### About a family of deformations

### of the Costa-Hoffman-Meeks surfaces

### Filippo Morabito

**Abstract.** We show the existence of a family of minimal surfaces obtained by de-
formations of the Costa-Hoffman-Meeks surface of genus*k* 1,*M** _{k}*.These surfaces
are obtained varying the logarithmic growths of the ends and the directions of the axes
of revolution of the catenoidal type ends of

*M*

*.Also we obtain a result about the non degeneracy property of the surface*

_{k}*M*

*.*

_{k}**Keywords:** deformation, Jacobi operator, Costa-Hoffmann-Meeks surfaces.

**Mathematical subject classiﬁcation:** 53A10.

**Introduction**

C. Costa in [1, 2] described a genus one minimal surface with two ends asymptotic to the two ends of a catenoid and a middle end asymptotic to a plane.

D. Hoffman and W.H. Meeks in [5], [6] and [7] proved the global embedded-
ness for the Costa surface, and generalized it for higher genus. We will denote
the Costa-Hoffman-Meeks surface of genus*k* 1 by *M** _{k}*.For each

*k*1 is a properly embedded minimal surface and has three ends of ﬁnite total curvature.

J. Pérez and A. Ros in [11] studied the space_{M}of minimal surfaces of ﬁnite
total curvature, genus*k*and*r*ends, properly immersed inR^{3}and with embedded
horizontal ends. Given*M* ∈M,the inﬁnitesimal deformations of*M*are gener-
ated by the elements of *J*(*M*),the space of the Jacobi functions*u*on*M*,that is
functions such that *Lu*=0,where *L* denotes the Jacobi operator of*M*,which
have logarithmic growth at the ends. They showed that dim *J*(*M*) *r* +3.
They denoted by_{M}^{∗} = {*M* ∈ M: dim *J*(*M*)=*r* +3}the subspace of non
degenerate surfaces and proved

Received 6 March 2008.

**Theorem 1** (th. 6.7, [11]). _{M}^{∗} *is an open subset of* _{M} *and is a* (*r* +3)*-*
*dimensional real analytic manifold.*

The dimension of the space *J*(*M*)just introduced is known for *M* = *M** _{k}* for

*k*1.Indeed thanks to the works [9] and [10] by S. Nayatani, dim

*J*(

*M*

*)=6, since*

_{k}*r*=3,but only for 1

*k*37.Recently this result has been proved also for

*k*38 (see [8]). The elements of

*J*(

*M*

*)are the Jacobi ﬁelds associated with the horizontal translations, the rotation about the vertical direction and three functions (one for each end) whose form in a neighbourhood of an end is*

_{k}*a*ln|w|,being

*a*the logarithmic growth. Thus, the one parameter family of deformations of these surfaces, described by D. Hoffman and H. Karcher in [4], contains all the embedded surfaces nearby

*M*

*with a symmetry group generated by*

_{k}*k*vertical planes, up to dilations preserving the vertical direction.

In this work, following [11], we show the existence of a bigger family of
immersed minimal deformations of*M** _{k}*for

*k*1 having three embedded ends.

These surfaces do not enjoy any property of symmetry. In fact we admit the possibility to rotate, translate and dilate any of the three ends of the surface and, in addition, to bend the two catenoidal type ends and to transform the middle end from a planar type end into a catenoidal type end (we recall that the planar end can be thought as a catenoidal type end with null vertical ﬂux). We will prove the following result.

**Theorem 2.** *For each possible choice of the limit values of the normal vectors*
*of the three ends, there is, up to isometries, a*1-dimensional real analytic family
*of smooth minimal deformations of M*_{k}*, for k* 1, letting the middle planar end
*horizontal.*

Our result is a consequence of the moduli space theory and of the implicit function theorem. We do not treat the case where also the middle planar end is not horizontal because it can be reconduced to the previous one by an isometry.

The family of surfaces described in the statement of the theorem here, contains
the 1-parameter family of deformations of *M** _{k}*, for 1

*k*37, obtained by L. Hauswirth and F. Pacard in [3] bending the top and the bottom end and letting horizontal the middle planar end. All the surfaces of this family are not embedded and are symmetric with respect to the vertical plane{

*x*

_{2}=0}that in particular contains the axis of the catenoidal type ends (it is assumed to be the same for the two ends). The parameter is the angle between this axis and the vertical direction. This family is used in the same work to construct some new examples of minimal surfaces by a gluing technique.

**Degeneracy and non degeneracy of the Costa-Hoffman-Meeks surfaces**
Let *K*_{0} denote theC^{2}^{,α} elements of the kernel of the Jacobi operator about the
compactiﬁcation of*M*∈M,*Killing*the space of the Jacobi ﬁelds induced by the
isometries of the ambient space and*r* the number of the ends. We set*Killing*0=
*Killing*∩*K*_{0}.In [11] the authors give the following deﬁnition of non degeneracy.

**Deﬁnition 3**([11]). *A minimal surface is non degenerate if Killing*0= *K*0*.*
J. Pérez and A. Ros show in Proposition 5.3 of [11] that the non degeneracy of
*M*is equivalent to the equality dim*J*(*M*)=*r* +3 and obtain Theorem 1. So if
a minimal surface*M*is non degenerate then the set of the minimal immersions
near*M*with horizontal ends has a nice behaviour.

**Remark 4.** The works [9] and [10] by S. Nayatani and [8] by the author about
the number of bounded Jacobi functions ensure that *M** _{k}* is non degenerate for
1

*k*<+∞.

In section 2.2 we will prove that *M** _{k}* is non degenerate with respect to the
Deﬁnition 3 but in a more general setting. We remind that the minimal surfaces
considered in [11] (where Deﬁnition 3 is introduced) have horizontal embedded
ends. On the contrary here we suppose the surfaces can have non horizontal
embedded ends.

Now we are going to explain why *M**k* is degenerate with respect to the dif-
ferent deﬁnition used by L. Hauswirth and F. Pacard in [3]. They studied the
mapping properties of the Jacobi operator of *M** _{k}* acting on the space of the

_{C}

_{δ}

^{2}

^{,α}functions deﬁned on

*M*

*k*and that are invariant under the action of the symmetry with respect to the vertical plane{

*x*2=0}.In particular if

*f*∈ C

_{δ}

^{2}

^{,α}(

*M*

*),then*

_{k}*f*=

*O*(

*e*

^{δ}

*)on the catenoidal type ends. The mapping properties of the Jacobi operator (denoted by*

^{s}*L*

_{δ}) acting on functions of

_{C}

^{2}

_{δ}

^{,α}(

*M*

*k*)depend on the choice ofδ.Their deﬁnition of non degeneracy of

*M*

*is the following one.*

_{k}**Deﬁnition 5** ([3]). *The surface M*_{k}*is non degenerate if the operator L*_{δ} *is*
*injective for all*δ <−1.

Thanks to the works [9] and [10] by S. Nayatani and [8] by the author, the
space *K* ⊂ *J*(*M**k*)of the bounded Jacobi functions, is known to be generated
by the functions*N*,*e*_{1},*N*,*e*_{2}and*N*,*e*_{3},*N*,*e*_{3}×*p*,where*N*denotes the
normal vector ﬁeld about*M** _{k}*, (

*e*

_{1},

*e*

_{2},

*e*

_{3})is the canonical basis ofR

^{3}and

*p*the position vector on

*M*

*k*.These functions are associated with 4 isometries of the ambient space: the three translations and the rotation about the

*e*3-axis. In [3] the authors remark that the Jacobi function

*N*,

*e*

_{3}×

*p*associated with the rotation

about the*e*3-axis and*N*,*e*2associated with the translation along the*e*2-axis do
not respect the mirror symmetry described above, that is they are not invariant
with respect to the action of the map(*x*_{1},*x*_{2},*x*_{3})→ (*x*_{1},−*x*_{2},*x*_{3}).So they did
not taken into account them and could conclude that *M** _{k}* is non degenerate, in
the sense of their deﬁnition.

The surfaces of the family described in our work do not enjoy any prop-
erty of symmetry, since we admit to bend the catenoidal type ends in arbitrary
directions. Then the Jacobi functions described above must be taken into ac-
count. Since the Jacobi function*N*,*e*_{3}× *p*belongs to the space_{C}_{δ}^{2}^{,α}(*M** _{k}*)for
δ = −

*k*−1 −2,the property of non degeneracy does not hold any more.

Actually the operator*L*_{δ}acting onC_{δ}^{2}^{,α}(*M** _{k}*)is no more injective for allδ <−1.

As consequence, we can state that for all *k* 1 the Costa-Hoffman-Meeks
surface*M** _{k}*is degenerate in the sense of Deﬁnition 5.

**1** **Preliminaries and notation**

We denote by *X*: *M** _{k}* → R

^{3}the conformal minimal immersion of the Costa- Hoffman-Meeks surface

*M*

*inR*

_{k}^{3}. If

*g*andηare the Weierstrass data of

*M*

*, we can write:*

_{k}*X*(*z*)=
1

2

*g*^{−}^{1}η− 1
2

*g*η,Re

η

∈C×R=R^{3}. (1)
The meromorphic function*g*is the stereographic projection from the north pole
of the Gauss map*N*: *M** _{k}* →S

^{2}. The total curvature is ﬁnite and

*M*

*is confor- mally diffeomorphic to*

_{k}*M*

*\ {*

_{k}*p*

*,*

_{t}*p*

*,*

_{b}*p*

*},being*

_{m}*M*

*a compact surface and*

_{k}*p*

*three points. The Weierstrass data extend in a meromorphic way at each puncture*

_{i}*p*

*.In particular the Gauss map of*

_{i}*X*(

*z*)is well deﬁned at

*p*

*.The points*

_{i}*p*

*are identiﬁed with the ends and a neighbourhood of a puncture will parametrize the corresponding end. In the sequel we will refer to various quantities related to the three ends of the surface using the index*

_{i}*t*for the top end, the index

*b*for the bottom end and the index

*m*for the planar end. The Gauss map

*N*takes the limit values(0,0,1)at the ends

*p*

*and*

_{t}*p*

*and(0,0,−1)at the end*

_{b}*p*

*.*

_{m}We parametrize the ends*p** _{i}*in the graph coordinate

*x*=

*x*

_{1}+

*i x*

_{2}on

*D*

_{i}^{∗}(ε

*i*)= {

*x*∈C;0<|

*x*|

*i*}by the immersions

*X** _{i}*(

*x*)= 1

*x*,−˜*a** _{i}*ln|

*x*| +

*h*

*(*

_{i}*x*)

∈C×R=R^{3}

for*i* =*t*,*b*,where*h**i*is a smooth real valued function on*D*_{i}^{∗}(ε*i*).The quantities

˜

*a** _{i}* and

*h*

*(0)are called the logarithmic growth and the height of the end. We can observe that, for the null ﬂux condition,*

_{i}*a*˜

*= −˜*

_{t}*a*

*.*

_{b}As for the planar end *p** _{m}*,we will use the following parametrization

*X*

*(*

_{m}*x*)=

1
*x*,*h** _{m}*(

*x*)

on *D*_{m}^{∗}(*m*).

So its logarithmic growth is zero.

**1.1** **The mean curvature operator at an end**

Let us consider a not necessarily minimal immersion *E*: *D*^{∗}(ε)→ R^{3} deﬁned
by

*E*(*x*)=
1

*x*,−*a*ln|*x*| +*h*(*x*)

.

We denote by *ds*_{0}^{2} the ﬂat metric of the *x*-plane. We set ρ = |*x*| and *f* =

−*a*lnρ+*h*.The induced metric*ds*^{2}=(*g** _{i j}*)is given by ([11])

*g*

*i j*(

*x*)= 1

ρ^{4}

δ*i j* +ρ^{4}∂*i* *f*∂*j* *f*

. (2)

If we denote by *d A* (respectively *d A*_{0}) the area measure associated with the
metric*ds*^{2}(respectively*ds*_{0}^{2}), then (2) implies that

*d A*= *Q*^{1}^{2}
ρ^{4}*d A*0

where *Q* = 1+ |*x*|^{2}(*a*^{2}+ |*x*|^{2}|∇0*h*|^{2}−2a*x*,∇0*h*)(∇0 denotes the gradient
computed with respect to the ﬂat metric*ds*_{0}^{2}).

The Gauss map of *E*is given by ([11])
*N*(*x*)=*Q*^{−}^{1}^{2}

−*ax*¯+ ¯*x*^{2}∇0*h*,1

, (3)

where *x*¯^{2}∇^{0}*h* means the product of the complex number *x*¯^{2} with the gradient

∇0*h.*

The mean curvature of the immersion*E* is
*H* = ρ^{4}

2 *di*v0

∇0*f*
1+ρ^{4}|∇0*f*|^{2}

.

**2** **The deformation of the surface and its Jacobi operator**

In this section we describe how we deform the surface *M** _{k}* following the ideas
of [11]. In subsection 2.1 we introduce the Jacobi operator of

*M*

*and we study its kernel and its range.*

_{k}We will construct a family of deformations of *M** _{k}* using in particular Theo-
rem 4.1 of [11]. The ﬁrst step is the construction of a new immersion of

*M*

*in R*

_{k}^{3}starting from

*X*(

*z*) given by (1). Using a smooth cut-off function we glue

*X*:

*M*

*\(*

_{k}*D*

_{t}^{∗}(ε

*t*)∪

*D*

_{b}^{∗}(ε

*b*)∪

*D*

_{m}^{∗}(ε

*m*)) → R

^{3}with the parametrizations of the three ends with a different value of the logarithmic growths (that we denote by

*a*

*,*

_{t}*a*

*,*

_{b}*a*

*). Furthermore we rotate the ends*

_{m}*p*

*and*

_{t}*p*

*, that is we change the directions of their axes of revolution. Secondly we consider small variations of the immersion just constructed with respect an appropriate smooth vector ﬁeld.*

_{b}Thanks to Theorem 4.1 of [11], each immersed minimal surface having prop-
erly embedded ends with ﬁnite total curvature and ﬁxed topology, that is in a
neighbourhood of*M** _{k}*,admits an immersion constructed in this way. We remark
this theorem has been proved for minimal surfaces with horizontal ends but it
holds also in our setting.

To give the details of the construction we need to introduce some notation.

We denote by *F*(θ1,*i*, θ2,*i*)the frame deﬁned by the following unit vectors:

*e*1(θ^{1},*i*, θ^{2},*i*)=cosθ^{1},*i**e*1+sinθ^{1},*i*sinθ^{2},*i**e*2+sinθ^{1},*i*cosθ^{2},*i**e*3,
*e*2(θ^{1},*i*, θ^{2},*i*)=cosθ^{2},*i**e*2−sinθ^{2},*i**e*3,

*e*_{3}(θ1,*i*, θ2,*i*)= −sinθ1,*i**e*_{1}+cosθ1,*i*sinθ2,*i**e*_{2}+cosθ1,*i*cosθ2,*i**e*_{3},
(4)

where(*e*_{1},*e*_{2},*e*_{3})denotes the canonical base ofR^{3}.

We deﬁne the immersions of the rotated catenoidal type ends on *D*^{∗}* _{i}*(ε

*i*)as

*X*

_{i}_{,θ}

_{1,i}

_{,θ}

_{2,i}(

*x*) =

*x*1

|*x*|^{2}*e*_{1}(θ1,*i*, θ2,*i*)− *x*2

|*x*|^{2}*e*_{2}(θ1,*i*, θ2,*i*)
+(−*a**i*ln|*x*| +*h**i*(*x*))*e*3(θ1,*i*, θ2,*i*),

for*i* = *t*,*b. As for the planar end, we consider on* *D*_{m}^{∗}(ε*m*)in the canonical
frame(*e*1,*e*2,*e*3)the immersion

*X*_{m}_{,}_{0}_{,}_{0}(*x*)=
1

*x*,−*a** _{m}*ln|

*x*| +

*h*

*(*

_{m}*x*)

.
We deﬁne *y*=(*a**t*,*a**b*,*a**m*, θ1,*t*, θ2,*t*, θ1,*b*, θ2,*b*).

Using a smooth cut-off function we can glue the three immersions above,
deﬁned on*D*_{i}^{∗}(ε*i*),*i* =*t*,*b*,*m*,to the restriction of*X*(*z*)to*M** _{k}*\

∪*i*=*t*,*b*,*m**D*_{i}^{∗}(ε*i*)

and obtain a new immersion we denote by*X** _{y}*.It is not necessarily minimal and
depends smoothly on

*y*.

Now let*N*(*y*)∈C^{∞}(*M** _{k}*,R

^{3})be a smooth vector ﬁeld such that

*N*(

*y*),

*N*= 1 on

*M*

*\(*

_{k}*D*

_{t}^{∗}(ε

*t*)∪

*D*

_{b}^{∗}(ε

*b*))and

*N*(*y*)= *e*3(θ1,*i*, θ2,*i*)
*N*,*e*3(θ^{1},*i*, θ^{2},*i*)

on *D*_{i}^{∗}(ε*i*) for*i* = *t*,*b*.We remark that we do not modify the normal vector
ﬁeld on *D*^{∗}* _{m}*(ε

*m*)because we keep the middle planar end horizontal. Let

_{A}be a neighbourhood of(

*a*˜

*,*

_{t}*a*˜

*,0)(the logarithmic growths of the ends of*

_{b}*M*

*),*

_{k}_{U}a neighbourhood of zero inC

^{2}

^{,α}(

*M*

*).For*

_{k}*y*∈A×[−ε, ε]

^{4}and a function

*u*∈U, we consider the family of immersions{

*X*

_{y}_{,}

*}such that*

_{u}*X*_{y}_{,}* _{u}* :=

*X*

*+*

_{y}*uN*(

*y*):

*M*

*−→R*

_{k}^{3}. (5) Such a family depends analytically on (

*y*,

*u*). As we have anticipated, Theo- rem 4.1 of [11] ensures that each immersed minimal surface having properly embedded ends with ﬁnite total curvature that is in a neighbourhood of

*M*

*, admits an immersion in R*

_{k}^{3}which is in this family. In other terms each of them is the graph of a function

*u*with respect the vector ﬁeld

*N*(

*y*)about the surface whose immersion is

*X*

*y*.Each immersion is determined by an element (

*y*,

*u*)∈A× [−ε, ε]

^{4}×U.In particular if

*y*=(

*a*˜

*,*

_{t}*a*˜

*,0,0,0,0,0)the immer- sion is the one of*

_{b}*M*

*(that is actually an embedding).*

_{k}Now we are going to introduce the mean curvature operator of the immer-
sion *X*_{y}_{,}* _{u}*and its “compactiﬁcation”.

Letλ∈ C^{∞}(*M** _{k}*)be a positive function which in terms of the graph coordi-
nate

*x*,is deﬁned by

λ(*x*)=

⎧⎨

⎩ 1

|*x*|^{4} on*D*_{t}^{∗}(ε*t*), *D*^{∗}* _{b}*(ε

*b*),

*D*

_{m}^{∗}(ε

*m*),

1 on*M** _{k}*\(

*D*

_{t}^{∗}(2ε

*t*)∪

*D*

^{∗}

*(2ε*

_{b}*b*)∪

*D*

_{m}^{∗}(2ε

*m*)). (6) If

*ds*

^{2}denotes the induced metric on

*M*

*,then (2), which gives the expression of the metric for a catenoidal type end, implies that*

_{k}*ds*¯

^{2}=(1/λ)

*ds*

^{2}is a Rieman- nian metric on

*M*

*.We denote the associated area mesure by*

_{k}*dA*¯.If

*H*(

*y*,

*u*)is the mean curvature operator of the immersion

*X*

_{y}_{,}

*=*

_{u}*X*

*+*

_{y}*uN*(

*y*), we deﬁne the operator

*H*(

*y*,

*u*)=λ

*H*(

*y*,

*u*). Since at the ends

*p*

*and*

_{t}*p*

*, the rotation does not change the value of the mean curvature, we can apply Lemma 6.4, proved*

_{b}in [11], at each end to conclude that there exist ε and neighbourhoods_{A}, U
such that the operator

*H*:A× [−ε, ε]^{4}×U→C^{0}^{,α}(*M** _{k}*)
is real analytic.

**2.1** **The Jacobi operator**

The Jacobi operator*L*of*M** _{k}*is given by

*L* =*ds*^{2} + |*A*|^{2}_{ds}^{2},

where*ds*^{2} and|*A*|^{2}* _{ds}*2are respectively the Laplacian and the norm of the second
fundamental form with respect to the metric

*ds*

^{2}(the metric on

*M*

*).*

_{k}The geometric meaning of *L* can be explained in the following way (see
Section 5 of [11]). Let{*M** _{k}*(

*t*)}|

*t*|<εdenote a family of smooth deformations of

*M*

*,such that*

_{k}*M*

*(0)=*

_{k}*M*

*and let*

_{k}*H*(

*t*)denote the mean curvature operator of

*M*

*(*

_{k}*t*).Ifψ

*t*:

*M*

*→R*

_{k}^{3}is the immersion inR

^{3}of

*M*

*(*

_{k}*t*),andw=

_{dt}

^{d}_{|}

_{t}_{=}

_{0}ψ

*t*,

*N*, then we have the equality

*d*
*dt*_{|}*t*=0

*H*(*t*)= 1

2*L*w. (7)

If *Lu* = 0 is satisﬁed,*u* is called Jacobi function on *M** _{k}* and it corresponds to
an inﬁnitesimal deformation of

*M*

*by minimal surfaces. The operator*

_{k}*L*can be “compactiﬁed” to obtain the operator

*L*=

*d*

*s*¯

^{2}+ |

*A*|

^{2}

_{d}

_{s}_{¯}2 =λ

*L*on

*M*

*(the functionλis deﬁned by (6)). It is related to the differential of*

_{k}*H*(

*t*) = λ

*H*(

*t*) by a relation similar to (7).

In the sequel we will consider the family of deformations of *M** _{k}* constructed
in the following way. We deﬁne

˜
*y*=

˜

*a**t*,*a*˜*b*,0,0,0,0,0

and *y*˙ =

˙

*a**t*,*a*˙*b*,*a*˙*m*,θ˙1,*t*,θ˙2,*t*,θ˙1,*b*,θ˙2,*b*

and consider a smooth curve

γ (*t*)=

*a** _{t}*(

*t*),

*a*

*(*

_{b}*t*),

*a*

*(*

_{m}*t*), θ1,

*t*(

*t*), θ2,

*t*(

*t*), θ1,

*b*(

*t*), θ2,

*b*(

*t*)

, (8)

for|*t*|< ε,such thatγ (0)= ˜*y*,with accelerationγ^{}(0)= ˙*y*and a function
*u*(*t*): {*t* ∈R,|*t*|< ε} →C^{2}^{,α}(*M** _{k}*)

such that*u*(0)=0.

We deﬁne the family of deformations{*M** _{k}*(

*t*)}|

*t*|<εof

*M*

*as the family of im- mersed surfaces whose immersion inR*

_{k}^{3}equals

*X*

_{γ (}

_{t}_{),}

_{u}_{(}

_{t}_{)}=

*X*

_{γ (}

_{t}_{)}+

*u*(

*t*)

*N*(γ (

*t*)) (see (5)). We remark that in general

*M*

*(*

_{k}*t*)is not a minimal surface.

We are going to give the expression of the Jacobi functions on *M** _{k}* deﬁned
by

_{dt}

^{d}_{|}

_{t}_{=}

_{0}

*X*

_{γ (}

_{t}_{),}

_{u}_{(}

_{t}_{)},

*N*. To do that we need to introduce additional notation.

Let *f*_{1}, *f*_{2}, *f*_{3}be the functions deﬁned by:

*f*_{1}(*x*,*i*)= *x*1

|*x*|^{2}*N*,*e*_{3} −(−˜*a** _{i}*ln|

*x*| +

*h*

*(*

_{i}*x*))

*N*,

*e*

_{1}, (9)

*f*

_{2}(

*x*,

*i*)=

*x*

_{2}

|*x*|^{2}*N*,*e*_{3} +(−˜*a** _{i}*ln|

*x*| +

*h*

*(*

_{i}*x*))

*N*,

*e*

_{2}, (10)

*f*

_{3}(

*x*,

*a*˙

*)= −˙*

_{i}*a*

*ln|*

_{i}*x*|

*N*,

*e*

_{3}(11) for

*x*∈

*D*

_{i}^{∗}(ε

*i*) with

*i*=

*t*,

*b*,

*m*, and

*f*

*= 0,*

_{n}*n*= 1,2,3, in

*M*

*\*

_{k}*D*

_{i}^{∗}(2ε

*i*).

*f** _{n}* are a smooth interpolation of previous values on the remaining part of

*M*

*. We recall that*

_{k}*a*˜

*,*

_{t}*a*˜

*are the logarithmic growths of the top and of the bottom end of*

_{b}*M*

*, and since the middle planar end is horizontal, we have*

_{k}*a*˜

*=0.*

_{m}**Proposition 6.** *The Jacobi functions about M*_{k}*have, in D*_{i}^{∗}(ε*i*),*the following*
*expression*

θ˙1,*i**f*_{1}(*x*,*i*)+ ˙θ2,*i* *f*_{2}(*x*,*i*)+ *f*_{3}(*x*,*a*˙* _{i}*)+

*u*

_{i}*for i* =*t*,*b*,*m*, *with* θ˙1,*m* = ˙θ2,*m* =0, *a*˜* _{m}* =0

*and u*

*∈C*

_{i}^{2}

^{,α}(

*D*

*(ε*

_{i}*i*)).

**Proof.** A Jacobi function is deﬁned by
*d*

*dt*_{|}*t*=0

*X*_{γ (}_{t}_{)}+*u*(*t*)*N*(γ (*t*))

,*N*. (12)

We observe that*X*_{γ (}_{t}_{)}in*D*_{i}^{∗}(ε*i*)is given by
*x*_{1}

|*x*|^{2}*e*1(θ^{1},*i*(*t*), θ^{2},*i*(*t*))− *x*_{2}

|*x*|^{2}*e*2(θ^{1},*i*(*t*), θ^{2},*i*(*t*))
+(−*a** _{i}*(

*t*)ln|

*x*| +

*h*

*(*

_{i}*x*))

*e*

_{3}(θ1,

*i*(

*t*), θ2,

*i*(

*t*)),

(13)

for*i* =*t*,*b*and in*D*_{m}^{∗}(ε*m*)by
1

*x*,−*a** _{m}*(

*t*)ln|

*x*| +

*h*

*(*

_{m}*x*)

.

To simplify the notation we will omit the dependence on the end wherever it is
possible, replacingθ*k*,*i*(*t*)byθ*k*(*t*)and*a** _{i}*(

*t*)by

*a*(

*t*).

To obtain _{dt}^{d}_{|}

*t*=0*X*_{γ (}_{t}_{)}we need to compute

˙

*e**j*(0)= *d*
*dt*_{|}*t*=0

*e**i*(θ1(*t*), θ2(*t*)), *j*=1,2,3.

So we suppose thatθ1(0)=θ2(0)=0. We observe that from equation (4) since
*e*_{1}(θ1(*t*), θ2(*t*))=cosθ1(*t*)*e*_{1}+sinθ1(*t*)sinθ2(*t*)*e*_{2}+sinθ1(*t*)cosθ2(*t*)*e*_{3},
we have

˙

*e*_{1}(*t*)= −θ1^{}(*t*)sinθ1(*t*)*e*_{1}
+

θ1^{}(*t*)cosθ1(*t*)sinθ2(*t*)+θ2^{}(*t*)sinθ1(*t*)cosθ2(*t*)
*e*_{2}
+

θ1^{}(*t*)cosθ^{1}(*t*)cosθ^{2}(*t*)−θ2^{}(*t*)sinθ^{1}(*t*)sinθ^{2}(*t*)
*e*3,
then using the initial conditions, we obtain

˙

*e*1(0)=θ1^{}(0)*e*3. (14)
In a similar way we obtain*e*˙* _{j}*(0),with

*j*=2,3.We ﬁnd

˙

*e*2(0)= −θ2^{}(0)*e*3, (15)

˙

*e*3(0)= −θ1^{}(0)*e*1+θ2^{}(0)*e*2. (16)
Then from equations (13), (14), (15) and (16)

*d*
*dt*_{|}*t*=0

*X*_{γ (}_{t}_{)} = *x*_{1}

|*x*|^{2}θ1^{}(0)*e*3− *x*_{2}

|*x*|^{2}(−θ2^{}(0)*e*3)

+(−*a*(0)ln|*x*| +*h*(*x*))(−θ1^{}(0)*e*_{1}+θ2^{}(0)*e*_{2})+(−*a*^{}(0)ln|*x*|)*e*_{3}.
Collecting the summands in terms of theθ_{k}^{}(0)≡θ_{k}^{}_{,}* _{i}*(0)and taking into account
the deﬁnitions (9), (10) and (11) of the

*f*

*functions, we get*

_{k}*d*
*dt*_{|}*t*=0

*X*_{γ (}_{t}_{)},*N* =θ1^{},*i*(0)*f*1(*x*,*i*)+θ2^{},*i*(0)*f*2(*x*,*i*)+ *f*3(*x*,*a*_{i}^{}(0)).

As for the last term of (12), we recall that*u*(0) = 0 and, on *D*^{∗}* _{i}*(ε

*i*),from the deﬁnition of

*N*˜(

*y*)it holds that

*N*(γ (*t*))= *e*3(θ1,*i*(*t*), θ2,*i*(*t*))
*N*,*e*3(θ1,*i*(*t*), θ2,*i*(*t*)).
Then_{dt}* ^{d}*(

*u*(

*t*)

*N*(γ (

*t*))),evaluated in

*t*=0,is equal to

*u*^{}(0)*N*(γ (0))+*u*(0)*d*
*dt*_{|}*t*=0

*N*(γ (*t*))= *u*^{}(0)
*N*,*e*_{3}*e*3.

If*u** _{i}* denotes the restriction of

*u*

^{}(0)to

*D*

^{∗}

*(ε*

_{i}*i*)for

*i*=

*t*,

*b*,

*m*,then the result is

obvious.

**Lemma 7.** *Let U*,*V* ∈ C^{2}^{,α}(*M** _{k}*)

*be the functions deﬁned in D*

^{∗}

*(ε*

_{i}*i*),

*for i*∈ {

*t*,

*b*,

*m*},

*by*

*U** _{i}*(

*x*)= ˙θ1,

*i*

*f*

_{1}(

*x*,

*i*)+ ˙θ2,

*i*

*f*

_{2}(

*x*,

*i*)+

*f*

_{3}(

*x*,

*a*˙

*)+*

_{i}*u*

*(*

_{i}*x*)

*and*

*V** _{i}*(

*x*)= ˙ϕ1,

*i*

*f*

_{1}(

*x*,

*i*)+ ˙ϕ2,

*i*

*f*

_{2}(

*x*,

*i*)+

*f*

_{3}(

*x*,

*b*˙

*)+v*

_{i}*i*(

*x*),

*with* θ˙*j*,*i*,ϕ˙*j*,*i* ∈ R *and* *a*˜* _{m}* = 0,θ˙

*j*,

*m*= ˙ϕ

*j*,

*m*= 0,

*j*= 1,2,

*and u*

*, v*

_{i}*i*∈ C

^{2}

^{,α}(

*D*

*(ε*

_{i}*i*)).

*Then we have*

*M**k*

(*UL V*¯ −*VLU*¯ )*dA*¯ =

*M**k*

(*U L V* −*V LU*)*d A*

=2π

*i*∈{*t*,*b*}

˙*i*,∇*u** _{i}*(0) − ˙

*i*,∇v

*i*(0)

+2π

*i*∈{*t*,*b*,*m*}

[ ˙*b*_{i}*u** _{i}*(0)− ˙

*a*

*v*

_{i}*i*(0)],

*with* ˙*i* =(θ˙1,*i*,θ˙2,*i*), ˙*i* =(ϕ˙1,*i*,ϕ˙2,*i*) *and* ∇· =(∂*x*_{1}·, ∂*x*_{2}·).

**Proof.** The proof makes use of the Green identity, so ﬁrst of all we obtain the
local expression of the Weiertrass representation of a properly embedded end
*p* with ﬁnite total curvature of a minimal immersion, ψ, inR^{3},in terms of a
conformal coordinate*z*=*r e*^{i}^{α} centered at *p*.

Let’s suppose that the Weiestrass representation ofψ is ψ=

1 2

*g*^{−}^{1}η−

*g*η

,Re

η

. (17)

It is known (see for example Section 2 of [11]) that properly embedded minimal
ends with ﬁnite total curvature must be asymptotic to the end of a catenoid or of
a plane. Indeed from the hypotheses on *p*it follows that the Weierstrass data of
ψin terms of the conformal coordinatewcentered at the end are given by:

*g*(w)= *q*(w)

w* ^{k}* , η(w)=

*s*(w)w

^{k}^{−}

^{2}

*d*w,

for w ∈ *D*^{∗}(ε), *k* ∈ N^{∗}. The functions*q*(w)and*s*(w)are holomorphic and
satisfy*q*(0)=0,*s*(0)= −*a* ∈R^{∗}.From (17) we obtain

ψ(w)= 1

2

*s*(w)

*q*(w)w^{2k}^{−}^{2}*d*w−

*q*(w)*s*(w)
w^{2} *d*w

,Re

*s*(w)w^{k}^{−}^{2}*d*w

,

If*k* =1, ψis asymptotic to the end of a vertical catenoid (under the additional
hypothesis(*qs*)^{}(0)=0, which ensuresψis well deﬁned). Indeed we can write

ψ(w)=
*l*(w)

w ,−*a*ln|w| +v(w)

. (18)

So*a*is the logarithmic growth of the end andvis a smooth function on *D*(ε).

Now we consider the change of coordinate*z* = −_{q}^{w}_{(}_{0}_{)}*e*^{−}^{v(w)}* ^{a}* .In the new con-
formal coordinate we can write:

*g*(*z*)= −1

*z* + ˜*t*(*z*), η(*z*)= −*ad z*
*z* .

Replacing*g*(*z*)andη(*z*)in (17) we get the expression ofψin terms of*z:*

ψ(*z*)=
*a*

2

¯
*z*+1

*z*

+*t*(*z*),−*a*ln|*z*| +*const*

. (19)

We denoted by*x* the graph coordinate around the catenoidal end *p** _{i}* of

*M*

*and by*

_{k}*a*˜

*its logarithmic growth. Then, for the catenoidal type ends of*

_{i}*M*

*,we get from (19) the following equality*

_{k}1
*x* = *a*˜_{i}

2z(1+ |*z*|^{2}+*zt** _{i}*(

*z*))=

*s*

*(*

_{i}*z*)

*z* (20)

with*s** _{i}*(0)=

^{a}^{˜}

_{2}

*,*

^{i}*i*∈ {

*t*,

*b*}.

In the case of the planar end *p** _{m}*, that is for

*k*2, the third coordinate function in (18) is bounded andψis asymptotic to the end of a horizontal plane.

Similar arguments lead us to 1

*x* = *s** _{m}*(

*z*)

*z*where

*s*

*(0)=0.*

_{m}The next step is to ﬁnd the expressions of*U* and*V* near the ends in terms of
(*r*, α)coordinates. From (3) we obtain that, for an end with logarithmic growth
*a*,it holds that:

*N*,*e*3 = *Q*^{−}^{1}^{2} =

1+ |*x*|^{2}(*a*^{2}+ |*x*|^{2}|∇0*h*|^{2}−2a*x*,∇0*h*)_{−}^{1}_{2}
,
*N*,*e*_{1} = *N*,*e*_{3}Re(−*ax*¯+ ¯*x*^{2}∇0*h*),

*N*,*e*_{2} = *N*,*e*_{3}Im(−*ax*¯ + ¯*x*^{2}∇0*h*).

Then in a neighbourhood of each end we can write:

*N*,*e*3 =

1+*O*(|*x*|^{2})_{−}^{1}_{2}

=1+*O*(|*x*|^{2}), (21)
*N*,*e*1 =

1+*O*(|*x*|^{2}) −*ax*1+*O*(*x*¯^{2})

= −*ax*1+*O*(|*x*|^{2}), (22)
*N*,*e*_{2} =

1+*O*(|*x*|^{2}) *ax*_{2}+*O*(*x*¯^{2})

=*ax*_{2}+*O*(|*x*|^{2}). (23)
In(*r*, α)coordinates,*U** _{i}* and

*V*

*have the following expressions:*

_{i}*U** _{i}*(

*r*)= ˙θ1,

*i*

*f*1(

*r*,

*i*)+ ˙θ2,

*i*

*f*2(

*r*,

*i*)+

*f*3(

*r*,

*a*˙

*)+*

_{i}*u*

*(*

_{i}*r*),

*V*

*i*(

*r*)= ˙ϕ1,

*i*

*f*1(

*r*,

*i*)+ ˙ϕ2,

*i*

*f*2(

*r*,

*i*)+

*f*3(

*r*,

*b*˙

*i*)+v

*i*(

*r*) where

*f*1(*r*,*i*)= *a*˜* _{i}*cosα

2r +*O*(*r*ln*r*),
*f*_{2}(*r*,*i*)= *a*˜* _{i}*sinα

2r +*O*(*r*ln*r*),
*f*3(*r*,*a*)= −*a*ln*r* +*O*(*r*).

If *D** _{i}*(0,

*r*)are conformal disks and

*M*(

*r*) =

*M*\(∪

*i*∈{

*t*,

*b*,

*m*}

*D*

*(0,*

_{i}*r*)),then the conformal invariance of the integral implies:

*I*(*r*) =

*M*(*r*)(*U L V* −*V LU*)*d A*=

∂*M*(*r*)

*U*∂*V*

∂η −*V*∂*U*

∂η

*ds*

= −

*i*∈{*t*,*b*,*m*}

∂*D**i*(0,*r*)

*U** _{i}*∂

*V*

_{i}∂*r* −*V** _{i}*∂

*U*

_{i}∂*r*

|*d z*|,

(24)

where*d A* is the area measure associated with*ds*^{2}, η is the exterior conormal
ﬁeld to the immersion along∂*M*(*r*)and|*d z*| = *r d*α.To get the lemma it will
be sufﬁcient to let*r* go to zero.

Of course we have for*i*∈ {*t*,*b*,*m*} :

∂*U*_{i}

∂*r* = ˙θ1,*i*∂*f*_{1}(*r*)

∂*r* + ˙θ2,*i*∂*f*_{2}(*r*)

∂*r* +∂*f*_{3}(*r*,*a*˙* _{i}*)

∂*r* + ∂*u** _{i}*(

*r*)

∂*r*
and a similar expression for ^{∂}_{∂}^{V}_{r}* ^{i}* :

∂*V*_{i}

∂*r* = ˙ϕ1,*i*

∂*f*_{1}(*r*)

∂*r* + ˙ϕ2,*i*

∂*f*_{2}(*r*)

∂*r* +∂*f*_{3}(*r*,*b*˙* _{i}*)

∂*r* +∂v*i*(*r*)

∂*r* .

Given a*C*^{2}^{,α}function*l*we will write it by its Taylor expansion in the coordinate
*z* =*z*_{1}+*i z*_{2}=*r e*^{i}^{α},i.e.,

*l*=*l*(0)+*r*cosα(∂*z*1*l*)(0)+*r*sinα(∂*z*2*l*)(0)+*O*(*r*^{2}). (25)
Now we proceed with the evaluation of the limit as *r* → 0 of each summand
that appears in (24). For*i* ∈ {*t*,*b*,*m*}we have (to simplify the notation, we will
omit the dependence on*r* and*i)*

lim

*r*→0

∂*D** _{i}*(0,

*r*)

*U** _{i}*∂

*V*

_{i}∂*r* −*V** _{i}*∂

*U*

_{i}∂*r*

|*d z*|

=lim

*r*→0

{|*z*|=*r*}

˙

ϕ^{1},*i**u** _{i}*(

*z*)∂

*f*

_{1}

∂*r* − ˙θ^{1},*i*v*i*(*z*)∂*f*_{1}

∂*r*

+
∂v*i*

∂*r* θ˙^{1},*i* *f*1−∂*u*_{i}

∂*r* ϕ˙^{1},*i* *f*1

+

˙

ϕ^{2},*i**u** _{i}*(

*z*)∂

*f*

_{2}

∂*r* − ˙θ^{2},*i*v*i*(*z*)∂*f*_{2}

∂*r*

+
∂v*i*

∂*r* θ˙^{2},*i* *f*2−∂*u*_{i}

∂*r* ϕ˙^{2},*i**f*2

+

*u** _{i}*(

*z*)∂

*f*

_{3}(

*b*˙

*)*

_{i}∂*r* −v*i*(*z*)∂*f*_{3}(*a*˙* _{i}*)

∂*r*

+

*u** _{i}*∂v

*i*

∂*r* −v*i*

∂v*i*

∂*r*

|*d z*|.

We deﬁne (the expression of*l*is given by (25)):

*G*(*l*)= lim

*r*→0

{|*z*|=*r*}*l*(*r*)∂*f*1

∂*r* |*d z*|

= −lim

*r*→0

{|*z*|=*r*}

*l*(0)+*r*(cosα(∂*z*_{1}*l*)(0)+sinα(∂*z*_{2}*l*)(0))+*O*(*r*^{2})

×

*a*˜*i*cosα

2r^{2} +*O*(ln*r*)

*r d*α

= −lim

*r*→0

{|*z*|=*r*}

*l*(0)+*r*(cosα(∂*z*1*l*)(0)+sinα(∂*z*2*l*)(0))*a*˜* _{i}*cosα

2r *d*α+*O*(*r*ln*r*)

= −lim

*r*→0

{|*z*|=*r*}

*l*(0)

*r* +sinα(∂*z*2*l*)(0)

*a*˜* _{i}*cosα
2

*d*α

−lim

*r*→0

{|*z*|=*r*}

˜

*a** _{i}*(∂

*z*1

*l*)(0)

2 cos^{2}α*d*α+*O*(*r*ln*r*).

Then, since lim

*r*→0

{|*z*|=*r*}

*l*(0)

*r* +sinα(∂*z*_{2}*l*)(0)

*a*˜* _{i}*cosα
2

*d*α

=lim

*r*→0

˜
*a*_{i}*l*(0)

2r 2π

0

cosα*d*α+*a*˜* _{i}*(∂

*z*2

*l*(0)) 2

2π 0

cosαsinα*d*α =0

and lim

*r*→0

{|*z*|=*r*}

˜

*a** _{i}*(∂

*z*

_{1}

*l*)(0)

2 cos^{2}α*d*α = *a*˜* _{i}*(∂

*z*

_{1}

*l*)(0) 2

2π 0

cos^{2}α*d*α= π

2*a*˜* _{i}*(∂

*z*1

*l*)(0), we obtain

*G*(*l*)= −π

2*a*˜* _{i}*(∂

*z*

_{1}

*l*(0)).

In a similar way:

*T*(*l*)=lim

*r*→0

{|*z*|=*r*}

*l*(*r*)∂*f*2

∂*r* |*d z*|

= −lim

*r*→0

{|*z*|=*r*}

*l*(0)+*r*(cosα(∂*z*1*l*)(0)+sinα(∂*z*2*l*)(0))+*O*(*r*^{2})

×

*a*˜* _{i}*sinα

2r^{2} +*O*(ln*r*)

*r d*α

= −lim

*r*→0

{|*z*|=*r*}

˜
*a** _{i}*sin

^{2}α

2 (∂*z*2*l*)(0)*d*α = −π

2*a*˜* _{i}*(∂

*z*2

*l*)(0).

Then we can conclude that for*i* ∈ {*t*,*b*,*m*} :

*r*lim→0

{|*z*|=*r*}

ϕ˙1,*i**u** _{i}*(

*z*)− ˙θ1,

*i*v

*i*(

*z*)∂

*f*1

∂*r* |*d z*|

= ˙ϕ1,*i**G*(*u** _{i}*)− ˙θ1,

*i*

*G*(v

*i*)= π

2*a*˜* _{i}*θ˙1,

*i*(∂

*z*1v

*i*)(0)− ˙ϕ1,

*i*(∂

*z*1

*u*

*)(0) . In the same way we get*

_{i}*r*lim→0

{|*z*|=*r*}

ϕ˙2,*i**u** _{i}*(

*z*)− ˙θ2,

*i*v

*i*(

*z*)∂

*f*2

∂*r* |*d z*|

= ˙ϕ2,*i**T*(*u** _{i}*)− ˙θ2,

*i*

*T*(v

*i*)= π

2*a*˜* _{i}*θ˙2,

*i*(∂

*z*2v

*i*)(0)− ˙ϕ2,

*i*(∂

*z*2

*u*

*)(0) . We deﬁne another couple of functions:*

_{i}*R*(*l*)=lim

*r*→0

{|*z*|=*r*}

∂*l*

∂*r* *f*_{1}|*d z*|

=lim

*r*→0

{|*z*|=*r*}

cosα(∂*z*1*l*)(0)+sinα(∂*z*2*l*)(0)+*O*(*r*)

×

*a*˜* _{i}*cosα

2r +*O*(*r*ln*r*)

*r d*α

=lim

*r*→0

{|*z*|=*r*}

˜
*a** _{i}*cos

^{2}α

2 (∂*z*_{1}*l*)(0)*d*α = π

2*a*˜* _{i}*(∂

*z*

_{1}

*l*)(0)