The space of properly embedded minimal surfaces with finite total curvature

The space of properly embedded minimal surfaces with finite total curvature

Joaqu´ın P´ erez

^∗

Antonio Ros

^∗

Departamento de Geometr´ıa y Topolog´ıa Facultad de Ciencias

Universidad de Granada 18071 - Granada - SPAIN

Abstract.

It is showed that the set of non degenerate properly embedded minimal surfaces with finite total curvature and fixed topology in

R

³ has a structure of finite dimensional real analytic manifold —the non degeneration is defined in terms of the space of Jacobi functions on the surface which have logarithmic growth at the ends—. As application we show that if a non degenerate minimal surface has a symmetry which fixes its ends, then any nearby minimal surface has the same kind of symmetry. Finally, we construct a natural Lagrangian immersion of the space of non degenerate minimal surfaces, quotiented by certain group of rigid motions, into a Euclidean space with its standard symplectic structure.

Mathematics Subject Classification: 53A10, 53C42.

Introduction.

It is known, see R. Bohme, F. Tomi, J. Tromba and B. White, [1,27,28,30,31] and refe- rences there in, that several natural families of compact minimal surfaces admit a structure

∗Research partially supported by a DGICYT Grant No. PB91-0731.

of —infinite dimensional— smooth manifold. This fact is one of the hits of the theory of minimal surfaces and allows the authors above to apply differential topology —Morse theory, degree theory,...— on this space. In this way they obtain important results about existence, number of solutions and qualitative description for these kind of surfaces. In this paper we want to introduce this point of view in a class of non compact minimal surfaces without boundary : the class of properly embedded minimal surfaces in

R

³ with finite total curva- ture. The simplest examples are the plane and the Catenoid. Some non trivial existence results are obtained in Costa [4], Hoffman and Karcher [9] and Hoffman and Meeks [10].

Known non existence results say that in this family there are neither surfaces with positive genus and one or two ends, Schoen [25], nor surfaces of genus zero and more than two ends, Lopez, Perez and Ros [15,22]. Costa [5] shows that the space of surfaces with genus one and three ends is parametrized by a halfline. Finally, Ros [24] describes the weak compactness of the space for arbitrary genus and number of ends, and shows the strong compactness —up to homotheties— of the space of surfaces of genus one and more than four ends. For more details about the actual situation of the theory for this family, see the survey [9].

Denote by M the space of minimal surfaces of finite total curvature, genus k and r ends

—to avoid trivial cases we will suppose throughout this paper that k≥ 1and r ≥ 3—, prop- erly immersed in

R

³ and with embedded horizontal ends —such an end must be asymptotic to a plane or to a half-Catenoid—. These restrictions are verified, in particular, for the embedded surfaces. We first show that the different natural topologies on M are equivalent.

Given M ∈ M, the infinitesimal deformations of M in M are given by the space J (M) of the Jacobi functions on M , i.e. functions u on M such that Lu = 0, L being the Jacobi operator of M , which have logarithmic growth at the ends. We will show that dimJ (M) ≥ r + 3 for any M . If we denote by M^∗ = {M ∈ M : dim J (M) = r + 3} the subspace of non degenerate surfaces, then we will see thatM^∗ is an open subset of M and

M^∗ is an (r + 3)-dimensional real analytic manifold.

If M ∈ M^∗, the tangent space at M is given byJ (M). The unique embedded surfaces where the dimension of J (M) is known are the Hoffman-Meeks [10] surfaces M_k of genus k ≤ 37 and three ends. In this case, using a result of Nayatani [20], we have that dimJ (M) = 6.

Thus the one-parameter deformation of these surfaces given in Hoffman and Karcher [9]

contains all the surfaces nearby M_k, up to dilatations preserving the vertical direction.

As application of the above result we show that

If M is non degenerate and invariant by an isometry of

R

³ preserving the upper halfspace, then all the surfaces nearby M have the same kind of symmetry.

Note that, as it was observed in [9], all known embedded examples admit symmetries of the above type.

If G denotes the subgroup of direct rigid motions of

R

³ preserving the upper halfspace, we define a natural continuous map f :M/G −→

R

^2r depending only on information about the ends of the surface, and we show that

f :M^∗/G −→

R

^2r is a real analytic Lagrangian immersion.

In addition, we provide the second fundamental form of this natural Lagrangian immersion.

The basic idea of the arguments we will develope is the following: We want to parametrize the set of minimal surfaces near a given one. So, we consider the constraint of vanishing the mean curvature on deformations of the original surface. As we are working on non compact surfaces with catenoid or planar type ends, we are obliged to take deformations with logarithmic growing at infinity. Then the results are obtained via the Implicit Function Theorem after a careful study of the mean curvature operator for graphs defined outside a disk in the horizontal plane {x3 = 0}. In order to get the surjectivity of the differential of the mean curvature operator, we need to control the space of logarithmically growing Jacobi fields. Fortunately, when the analysis is transferred to the underlying compact surface obtained from the finite total curvature assumption, the allowed singularities turn out to have a nice behaviour in terms of the above operator.

Also note that, thank to the results of Meeks and Rosenberg [18] and Collin [3], when the number of ends is greater than one, the finite total curvature assumption is equivalent to the finiteness of the topology.

Finally we remark that Mazzeo, Pollack and Uhlenbeck [16], and also Kusner, Mazzeo and Pollack [12] have obtained recently finite dimensional smoothness results for the moduli spaces of others noncompact geometric objects: the space of solutions to the singular Yamabe problem on a finitely punctured n-dimensional sphere, and the space of properly embedded non minimal constant mean curvature surfaces in

R

³ with finite topology. Our general strategy and some of our statements are of the same type that those in [16] and [12] —they also have a Lagrangian structure—. ¿From the analytic machinery point of view, the main difference between both approaches is that they use weighted Sobolev spaces while we are able to work with functional spaces on the compactified surface whose elements are regular, up to a finite dimensional subspace of functions with given singularities.

1 Embedded minimal ends.

First of all, we will expose some well-known facts about minimal surfaces, for more details, see Osserman [21] and Hoffman and Karcher [9], and as a direct consequence we introduce the graph coordinate for properly embedded minimal ends with finite total curvature. Con- sider a conformal minimal immersion ψ : M −→

R

³ of an orientable surface M into the three-dimensional euclidean space. Denote by g and ω the meromorphic function and the holomorphic one-form determinated by the Weierstrass representation of ψ,

ψ =

1 2



ω−^ g²ω



, Real



gω



∈

C

^×

R

^≡

R

^{3. (1)}

Recall that g is the stereographic projection from the North Pole of the Gauss map of ψ. If ψ is complete and its total Gaussian curvature ^_MKdA is finite, then M has the conformal type of a finitely punctured compact surface M − {p1, . . . , p_r}, and the Weierstrass data extend in a meromorphic way to M . In particular, the Gauss map of ψ is well-defined at each p_i. The points p₁, . . . , p_r will be identified with the ends of ψ. We remark that in this family, completeness is equivalent to the properness of ψ —recall that ψ is said to be proper if {ψ ≤ R} is compact for any R > 0—. Assume that ψ has parallel embedded ends

—these conditions are verified, in particular, when ψ is an embedding—. Suppose also that, after a rotation in

R

³ if necessary, its Gauss map N takes vertical values at the punctures p_i. Properly embedded minimal ends with finite total curvature are characterized in terms of their Weierstrass Representation: such an end must be asymptotic to an end of a Catenoid or of a plane. If we suppose that the normal limit vector at p_i is (0, 0, 1), the Weierstrass data of ψ in terms of a conformal coordinate z centered at the end p_i are given by

g(z) = g₁(z)

z^k , ω = f (z)z^2(k−1)dz, |z| < ε,

where k ∈

N

^{and g}1, f are holomorphic functions such that g₁(0) = 0, f(0) = 0. So, (1 ) yields

ψ(z) =

1 2



f₁(z)−^ g₁²f z² dz



, Real



g₁f z^k−2 dz



,

where f₁ is holomorphic in |z| < ε. We should impose (g²₁f )^(0) = 0 and, when k = 1 , (g₁f )(0)∈

R

in order to insure that the last expression is well-defined. Thus we can write

ψ(z) =

t(z)

z ,−a log |z| + t1(z)



, (2)

where t is a smooth non vanishing complex valued function in {|z| < ε}, a ∈

R

^{and t}1 is harmonic in {|z| < ε}. When k = 1 , a does not vanishes and ψ is asymptotic to an end of a vertical Catenoid with logarithmic growth a. If k ≥ 2, the third coordinate function is bounded and ψ is asymptotic to the end of a horizontal plane. These cases are usually referred as Catenoid end or planar end, respectively. Symmetrically, if the limit normal at the end is (0, 0,−1), the Weierstrass data are g(z) = g1(z)z^k, ω = f (z)z⁻²dz and a similar argument gives ψ(z) =^t(z)

z ,−a log |z| + t1(z)^, with t, a, t₁ as above.

Take an end p_i of ψ where N points to (0, 0, 1). We have from (2) that, with the notation ψ = (x₁ + ix₂, x₃), after inversion in

C

^{of x}1 + ix₂ we obtain a new —non conformal—

coordinate w on M , satisfying w = ¹

x1+ix2 = ^z

t(z) such that ψ(w) =

1

w,−a log |w| + h(w)^, (3)

with h a smooth real valued function in |w| < δ. We will say that w is the graph coordinate at the end p_i. Symmetrically, if N (p_i) = (0, 0,−1) we had obtained ψ(z) =

t(z)

z ,−a log |z| + t1(z)^, and in terms of the graph coordinate w = _x ¹

1+ix2 = _t(z)^z , the same expression (3) holds. We can associate to each end p_i of M the logarithmic growth a and the height h(0): If p_i is a Catenoid end, its logarithmic growth coincides with the one of the asymptotic Catenoid, and its height differs of the third coordinate of the cen- ter of the asymptotic Catenoid in the term a log|^a₂|. When p_i is a planar end, its loga- rithmic growth vanishes and its height coincides with the height of the asymptotic plane.

Also note that if γ is a closed curve around p_i, the logarithmic growth can be obtained by ^_γηds = 2πaN(p_i), e₃e₃, η being the unit conormal vector along γ and e₃ = (0, 0, 1).

We will denote by log(ψ) = (a₁, . . . , a_r), height(ψ) = (b₁, . . . , b_r), the lists of logarithmic growths and heights of the ends of ψ.

We remark that any minimal surface which is a graph on the domain{|x₁+ ix₂| > ¹_δ} has finite total curvature at the end —this follows, for instance, from Fischer-Colbrie [7] because the end is complete and stable—, so it admits the representation given in (3).

If M ⊂

R

³ is a properly embedded minimal surface with finite total curvature, then its ends p₁, . . . , p_r are naturally ordered with respect to the x₃ linear coordinate. This order coincides with the lexicographical order of the pairs (logarithmic growth, height) at the punctures. The maximum principle at infinity, see Langevin and Rosenberg [14] or Meeks and Rosenberg [17], implies that for different ends of M the above pairs are different.

2 The mean curvature operator at an end.

Let ε > 0. Consider a not necessarily minimal immersion ψ :{0 < |w| < ε} −→

R

^{3 of the}

type

ψ(w) =

1

w,−a log |w| + h(w)^, (4)

where a ∈

R

^{and h} ∈ C^2,α({|w| < ε}), the usual H¨older space. Put ρ = |w| and u =

−a log ρ + h. The induced metric ds² = (g_ij) is given by g_ij(w) = 1

ρ⁴



δ_ij + ρ⁴D_iuD_ju^, (5)

where δ_ij = 1 if i = j, 0 otherwise, and D_iu denotes the corresponding first derivative of u.

If we denote by dA —resp dA₀— the measure associated to the metric ds² —resp. |dw|²—, (5) implies that

dA = Q¹2

ρ⁴ dA₀, where Q = 1 + ρ⁴∇₀u², (6) and the subindex •0 denotes that the corresponding object is computed respect to the flat metric of the w-plane. As u = −a log ρ + h, we get that ρ²D_iu = −aw, e_i + ρ²D_ih lies in C^1,α({|w| < ε}) and depends analytically on its three variables a, w and ∇0h. Also, Q = 1 + ρ²Q₁, where

Q₁ = ρ²∇₀u²= a² + ρ²∇₀h²− 2aw, ∇₀h.

Hence, Q, Q₁ ∈ C^1,α, and Q, Q₁, g_ij, dA are analytic in its three variables.

The Gauss map of ψ is given by

N = Q^{−12 }w²∇0u, 1^= Q^{−12 }−aw + w²∇0h, 1^, (7) where w²∇₀u means the product of the complex numbers w² and ∇₀u.

Lemma 1 In the above situation,

1. The Gauss map N is a C^1,α({|w| < ε},

R

³)-valued map which depends analytically on its variables (a, w,∇0h)∈

R

× {|w| < ε} ×

C

^.

2. If {e₁, e₂, e₃} is the usual basis in

R

³, the functions N, e₁, N, e₂, det(ψ, N, e₃) lie in C^1,α({|w| < ε}), depend analytically on its variables a, w and ∇0h and vanish at the puncture w = 0.

3. The support function of ψ is given byψ, N = −aN, e₃ log ρ+ f(a, w, h, ∇₀h), where f ∈ C^1,α({|w| < ε}), takes the value −a + h(0) at w = 0 and depends analytically on its variables (a, w, h,∇0h)∈

R

× {|w| < ε} ×

R

×

C

^.

Proof. 1 is a direct consequence of (7) and the above comment about Q. Concerning 2, we only have to compute det(ψ, N, e₃) from (4),(7):

det(ψ, N, e₃) = det

(1

w, 0), Q⁻¹²(−aw + w²∇₀h, 0), (0, 1)



=

= Q⁻¹²i

w,−aw + w²∇0h = −Q⁻¹²Real (iw∇0h).

Finally, from (4) and (7) it follows that

ψ, N = −aN, e3 log ρ + N, e3 (−a + Real(w∇0h) + h) . This concludes the proof.

On the other hand, as ψ can be viewed as the graph of the function u(x₁+ ix₂) with x₁+ ix₂ = ¹

w, its mean curvature is given by 2H(x₁+ ix₂) = div₁



 ∇1u



1 +∇₁u²



,

where the subindex •1 means that the object is computed in the Euclidean geometry of the parameters x₁ + ix₂. Using that

ρ⁴(∇0u)(w) = (∇1u)(x₁+ ix₂), and div₁(ρ⁴Y ) = ρ⁴div₀Y

for any vector field Y on{0 < |w| < ε}, we conclude that the mean curvature of ψ in terms of the parameter w is given by

2H = ρ⁴ div₀



 ∇₀u



1 + ρ⁴∇₀u²



= ρ⁴ div₀^Q⁻¹²∇0u^. (8)

The next structure result for the mean curvature operator at one end will play a key role in what follows.

Proposition 1 Let ε > 0, a∈

R

^{and h∈ C2,α(}{|w| < ε}). Denote by H the mean curvature function of the immersion ψ :{0 < |w| < ε} −→

R

³ given by ψ(w) = ^_w¹,−a log ρ + h(w)^, where ρ =|w|. Then, _ρ¹4H is a operator of the type

1

ρ⁴H =^

i,j

a_ij(w, a,∇0h)D_ijh + b(w, a,∇0h), (9) which depends analytically on its variables a∈

R

^, |w| < ε, ∇₀h∈

C

^{and ∇2}₀^{h∈ S}2(

R

^{), the}

space of real symmetric matrices of order 2. In particular, it is C^α({|w| < ε})-valued.

Proof. ¿From (8) we have 2

ρ⁴H = div₀^Q⁻¹²∇0u^= Q⁻¹²∆₀u+∇0(Q⁻¹²),∇0u = Q⁻¹²∆₀h−1

2Q⁻³²∇0Q,∇0u. (10) To prove the analyticity of _ρ¹₄H we only need to check this fact for the last inner product of the expression above. But

∇₀Q,∇₀u = ∇₀(ρ²Q₁),∇₀(−a log ρ + h) = 2Q₁w + ρ²∇₀Q₁,−aw

ρ² +∇₀h =

=−2aQ₁− a∇₀Q₁, w + 2wQ₁+ ρ²∇₀Q₁,∇₀h, (11) which is analytic on its variables, as we claimed. Hence, ¹

ρ⁴H lies in C^α, and the expression (9) follows directly from (10), (11). This completes the proof.

Remark 1 By simple computation we can show that given C > 0, the operator ρ⁻⁴H is uniformly elliptic on the pairs (a, h) such that |a|, ρ²∇₀h ≤ C, and that b(w, a, ∇₀h) is uniformly bounded on the pairs (a, h) with |a|, ρ∇0h ≤ C.

Lemma 2 Consider ε^ > ε > 0, a ∈

R

^{and h} ∈ C^∞({|w| < ε^}) such that the immersion ψ :{0 < |w| < ε^} −→

R

³ given by ψ(w) =^1

w,−a log ρ + h(w)^is minimal. Then, the map E :

R

^{× Ck,α(}{|w| ≤ ε}) −→

R

^{× Ck−2,α(}{|w| ≤ ε}) × C^k,α({|w| = ε})

(b, f ) −→ ^b,_ρ¹₄H(b, f ), f|_{|w|=ε}^,

where H(b, f ) is the mean curvature of the immersion w → ^_w¹,−b log ρ + h(w) + _N,e^{f (w)}₃ ^, 0 <|w| ≤ ε, is a real analytic local diffeomorphism around (a, 0), for any k ≥ 2.

Proof. From Proposition 1we conclude directly that E is real analytic. In order to compute the differential of E at (a, 0), we need the following fact that will be proved in section 5:

D

1 ρ⁴H



(a,0)

(b, f ) = 1

2L (−bN, e3 log ρ + f) , (12) for any (b, f ) ∈

R

× C^k,α({|w| ≤ ε}), where L = ∆ + ∇N² is the Schr¨odinger operator associated to the extended Gauss map N : {|w| < ε^} −→ S²(1) of ψ, ∆,∇N² being the laplacian and the square length of the gradient of N respect to the Riemannian metric ds² = ρ⁴ds² obtained from (5). Hence, (12) yields

DE_(a,0)(b, f ) =

b, 1

2L (−bN, e₃ log ρ + f) , f|_{|w|=ε}^. If (b, f )∈ Kernel^DE_(a,0)^, then b = 0 and f is a solution of the problem

 Lf = 0 in {|w| ≤ ε}

f = 0 in {|w| = ε}.

As N, e3 is positive in {|w| ≤ ε} and LN, e3 = 0, it is a standard fact that the first eigenvalue for the Dirichlet problem

 Lu + λu = 0 in {|w| ≤ ε}

u = 0 in {|w| = ε}

is positive. Hence f vanishes identically and so, DE_(a,0) is injective. ¿From Proposition 1and (12) we have that L (−bN, e₃ log ρ) ∈ C^k−2,α({|w| ≤ ε}). Hence, given b ∈

R

^,

v ∈ C^k−2,α({|w| ≤ ε}) and ϕ ∈ C^k,α({|w| = ε}), linear elliptic theory [8] insures that there exists f ∈ C^k,α({|w| ≤ ε}) satisfying

 Lf = 2v− L (−bN, e₃ log ρ) in {|w| ≤ ε}

f = ϕ in {|w| = ε},

hence DE_(a,0)(b, f ) = (b, v, ϕ). Now lemma 2 follows directly from the Inverse function theorem.

Also note that the maximum principle at infinity, see [14], implies that given b ∈

R

^,

ϕ∈ C^k,α({|w| = ε}), there is at most one function f ∈ C^k,α({|w| ≤ ε}) such that E(b, f) = (b, 0, ϕ).

3 The topology of the space of minimal surfaces.

Let E the space of properly embedded minimal surfaces in

R

³ with finite total curvature, horizontal ends and fixed topology —genus k ≥ 1and r ends, r ≥ 3—. The absolute total curvature of the surfaces in E is 4π(k + r − 1), [11]. In this section, we will show that the different natural notions of convergence in the space E are equivalent. Let Mn ∈ E, n = 1, 2, . . ., and M_∞∈ E.

1. We say that {Mn}n converges to M_∞ as subsets if M_∞ = {p ∈

R

³ : there exists a sequence p_n ∈ M_n with p_n → p}. This is the same that the convergence for the Hausdorff distance on compact subsets of

R

^3.

2. We say that {M_n}_nconverges to M_∞smoothly if for any relatively compact subdomain Ω of M_∞, for any embedded tubular neighbourhood of Ω in

R

^{3, Ω(ε) ={p + tN}∞(p) : p ∈ Ω, |t| < ε}, ε > 0 small enough, N∞ being the Gauss map of M_∞, and for each n large enough we have that Ω_n = M_n∩ Ω(ε) consists in a —single— graph over Ω, p ∈ Ω −→ p + u_n(p)N_∞(p), and the sequence {u_n}_n converges to zero in C^k(Ω), for any k≥ 0.

3. Let M_n, M_∞ be the compactified surfaces obtained from M_n, M_∞, respectively. Con- sider the r-dimensional vector space, r being the number of ends of M_∞, spanned by r functions f₁, . . . , f_r defined as follows:

3.a. f_i ∈ C^∞(M_∞− {pi}), where p1, . . . , p_r are the ends of M_∞.

3.b. In the graph coordinate w for M_∞ around p_i, f_i(w) = log|w|, and f_i vanishes in a neighbourhood of each p_j, for each j = i.

We say that the sequence {M_n}_n converges to M_∞ smoothly if there exists a vector field N^ ∈ C^∞(M_∞,

R

³) transverse to M_∞ which takes the values ±e₃ outside some compact subset of M_∞, such that for each n large enough, M_nis a globalN -graph over^ M_∞, p ∈ M∞ −→ p + un(p)N (p), where u^ _n= ϕ_n+ v_n, ϕ_n ∈ Span{f1, . . . , f_r}, vn ∈ C^∞(M_∞), ϕ_n → 0 in Span{f₁, . . . , f_r} and v_n → 0 in C^k(M_∞), k ≥ 0. We remark that this convergence is independent of the choice of N .^

The relation beetween these three types of convergence is stated at the following result:

Theorem 1 Given {Mn}n⊂ E and M∞ ∈ E, the following assertions are equivalent:

1. M_n→ M∞ as subsets.

2. M_n→ M_∞ smoothly.

3. M_n → M_∞ smoothly.

Proof. Clearly 3 =⇒2 =⇒1. Suppose we have 1. It follows from the arguments in Choi and Schoen [2], and White [29], see also Ros [24], that there exists a subsequence {M_n
}_n
⊂ {M_n}_n such that M_n
converges smoothly with finite multiplicity on compact subsets of

R

³ − X, X being a finite subset, to a properly embedded minimal surface M_∞^ . It is clear that we must have M_∞^ = M_∞. Moreover, if X = ∅, we know that given a small neighbourhood U of X, the total curvature on U∩ M_n
is near a positive multiple of 4π for n^ large enough. As C(M_n
) = C(M_∞), where C(•) denotes the absolute total curvature enclosed in the corresponding domain, we conclude that X = ∅ and Mn
converges to M_∞ uniformly on compact sets of

R

³ with multiplicity one, i.e. M_n
−→ M_∞ smoothly. Note that this argument also proves that every subsequence of{Mn}n has a smoothly convergent subsequence to M_∞. This fact is equivalent to M_n→ M∞ smoothly, so we have proved that 1 implies 2.

Assume now that assertion 2 holds. Note that all the surfaces M_n, M_∞ have the same absolute total curvature. Take δ > 0 small and consider the relatively compact subdomain Ω⊂ M_∞ obtained by cutting M_∞ with a vertical solid cylinder{|x₁+ ix₂| ≤ R}×

R

^{, with R}

large enough in such a way that C(M_∞)− C(Ω) < ^δ₂. Hence, for n large enough, the domain Ω_n obtained by the same procedure with M_n instead of M_∞ will verify C(M_n)− C(Ωn) < δ, and Ω_n is a graph over Ω. We can also suppose that along ∂Ω_n, the relation|N_n, e₃| ≥ 1−δ holds, where N_nis the Gauss map of M_n, and that each component of ∂Ω_nprojects injectively into the (x₁+ ix₂)-plane. As N_n is an open map, it follows that either |Nn, e₃| ≥ 1 − δ on M_n− Ωn or the spherical image N_n(M_n− Ωn) has area near 4π. As the second possibility cannot happen, we conclude that the projection of each component of M_n− Ω_n into{|x₁ + ix₂| > R} is a proper local diffeomorphism, so this projection is necessarily globally injective on each component, or in other words, for n large enough, M_n is a globalN -graph over M^ _∞ for a suitable chosen transversal section N verifiying the conditions of definition 3. Let u^ _n be the function defined by this graph. From the smooth convergence of M_n to M_∞ we get the convergence of the logarithmic growths of the ends of M_n to the logarithmic growths of the ends of M_∞ —this holds because that growths can be computed as the integral along each component of ∂Ω_n of its conormal vector—. Hence, with the notation of definition 3, the component of u_nin Span{f1, . . . , f_r} converges to zero. In the notation of lemma 2, this implies that for n large enough, if M_nis written as w→^_w¹,−b_nlog ρ + h(w) + _N,e^fn(w)

3



around an end p_i, where b_n ∈

R

^{and f}n ∈ C^∞({|w| ≤ ε}), then {bn}n → 0 and fn|_{|w|=ε} → 0 in C^k({|w| = ε}) for any k ≥ 0. Thus, E(bn, f_n) =^b_n, 0, f_n|_{|w|=ε}^lies in a neighbourhood of

(a, 0, 0) in

R

^×Ck−2,α({|w| ≤ ε})×C^k,α({|w| = ε}) where the map E in lemma 2 is bijective.

As the maximum principle at infinity [14] insures that b_n, f_n|_{|w|=ε} determine uniquely to f_n(w),|w| ≤ ε, it follows from lemma 2 that {fn}n converges to zero in C^k,α({|w| ≤ ε}) for any k ≥ 2. This concludes the proof.

Although we have stated theorem 1for the case of embedded minimal surfaces for the sake of clarity, the result extends to the spaceM of simple proper minimal immersions with finite total curvature, embedded horizontal ends and fixed topology. An immersion is called simple, following [31], if it gives an embedding when restricted to some open dense subset

—for proper minimal immersions this is the same that to assume that the immersion does not factorizes through a non trivial covering map—. We will consider onM the above topology.

So, E ⊂ M and it can be shown, use for instance theorem 1of [24], that E is closed in M.

In spite of we are mainly interested in the embedded case, it could happen that arbitrary nearby surfaces to an embedded one were not embedded but only simply immersed —this is possible only if two different ends of the embedded surface have the same logarithmic growth—. So, we will also consider this kind of immersions. For the remainder of the paper, by a proper minimal immersion we will mean a simple proper minimal immersion, and this surface will be represented by M 4→

R

^{3 or ψ : M −→}

R

³ indistinctly. As there is not a natural order for the ends of a surface in M we will consider all possible orders and we will view different orders as different surfaces. The topology ofM will be modified, correspondly, in an obvious way to be sensitive to the order of the ends.

4 The Jacobi operator.

Let M ∈ M, and M = M ∪ {p1, . . . , p_r} the associated compact Riemann surface. Put N : M −→ S²(1) the extended Gauss map and ds² the induced metric on M . Take a function λ ∈ C^∞(M ) such that around each end p_i, λ(w) = ρ⁻⁴ in terms of the graph coordinate at p_i. Then (5) implies that ds² = ¹

λds² is a Riemannian metric on M compatible with its complex structure. The Jacobi operator of M is L = ∆ +σ² = ∆ +∇N², where

∆ is the Laplacian of ds² and σ² the square length of the second fundamental form of the embedding. In certain sense, this operator is the Hessian of the Area functional. A way to understand the geometric role of L is the following: take {Mt}|t|<ε, M₀ = M a smooth deformation of M and denote by H(t) the mean curvature of M_t. Calling u =_dt^d

t=0ψ_t, N, where ψ_t : M −→

R

³ is a smooth family of immersions such that ψ_t represents the surface

M_t, we have —see for instance [30]—

d dt



t=0

H(t) = 1

2Lu. (13)

In particular, the equation Lu = 0 corresponds to an infinitesimal deformation of M by minimal surfaces. A function u satisfying the last equation is called a Jacobi function on M . The operator L can be “compactified” to a Schr¨odinger operator L = λL = ∆ +∇N² on M , where • means that the corresponding object is computed respect to ds².

Let B = B(M) ⊂ C^2,α(M ) the space of functions v such that in the graph coordinate w around the end p_i, i = 1, . . . , r, are given by

v(w) =−aiN, e3 log ρ + u(w), (14) with a_i ∈

R

^{and u} ∈ C^2,α({|w| < ε}). Consider the r-dimensional space V = V (M) = {u_a : a ∈

R

^r} ⊂ B whose functions are given by u_a = −N, e₃^r_i=1a_if_i, where a = (a₁, . . . , a_r)∈

R

^{r and f}1, . . . , f_r are the functions which appear in the definition 3 of section 3. Thus around each p_i, u_a is given by u_a(w) = −aiN, e3 log ρ. Clearly B = V ⊕C^2,α(M ) = {u_a+ u : a∈

R

^{r, u∈ C2,α(M )}}. Hence, B becomes a Banach space with the natural norm on each component, and the topology of B is independent of the choice of f₁, . . . , f_r. By identification of B with

R

^r⊕ C^2,α(M ), we will see in section 5 that L is the differential of a real analytic operator defined in a neighbourhood of (log(M ), 0) in

R

^{r ⊕ C2,α(M ) with}

values in C^α(M ), hence L :B −→ C^α(M ) is a bounded linear operator. In particular, uLv

—resp. uLv— is ds²-integrable —resp. ds²-integrable—, for any pair of functions u, v ∈ B.

Given a function v∈ B, v = ua+ u∈ V ⊕ C^2,α(M ) we define

Log(v) = a, Height(v) = (u(p₁)N(p₁), e₃, . . . , u(p_r)N(p_r), e₃).

Hence we have two bounded linear operators Log, Height:B −→

R

^r. Lemma 1says that

N, e1, N, e2, det(p, N, e3), p being the position vector on M , lie in B and Log, Height vanish on these functions. It is clear that also N, e3 ∈ B and Log(N, e3) = (0, . . . , 0), Height(N, e₃) = (1, . . . , 1) ∈

R

^r. Furthermore, it follows also that the support function

p, N ∈ B and Log(p, N) = log(M), Height(p, N) = height(M) − log(M), where log, height are defined in section 1. These linear operators Log, Height be related with the differential operator L by means of a skew-symmetric bilinear form:

Lemma 3 Given u, v∈ B, we have



M

(uLv− vLu) dA =^

M

(uLv− vLu) dA =

=−2π [Log(u), Height(v) − Log(v), Height(u)] , (15)

Proof. Assume that in the graph coordinate around the end p_i, i = 1, . . . , r, the functions u and v are given by u(w) =−aiN, e3 log ρ + u^_i(w), v(w) = −biN, e3 log ρ + v_i^(w), u^_i, v_i^ being C^2,α-functions around w = 0. So,

Log(u) = (a₁, . . . , a_r), Height(u) = (u^₁(0)N(p1), e₃, . . . , u^_r(0)N(pr), e₃) , Log(v) = (b₁, . . . , b_r), Height(v) = (v^₁(0)N(p1), e₃, . . . , v^_r(0)N(pr), e₃) .

Consider a conformal coordinate z centered at p_i. Thus, w = t(z)z or w = t(z)z for some smooth complex valued function t with t(0) = 0. In the new parameter we have around p_i, u(z) = −a_iN, e₃ log |z| + u^_i(z), v(z) = −b_iN, e₃ log |z| + v_i^(z), where u^_i = u^_i− aiN, e3 log |t(z)| and v_i^= v_i^− biN, e3 log |t(z)|. Note that the sum in the right-hand side of (15) is equal to

−2π^r

i=1

(a_iv_i^(0)− biu^_i(0))N(pi), e₃.

Denote by D(p_i, r) the conformal disk {|z| < r} around pi, and M (r) = M − ∪^r_i=1D(p_i, r).

Then _

M (r)

(uLv− vLu) dA =^

∂M (r)



u∂v

∂η − v∂u

∂η



ds,

where dA, ds are measured in the induced metric by ψ, and η is the exterior conormal field to ψ along ∂M (r). As the last integral is conformal-invariant, it can be written on each disk as −^_{|z|=r}^u^∂v_∂r − v^∂u_∂r^|dz|, where z = re^iθ. By using the above local expression of v we

have ∂v

∂r =−biN, e31

r + lower order terms,

where the not specified terms grow at most logarithmically. A similar expression can be obtained for u. Thus,

−^

{|z|=r}



u∂v

∂r − v∂u

∂r



|dz| =

=−^

{|z|=r}

−u(z)biN, e31

r + v(z)a_iN, e31

r + lower order terms



|dz| =

= 1 r



{|z|=r}(u^_i(z)b_iN, e3 − v_i^(z)a_iN, e3) |dz|+o(1) = 2π (u^_i(0)b_i− v^_i(0)a_i)N, e3+o(1), o(1) being an expression whis converges to zero as r goes to zero. This completes the proof of the lemma.

We are now interested in the Kernel and Image of L :B −→ C^α(M ). For any X ⊂ C^α(M ) we will represent its L²-orthogonal in C^α(M ) respect to the metric ds² by X^⊥ ⊂ C^α(M ).

Note that for any finite dimensional subspace W ⊂ C^α(M ) we have that C^α(M ) = W⊕ W^⊥ and W^⊥⊥ = W . Moreover, if W^ ⊂ C^α(M ) is a subspace with W^⊥ ⊂ W^, then C^α(M ) = W^⊕ W^⊥ and W^⊥⊥ = W^ are also true. Consider in the Banach spaceB the subspaces

J = J (M) = Kernel(L), K = K(M) = C^2,α(M )∩ J , and K0 =K0(M ) = L(B)^⊥. By linear elliptic theory [8] we know that K = Kernel^L |_C2,α(M)

 ⊂ C^∞(M ) has finite dimension and that L(C^2,α(M )) = K^⊥. Thus, K₀ = L(B)^⊥ ⊂ L(C^2,α(M ))^⊥ = K^⊥⊥ = K.

Moreover, as L(B) contains K^⊥, it follows that L(B) = L(B)^⊥⊥ =K₀^⊥. The space K consists of the Jacobi functions on M which are bounded at the ends.

Lemma 4 In the above situation,

1. K₀ ={v ∈ K : v(p_i) = 0, 1≤ i ≤ r}, 2. dimJ = r + dim K0.

Proof. Given v∈ K, we have v ∈ K0 if and only if ^_MvLudA = 0, ∀u ∈ B. Using (15), the last equation is equivalent to

0 =



M

uLv dA + 2π [Log(u), Height(v) − Log(v), Height(u)] =

= 2πLog(u), Height(v),

for each u∈ B. This gives 1. Concerning 2, consider on the r-dimensional space V defined just after lemma 3, the decomposition V = V₁⊕V2, where V₁ ={ua: Lu_a ∈ L(C^2,α(M ))} and V₂ is a suplementary subspace. Hence, L is injective on V₂ and L(B) = L(V₂)⊕ L(C^2,α(M )).

In particular, dimK0 = codim L(B) = codim L(C^2,α(M ))− dim L(V2) = dimK − dim V2, that is

dimK₀ = dimK − r + dim V₁. (16)

Now consider the natural projection B −→ V restricted to J , π : J −→ V . It is clear that Kernel(π) =K. Furthermore, given v ∈ J , v = u_a+ u, u_a∈ V , u ∈ C^2,α(M ), it follows that 0 = Lv = Lu_a+ Lu and so, π(v) = u_a ∈ V1. Also, for any u_a∈ V1 there exists u∈ C^2,α(M ) such that Lu = Lu_a, that is, u_a− u ∈ J . Thus π(J ) = V1 and then

dimJ = dim K + dim V₁. (17)

¿From (16) and (17) we conclude the proof of the lemma.

It is well-known that vector fields in

R

³ whose flow consists on isometries —Killing Fields— or dilatations induce on M Jacobi functions. As our surface has horizontal ends, this insures the following set of bounded Jacobi functions, see lemma 1:

Killing = Span{N, e1, N, e2, N, e3, det(p, N, e3)} ⊂ K,

p being the position vector on M . We will call Killing₀ = Killing ∩ K₀. Then, it follows that Killing₀ = Span{N, e₁, N, e₂, det(p, N, e₃)} ⊂ K₀.

Moreover, the support function p, N lies in J . If Killing₀ = K₀, we will say that M is a non degenerate minimal surface. This condition insures a nice behavior of the set of minimal immersions near M , as we will see in theorem 2. As we are assuming that the genus of M is as least one and the number of ends is as least three, it follows that the symmetry group of M is always finite and thus, dim Killing = 4 and dim Killing₀ = 3. Note also that the assumption K = Killing implies that K0 = Killing₀. As a direct consequence of lemma 4 we have

Proposition 2 With the notation above, dimJ (M) ≥ r + 3, and the equality holds if and only if M is non degenerate.

Soret [26] has shown that dimJ (M) ≥ r − 5 − 2k, k being the genus of the surface.

5 The smoothness of the space of minimal surfaces.

Let us return to our orientable proper minimal immersion ψ : M = M− {p₁, . . . , p_r} −→

R

³

with finite total curvature and embedded horizontal ends, M ∈ M. Put c = (c1, . . . , c_r) = log(M ) the list of logarithmic growths of ψ. As we saw in section 1, ψ can be written in the graph coordinate w around each end p_i as

ψ(w) =

1

w,−cilog|w| + h(w)^, (18) with h∈ C^∞({|w| < ε}). Consider a family ψa : M −→

R

³, where a = (a₁, . . . , a_r) varies in a neighbourhoodA ⊂

R

^r of c, of immersions such that

1. The family is smooth in A × M,

2. for each end p_i, the expression of ψ_a in terms of the graph coordinate w,|w| < ε, is given by ψ_a(w) = (¹

w,−ailog|w| + h(w)).

3. ψ_c= ψ.

Let N : M^ −→

R

³ be a smooth map such that N,N^ = 1 on M, and N =^ _N,e¹

3 e₃ in {|w| < ε} for each end. Theorem 1insures that given M ∈ M, there exists a neighbourhood of M inM such that each minimal surface in this neighbourhood is represented by ψa+ uN^ for suitable a∈

R

^{r, u∈ C∞}(M ) near c = log(M ), zero, respectively.

Given u ∈ C^2,α(M ) small enough and a near c, consider the immersion ψ_a + uN . Let^ H(a, u) = λH(ψ_a+ uN ), where H(ψ^ _a+ uN )^ ∈ C^α(M ) is the mean curvature of ψ_a+ uN ,^ and λ∈ C^∞(M ) is defined at the beginning of section 4. Note that around an end p_i of ψ, ψ_a+ uN can be written in terms of the graph coordinate w as^

(ψ_a+ uN )(w) =^

1

w,−ailog ρ + h + u

N, e₃



. (19)

As H(a, u) = ρ⁻⁴H(ψ_a+ uN ), proposition 1yields^

Lemma 5 There exist neighbourhoods U of the origin in C^2,α(M ) and A of c in

R

^{r such}

that the map H :A × U −→ C^α(M ) is a real analytic operator.

We will denote by M_a,u the surface determinated by the immersion ψ_a+ uN : M^ −→

R

^3,

(a, u)∈ A × U. Nearby minimal surfaces to M in M can be represented by points of A × U, see section 3, and in this neighbourhood, which will be denoted by M ∩ (A × U), both topologies coincide. Given M_a,u ∈ M ∩ (A × U) we can realize the r-dimensional space V (M_a,u) = {u˙a : ˙a ∈

R

^r} in section 4 as follows: consider the curve a(t) = a + t ˙a with

˙a = ( ˙a₁, . . . , ˙a_r) ∈

R

^r, and the function on M , u_˙a = _dt^d

t=0ψ_a(t), N_a,u, where Na,u is the Gauss map of M_a,u. By using the form of the deformation ψ_a at the ends we deduce that in the graph coordinate for M around p_i,

d dt



t=0

ψ_a(t) = (0, 0,−˙ailog ρ).