvature*1*hypersurface*
*with* *k* *ends of Delaunay type. We will say that the end* *E* *of* *M, which is*
*asymptotic to some Delaunay hypersurface* *D**τ**, is regular if there exists a*
*Jacobi ﬁeld* Ψ *which is globally deﬁned on M and which is asymptotic to*
Φ^{0,−}_{τ}*(the Jacobi ﬁeld on* *D**τ* *which corresponds to the the change of the*
*Delaunay parameter* *τ) on* *E.*

Of course,*n-Delaunay hypersurfaces have regular ends. In the case where*
*M* is a nondegenerate element of the moduli space of*k*ended constant mean
curvature hypersurfaces, we have the following characterization of regular
ends:

Proposition 3.5*. — Assume thatM* *is a nondegenerate element ofM**k*

*and let* *E* *be one of the ends of* *M. Then, the following propositions are*
*equivalent :*

*(i) The endE* *is regular.*

*(ii) There exists* *t*0 *>* 0 *and* (M*t*)_{τ∈(−t}_{0}_{,t}_{0}_{)} *a one parameter family of*
*constant mean curvature*1 *hypersurfaces in* *M**k* *such that*

*M*_{0}=*M*

*and the Delaunay parameter of the end ofM*_{t}*which is close toE* *is*
*equal toτ−t.*

*Proof. — The fact that (ii) implies (i) should be clear. The fact that*
(i) implies (ii) just follows from the application of the implicit function
Theorem and the analysis of the moduli space *M**k*. Indeed, as explained
in the end of the paper [5], the Jacobi ﬁelds span the tangent space to the
moduli space*M**k* at the point*M*. Therefore, we simply have

*M** _{t}*= exp

*(t ψ)*

_{M}as the one parameter family of constant mean curvature 1 hypersurfaces
close to*M* which have the right properties.

**3.3. Connecting two constant mean curvature hypersurfaces to-**
**gether**

Assume that we are given two nondegenerate complete noncompact con-
stant mean curvature hypersurfaces *M**i* *∈ M**k**i*, for *i*= 1,2. We denote by

*E*_{i,1}*, . . . , E*_{i,k}* _{i}* the ends of the hypersuface

*M*

*, each of which are assumed to be asymptotic to a*

_{i}*n- Delaunay hypersurfaces*

*D*

*τ*

*i,j*,

*j*= 1, . . . , k

*.*

_{i}We further assume that*M*1and*M*2have one end with the same Delau-
nay parameter. For example, let us assume that the Delaunay parameter of
*E*1,1and*E*2,1 are the same, both equal to

*τ*:=*τ*1,2=*τ*2,1*.*

Further more we assume that*τ* *=τ** _{∗}* i.e.

*E*1,1and

*E*1,2 are not asymptotic to a cylinders.

Given*m∈*N, we can use a rigid motion to ensure that the end *E*_{1,1} is
parameterized by

*Y*_{1,1}(s, θ) :=*X** _{τ}*(s, θ) +

*w*

_{1}(s+

*ms*

_{τ}*, θ)N*

*(s, θ),*

_{τ}for all (s, θ)*∈*(*−m s*_{τ}*,*+*∞*)*×S** ^{n−1}*and that the end

*E*

_{2,1}is parameterized by

*Y*_{2,1}(s, θ) :=*X** _{τ}*(s, θ) +

*w*

_{2}(s

*−ms*

_{τ}*, θ)N*

*(s, θ),*

_{τ}for all (s, θ)*∈*(*−∞, m s** _{τ}*)

*×S*

*. Though this is not explicit in the notation,*

^{n−1}*Y*

*does depend on*

_{i,1}*m. In addition, we know from (3.12) and (3.13) that*the functions

*w*1and

*w*2are exponentially decaying as

*s*tends to +∞(resp.

*−∞). More precisely,*

*w*_{1}*∈ E*_{−γ}^{2,α}_{n+1}_{(τ)}((0,+*∞*)*×S*^{n}^{−}^{1})
and also that

*w*_{2}*∈ E*_{γ}^{2,α}_{n+1}_{(τ)}((*−∞,*0)*×S*^{n}^{−}^{1}),

where the function spaces*E*_{δ}^{2,α} are introduced in Deﬁnition 3.2.

Given*s >−m s** _{τ}*, we deﬁne the truncated hypersurface

*M*

_{1}(s) :=

*M*

_{1}

*−Y*

_{1,1}((s,+∞)

*×S*

^{n}

^{−}^{1})) and given

*s < m s*

*we deﬁne the truncated hypersurface*

_{τ}*M*2(s) :=

*M*2

*−Y*2,1((−∞, s))

*×S*

*).*

^{n−1}Again, *M**i*(s) depends on*m. Now let* *s−→ξ(s) be a cutoﬀ function such*
that *ξ≡*0 for*s*1 and*ξ≡*1 for*s−1. We deﬁne the hypersurface ˜M**m*

to be

*M*˜*m*:=*M*1(−1) *∪* *C(1)* *∪M*2(1), (3.14)

where, for all *s* *∈* (0, m s* _{τ}*), the cylindrical type hypersurface

*C(s) is the*image of [

*−s, s]×S*

^{n}

^{−}^{1}by

(s, θ)*−→ξ(s)Y*1,1(s, θ) + (1*−ξ(s))Y*2,1(s, θ). (3.15)
By construction ˜*M**m*is a smooth hypersurface whose mean curvature is
identically equal to 1 except in the annulus*C(1) where the mean curvature*
is close to 1 and tends to 1 as *m* tends to +*∞*. Indeed, in *C(m s** _{τ}*), the
hypersurface ˜

*M*

*is a normal graph over the*

_{m}*n-Delaunay hypersurface*

*D*

*τ*

for some function*w*_{3}. But the estimates we have on*w*_{1}and*w*_{2} imply that
*w*3=*O*_{C}^{2,α}(e^{−γ}^{n+1}^{(τ) (s+m s}^{τ}^{)}) +*O*_{C}^{2,α}(e^{γ}^{n+1}^{(τ) (s−m s}^{τ}^{)}) (3.16)
on*C(m s**τ*). The mean curvature of ˜*M**m*in*C(m s**τ*) can then be computed
using Taylor’s expansion

*H**M*˜*m* =*H*_{D}* _{τ}* +

*O*

_{C}^{0,α}(w

_{3}).

It then follows from (3.16) that the mean curvature of ˜*M**m*can be estimated
by

*H**M*˜*m**−*1_{C}^{0,α}(C(1))*c e*^{−m γ}^{n+1}^{(τ)}^{s}^{τ}*.* (3.17)
**3.4. Mapping properties of the Jacobi operator about** *M*˜_{m}

In this subsection, we develop the linear analysis which will be needed in
the next subsection in order to perturb ˜*M**m*into a constant mean curvature
hypersurface. In particular, our aim is to ﬁnd function spaces on which the
mapping properties of the Jacobi operator about ˜*M**m* do not depend (too
much) on *m.*

We further assume that the Delaunay parameters of the ends of*M*1and
*M*_{2}are greater than or equal to*τ** ^{∗}*which has been introduced in Proposition
2.3. In particular, we can deﬁne

*δ*˜:= inf

*i=1,2* inf

*j=1,...,k**i*

*γ**n+1*(τ*i,j*),

and we know that ˜*δ >*0. Furthermore, the moduli space theory, as described
in subsection 3.1 applies. We ﬁx

*−δ < δ <*˜ 0.

We have already assumed that the hypersurfaces *M**i* are nondegenerate.

Using the notations of subsection 3.1, we deﬁne the deﬁciency subspace
*W*(M* _{i}*) associated to

*M*

*and we decompose*

_{i}*W(M**i*) =*N*(M*i*)*⊕ K(M**i*),

where *N*(M* _{i}*) and

*K*(M

*) are both (n+ 1)*

_{i}*k*

*dimensional, in such a way that the Jacobi operator about*

_{i}*M*

_{i}*L**M**i* :*C*_{δ}^{2,α}(M*i*)*⊕ K(M**i*)*−→ C*_{δ}^{0,α}(M*i*) (3.18)
is an isomorphism. The elements of*K(M**i*) are supported in the complement
of a compact set of *M**i*. Recall that, on each end *E**i,j* there are 2 (n+ 1)
locally deﬁned Jacobi ﬁelds Φ^{k,±}_{E}* _{i,j}*, where

*k*= 0, . . . , n. These Jacobi ﬁelds are

*a priori*only deﬁned for

*s*large enough, say

*ss*

*i,j*and are asymptotic to the corresponding Jacobi ﬁelds on

*D*

*τ*

Φ^{k,}_{E}_{i,j}* ^{±}* = Φ

^{k,}

_{τ}*+*

^{±}*O*

_{C}^{2,α}(e

*)*

^{δs}for any *δ* *∈* (−γ*n+1*(τ*i,j*),0). By deﬁnition, *W(M**i*) is the vector space
spanned by *χ** _{i,j}*Φ

^{k,±}

_{E}*i*

*j*

for *j* = 1, . . . , k* _{i}* and

*k*= 0, . . . , n, where

*χ*

*are cutoﬀ functions which are equal to 0 on*

_{i,j}*M*

*i*

*−E*

*i,j*, still equal to 0 on

*E*

*i,j*

for *s* *∈* (0, s*i,j*) and identically equal to 1 on the *E**i,j*, when *s* *s**i,j*+ 1.

Finally, recall that any globally deﬁned Jacobi ﬁeld which is at most linearly
growing on the ends of*M**i* belongs to

*C*^{2,α}* _{δ}* (M

*)*

_{i}*⊕ N*(M

*).*

_{i}Observe that, on a given end *E**i,j*, all Jacobi ﬁelds, except the Jacobi
ﬁeld Φ^{0,}_{E}^{−}

*i,j* which corresponds to a change of Delaunay parameter, come
from the action of the group of rigid motions. Hence, these Jacobi ﬁelds
are all globally deﬁned on *M** _{i}*. By assumption, the Jacobi ﬁeld Φ

^{0,}

_{E}_{1,1}

*also corresponds to a globally deﬁned Jacobi ﬁeld on*

^{−}*M*

_{1}, this is precisely the meaning of the fact that

*E*

_{1,1}is a regular end. Hence, for each

*k*= 0, . . . , n, the space

*N*(M

_{1}) contains an element which is equal to

*χ*

_{1,1}Φ

^{k,}

_{E}

^{±}1,1 on*E*_{1,1}.
As a consequence, one can choose *K*(M_{1}) in such a way that it does not
contain any *χ*1,1Φ^{k,±}_{E}_{1,1}, for *k* = 0, . . . , n. Since elements of *K(M*1) are not
supported on the end*E*1,1, they can be extended to ˜*M**m*.

A similar analysis can be performed on*M*2, with the Jacobi ﬁelds deﬁned
on the end*E*2,1. However, this time we have not assumed that the end*E*2,1

was a regular end. Hence, one can still choose*K(M*2) in such a way that it
does not contain any*χ*2,1Φ^{k,±}_{E}_{2,1}, for*k*= 1, . . . , nand also does not contain
*χ*_{2,1}Φ^{0,+}_{E}_{2,1}. But there might exist an element of *K*(M_{2}) which is equal to
*χ*2,1Φ^{0,−}_{E}_{2,1} on *E*2,1. Therefore, all elements of *K(M*2) can be extended to
*M*˜*m*except possibly the elements which are collinear to*χ*2,1Φ^{0,−}_{E}_{2,1} on*E*2,1.

We now explain how to extend the function*χ*_{2,1}Φ^{0,}_{E}^{−}

2,1to a function which
is globally deﬁned on ˜*M**m* and is equal to a Jacobi ﬁeld on *M*1(−1). This
construction again uses the fact that*E*1,1is a regular end. We parameterize
*C(m s**τ*) by (3.15). As already mentioned, Φ^{0,−}_{E}_{2,1} is asymptotic to Φ^{0,}_{τ}* ^{−}*. This
means that

*∀*(s, θ)

*∈*[−m s

*τ*

*, m s*

*τ*]

*×S*

^{n}

^{−}^{1}

*,*

Φ^{0,−}_{E}_{2,1}(s, θ) = Φ^{0,−}* _{τ}* (m s

_{τ}*−s) +O*

_{C}^{2,α}(e

^{δ}^{(m s}

^{τ}*).*

^{−s)}Similarly, Φ^{0,}_{E}_{1,1}* ^{−}* is asymptotic to Φ

^{0,−}

*. This means that*

_{τ}*∀*(s, θ)*∈*[−m s*τ**, m s**τ*]×S^{n−1}*,* Φ^{0,−}_{E}_{1,1}(s, θ) = Φ^{0,−}* _{τ}* (m s

*τ*+s)+O

_{C}^{2,α}(e

^{δ}^{(m s}

^{τ}^{+s)}).

Now, there exists*c*_{m}*∈*Rsuch that

Φ^{0,−}* _{τ}* (m s

*τ*

*−s)−*Φ

^{0,−}

*(m s*

_{τ}*τ*+

*s) =c*

*m*Φ

^{0,+}

*(s)*

_{τ}since the function of the left hand side is a Jacobi ﬁeld on *D**τ* which is
bounded and only depends on *s. In addition, it is easy to see that*

*|c*_{m}*|c*_{τ}*m,*
where the constant*c**τ* only depends on *τ.*

Now, on*M*_{1} the Jacobi ﬁelds Φ^{0,}_{E}^{−}

1,1 and Φ^{0,+}_{E}

1,1 are globally deﬁned. Ob- serve that we also have

Φ^{0,+}_{E}_{1,1}(s, θ) = Φ^{0,+}* _{τ}* (s) +

*O*

_{C}^{2,α}(e

^{δ}^{(m s}

^{τ}^{+s)})

on [*−m s*_{τ}*, m s** _{τ}*]

*×S*

*. This being understood, we use the cutoﬀ function*

^{n−1}*ξ*which has already been deﬁned in (3.15) to deﬁne the function ˜Φ

^{0,}

_{E}

^{−}_{2,1}which is equal to Φ

^{0,−}

_{E}_{2,1}on

*E*2,1

*∩M*2(1), which is equal to Φ

^{0,−}

_{E}_{1,1}+

*c*

*m*Φ

^{0,+}

_{E}_{1,1}on

*M*1(−1) and which interpolates smoothly between these deﬁnitions in

*C(1),*namely

Φ˜^{0,−}_{E}_{2,1}:=*ξ*(Φ^{0,−}_{E}_{1,1}+*c**m*Φ^{0,+}_{E}_{1,1}) + (1*−ξ) Φ*^{0,−}_{E}_{2,1}
on*C(1). Granted (3.15) and (3.17) one easily checks that*

Lemma 3.6*. — LetL**M*˜*m* *denote the Jacobi operator aboutM*˜_{m}*. Then,*
*L**M*˜*m*

Φ˜^{0,}_{E}_{2,1}* ^{−}* =

*O*

_{C}^{0,α}(C(1))(e

^{δ m s}*)*

^{τ}*onC(1).*

We now deﬁne ˜*K*(M_{2}) to be the space of functions of*K*(M_{2}) which have
been extended on all ˜*M** _{m}*either by 0 or by replacing Φ

^{0,}

_{E}

^{−}2,1 by ˜Φ^{0,}_{E}^{−}

2,1. We set
*K( ˜M**m*) :=*K(M*1)*⊕K(M*˜ 2).

We agree that the norm on*K( ˜M** _{m}*) is the product norm on

*K(M*1)

*×K(M*˜ 2).

Granted the decomposition of ˜*M**m*given in (3.14), we deﬁne the weighted
spaces :

Definition 3.7*. — Given* *δ* *∈* R, *k* *∈* N *and* *α* *∈* (0,1), we deﬁne
*D*^{k,α}* _{δ}* ( ˜

*M*

*)*

_{m}*to be the subspace of functions*

*u*

*∈ C*

_{loc}*( ˜*

^{k,α}*M*

*)*

_{m}*for which the*

*following norm*

*u*_{D}^{k,α}

*δ* ( ˜*M**m*) := *u*_{C}^{k,α}

*δ* (M1(−1))+*u*_{C}^{k,α}

*δ* (M2(1))

+ *e*^{−}^{δ m s}^{τ}*u*_{C}^{k,α}_{((−2,2)×S}* ^{n−1}*)

*is ﬁnite. Here,* *C*_{δ}* ^{k,α}*(M

*i*(s))

*is the restriction of functions of*

*C*

_{δ}*(M*

^{k,α}*i*)

*to*

*M*

*i*(s)

*and this space is endowed with the induced norm.*

We can now state the

Proposition 3.8*. — Assume that* *δ* *∈* (−*δ,*˜ 0) *is ﬁxed and* *E*_{1,1} *is a*
*regular end ofM*1*. There existm*0*>*0 *andc >*0 *and, for all* *mm*0*, one*
*can ﬁnd an operator*

*G**m*:*D*^{0,α}* _{δ}* ( ˜

*M*

*m*)

*−→ D*

_{δ}^{2,α}( ˜

*M*

*m*)

*⊕ K( ˜M*

*m*),

*such that*

*w*:=

*G*

*m*(f)

*solves*

*L*

*M*˜

*m*

*w*=

*f*

*onM*˜

*m*

*. Furthermore,*

*w*_{D}^{2,α}

*δ* ( ˜*M**m*)*⊕K*( ˜*M**m*)*c* *f* _{D}^{0,α}

*δ* ( ˜*M**m*)*.*

*Proof. — Given* *f* *∈ D*_{δ}^{0,α}( ˜*M**m*), we want to solve the equation
*L**M*˜*m**w*=*f*

on ˜*M** _{m}*. To this aim, we use the cutoﬀ function

*ξ*deﬁned in (3.15) and we start to solve

*L**M*1*w*1=*ξ f*
on*M*1 and also

*L**M*2*w*2= (1*−ξ)f*

on *M*_{2}. The existence of *w** _{i}* follows at once from (3.18) and we have the
estimate

*w*_{i}_{C}^{2,α}

*δ* (M*i*)*⊕K*(M*i*)*c* *f* _{D}^{0,α}

*δ* ( ˜*M**m*)*,* (3.19)
where the constant*c >*0 does not depend on*m. Observe that onE*2,1 the
function*w*2 can be decomposed as

*w*2:=*v*2+*a*2Φ^{0,−}_{E}_{2,1}*,* (3.20)
with *v*2 decays exponentially. We deﬁne on *C(m s**τ*) a cutoﬀ function *χ*1

which is identically equal to 1 for*sms**τ**−*2 and identically equal to 0 for
*sms**τ**−*1. We also deﬁne a cutoﬀ function*χ*2 which is identically equal
to 1 for *s−ms**τ*+ 2 and identically equal to 0 for *s−ms**τ*+ 1. This
being understood, we deﬁne a function*w*as follows :

*•* *w*is equal to*w*2 on*M*2(m s*τ*),

*•* *w*is equal to*w*_{1}+*a*_{2}Φ˜^{0,}_{E}^{−}

2,1 on*M*_{1}(*−m s** _{τ}*),

*•* *w*:=*χ*1*w*1+*χ*2*v*2+*a*2Φ˜^{0,−}_{E}_{2,1}, on*C(m s**τ*).

To begin with observe that
*w*_{D}^{2,α}

*δ* ( ˜*M**m*)⊕K( ˜*M**m*)*cf*_{D}^{0,α}

*δ* ( ˜*M**m*)

for some constant which does not depend on *m. This estimate follows at*
once from (3.19) and (3.20). We now estimate

*L**M*˜*m**w−f.*

Obviously, this quantity is equal to 0 on*M*1(1−m s*τ*)∪M2(m s*τ**−1). Hence,*
it remains to evaluate it on*C(m s**τ**−*1).

**Case 1**First assume that *−m s**τ*+ 2*s−1, then*
*L**M*˜*m**w−f* =*L**M*1*w−f* =*L**M*1*v*_{2}
since *L**M*1*w*_{1} = *f* and *L**M*1Φ˜^{0,}_{E}^{−}

2,1 = 0 in this set. Further observe that
*L**M*2*v*_{2}= 0 in this set, hence we conclude that

*L**M*˜*m**w−f* =*L**M*1*w−f* = (*L**M*1*− L**M*2)*v*_{2}*.*

We use the fact that *E*1,2 and *E*2,1 are graphs over the same Delaunay
hypersurface*D**τ*, hence we can write

*L**M*1=*L**τ*+*L*1*,*

where *L*_{1} is a second order partial diﬀerential operator whose coeﬃcients,
computed at*s, are bounded by a constant timese*^{−}^{γ}^{n+1}^{(τ)(s+m s}^{τ}^{)}in*C*^{0,α}((s*−*
1, s+ 1)*×S** ^{n−1}*). Similarly, we can write

*L**M*2=*L**τ*+*L*2*,*

where *L*2 is a second order partial diﬀerential operator whose coeﬃcients,
computed at*s, are bounded by a constant timese*^{γ}^{n+1}^{(τ)(s−m s}^{τ}^{)}in*C*^{0,α}((s, s+

1)*×S** ^{n−1}*). Using this, it is easy to show that, there exists a constant

*c >*0 which does not depend on

*m*such that

*||L**M*˜*m**w−f||*_{C}^{0,α}((s*−*1,s+1)*×**S*^{n}^{−}^{1})*c e*^{−}^{γ}^{n+1}^{(τ)(s+m s}^{τ}^{)}*e*^{−}^{δ(s}^{−}^{m s}^{τ}^{)} (3.21)
for all*s∈*[−m s*τ*+ 2,*−2].*

**Case 2**Now, assume that*−m s** _{τ}*+ 1

*s−m s*

*+ 2. In this set, we have to take into account the eﬀect of the cutoﬀ function*

_{τ}*χ*

_{2}, in addition to the estimate established in the previous case. This yields

*||L**M*˜*m**w−f||*_{C}^{0,α}_{((−m s}*τ*+1,−m s*τ*+2)×S* ^{n−1}*)

*c e*

^{2}

^{δ m s}

^{τ}*.*(3.22)

**Case 3**Finally, assume that

*s*

*∈*[−1,0]. Then, we have to take into account the eﬀect of the cutoﬀ function

*ξ, in addition to the estimate es-*tablished in the ﬁrst case. This yields

*||L**M*˜*m**w−f||*_{C}^{0,α}_{((−1,0)×S}* ^{n−1}*)

*c e*

^{−(γ}

^{n+1}^{(τ)−δ)}

^{m s}

^{τ}*.*(3.23)

**Case 4**The cases where

*s∈*[0, m s

_{τ}*−*1] can be treated similarly.

Collecting the previous estimates, we conclude that

*||L**M*˜*m**w−f||*_{D}^{0,α}

*δ* ( ˜*M**m*)*c*(e^{−}^{γ}^{n+1}^{m s}* ^{τ}*+

*e*

^{2}

^{δ m s}*)*

^{τ}*f*

_{D}^{0,α}

*δ* ( ˜*M**m*)*.*
So far, we have produced an operator *G*0, deﬁned by *G*0(f) =*w, which is*
uniformly bounded as*m*tends to +∞and which satisﬁes

*||L**M*˜*m**◦G*_{0}*−I||c*(e^{−}^{γ}^{n+1}^{m s}* ^{τ}* +

*e*

^{2}

^{δ m s}

^{τ}*).*

^{δ}The result then follows from a simple perturbation argument, provided *m*
is chosen large enough.

**3.5. The perturbation argument**

In this subsection, we construct a constant mean curvature hypersurface
by perturbing ˜*M**m*.

Proposition 3.9*. — There exists* *m*_{0} *∈* N *such that, for all* *m* *m*_{0}
*the hypersurface* *M*˜_{m}*can be perturbed into a constant mean curvature* 1
*hypersurface.*

*Proof. — As usual, we considerM** _{w}* a normal graph over ˜

*M*

*for some small function*

_{m}*w*

*∈ D*

_{δ}^{2,α}( ˜

*M*

*)*

_{m}*⊕ K*( ˜

*M*

*). Now, on each end*

_{m}*E*

*of ˜*

_{i,j}*M*

*, when*

_{m}*j*= 1, we decompose

*w*=*v*+*χ*_{i,j}*n*

*k=0*

*a** _{k,±}*Φ

^{k,}

_{E}

_{i,j}

^{±}and consider the normal graph for the function*v* over the hypersurface
exp_{E}* _{i,j}*(χ

*i,j*

*n*

*k=0*

*a** _{k,±}*Φ

^{k,±}

_{E}*).*

_{i,j}Now, the equation we have to solve reads

*H( ˜M**m*) +*L**M*˜*m**w*+*Q**m*(w) = 1, (3.24)
where *L**M*˜*m* is the Jacobi operator about ˜*M**m* and *Q**m* contains all the
nonlinear terms resulting from the Taylor expansion of the mean curvature
*H*(M*w*). We ﬁx*−δ < δ <*˜ 0. Using the result of Proposition 3.8, our problem
reduces to ﬁnding a ﬁxed point for

*w−→G*_{m}

1*−H( ˜M** _{m}*)

*− Q*

*m*(w)

*,* (3.25)

in the space*D*_{δ}^{2,α}( ˜*M** _{m}*)

*⊕K*˜( ˜

*M*

*). Using Proposition 3.8 and (3.17), we see that*

_{m}*||G**m*(1*−H*( ˜*M** _{m}*))||

_{D}^{0,α}

*δ* ( ˜*M**m*)*c e*^{−}^{(γ}^{n+1}^{(τ)+δ)m s}^{τ}*.*
Now, we claim that

*||G**m*(Q*m*(w))||_{D}^{0,α}

*δ* ( ˜*M**m*)*c m*^{2}*e*^{−m δ s}^{τ}*||w||*^{2}_{C}^{2,α}

*δ* ( ˜*M**m*)*.* (3.26)
This follows at once from the fact that the operator*Q**m*is quadratic in
*w*and from the fact that*w*can be decomposed as

*w*=*v*+ Φ,

where *v* *∈ D*^{2,α}* _{δ}* ( ˜

*M*

*m*) and Φ

*∈ K( ˜M*

*m*). Now, because of the modiﬁca- tion of

*χ*2,1Φ

^{0,−}

_{E}_{2,1}into

*χ*2,1Φ˜

^{0,−}

_{E}_{2,1}, we see that on

*M*1 and away from the