Affine Maximal Surfaces with Singularities1 1 Introduction

Affine Maximal Surfaces with Singularities1 Juan A. Aledo a , Antonio Mart´ınezb and Francisco Mil´anc

This paper is dedicated to the memory of Prof. Katsumi Nomizu

a

Departamento de Matem´aticas, Universidad de Castilla-La Mancha, E-02071 Albacete, Spain. e-mail: [email protected] b c

, Departamento de Geometr´ıa y Topolog´ıa, Universidad de Granada, E-18071 Granada, Spain. e-mail: [email protected] ; [email protected] Keywords: affine area, maximal surfaces, improper affine maps, isolated singularities, singular set. Abstract The paper deals with the study of affine maximal surfaces with admissible singularities. We shall give some fundamental notions, results, tools and open problems that keep unsolved up to date.

MSC: 53A15

1

Introduction

In affine surfaces theory, Blaschke (see [Bla]) found that the Euler-Lagrange equation of the equiaffine area functional is of fourth order and nonlinear. He also showed that this equation is equivalent to the vanishing of the affine mean curvature, which led to the notion of affine minimal surface without a previous study of the second variation formula. But Calabi proved in [Cal] that, for locally strongly convex surfaces, the second variation is always negative and since then, locally strongly convex surfaces with vanishing affine mean curvature are called affine maximal surfaces. 1

Research partially supported by Ministerio de Educaci´on y Ciencia Grant No. MTM2007-65249, Junta de Andaluc´ıa Grants No. FQM325, No. P06-FQM-01642 and Junta de Comunidades de CastillaLa Mancha Grant No. PCI-08-0023

1

After Calabi’s work this class of surfaces has become a fashion research topic and it has received many interesting contributions that may help us to understand its geometry as well as others functionals, involving curvatures, whose Euler-Lagrange equation is also a fourth order partial differential equation, such as the Willmore’s functional [GLW, Sim] and the Calabi’s functional [Cal2, Ch, WZ]. Recent celebrated contributions have been the following: • The entire convex solutions of the fourth-order affine maximal surface equation

−3/4 L[φ] := φyy ωxx − 2φxy ωxy + φxx ωyy = 0, ω = det ∇2 φ , where φ(x, y) is defined in a planar domain and ∇2 φ is its positive definite Hessian matrix, are always quadratic polynomials [TrWa, TrWa2]. • The only affine complete affine maximal surface is the elliptic paraboloid [LiJi, TrWa3]. This lack of global examples has motivated a recent study of singularities of affine maximal surfaces [ACG, AMaM, AMaM2, AMaM3, GMaMi, IM, Ma] which has revealed an interesting global theory for this class of surfaces where many questions and problems keep unsolved until now. This paper concerns with the notions, results, tools and open problems involved into this emergent theory. It is organized as follows: In Section 2 we introduce basic notations and the notion of affine maximal map. We also investigate how affine maximal maps can be determined by its set of singular points. Section 3 deals with the study of a kind of singularities which do not take place for Euclidean minimal surfaces but that play an interesting role in the affine case. Finally, Section 4 is devoted to global questions. Thus, we analyze global problems regarding to affine maximal surfaces admitting some natural singularities.

2

Recovering affine maximal maps from their singular set

Let ψ : Σ → R3 be a locally strongly convex immersion of a surface Σ, oriented so that its second fundamental form, σe , is positive definite everywhere. Denote by Ke and dAe its Gaussian curvature and the element of euclidean area, respectively. 1 −1 Then, the positive density dA := Ke4 dAe , the positive quadratic form g := Ke 4 σe and the transversal vector field ξ := 12 ∆g ψ, where ∆g is the Laplace-Beltrami operator associated to g, are unchanged if one replaces the given euclidean structure on R3 by any affinely equivalent one inducing the same orientation and volume form. In fact they are the most elementary unimodular affine invariants of the immersion and they are called the equiaffine area element, the Berwald-Blaschke metric, and the Blaschke normal or affine normal, respectively. 2

On Σ we have a canonical Riemann surface structure such that g is Hermitian. Moreover, the equiaffine normalization of ψ is given by the affine normal ξ and the −1/4 affine co-normal vector field N := Ke Ne , where Ne is the unit normal vector field to the immersion (see [LSZ, NS]). With this normalization 1 = hN, ξi, hN, dψi = 0, g = −hdψ, dN i = det[N, ∗dN, dN ], dψ = −N ∧ ∗dN, ∗dN ∧ dN , ξ = det[N, ∗dN, dN ]

(2.1) (2.2)

where h., .i is the usual inner product in R3 , ∧ denotes the cross product in R3 and ∗ is the standard conjugation operator acting on 1-forms. In affine differential geometry, it is natural to ask for extremals for the affine invariant area functional Z Z 1 dA = Ke4 dAe . The Euler Lagrange equation for this variational problem is equivalent to the following system of PDE’s: ∆g N = 0. So, when Σ is simply-connected, 12 N is locally the real part of a holomorphic curve Φ : Σ → C3 determined by ψ up to a real translation which satisfies N = Φ + Φ,   g = −2 i det Φ + Φ, dΦ, dΦ .

(2.3) (2.4)

Conversely, (2.2) allows us to recover ψ from its affine co-normal and the conformal class of g, ([Li]), as Z ψ = − N ∧ ∗dN, (2.5) which, along with (2.3) and (2.4), says that ψ is uniquely determined, up to a real  translation, by a holomorphic curve Φ satisfying that −2 i det Φ + Φ, dΦ, dΦ is a Riemannian metric (see also [Cal3]). To be precise,   Z Z Z
ψ = 2 Re i Φ + Φ ∧ dΦ = − i Φ ∧ Φ − Φ ∧ dΦ + Φ ∧ dΦ . (2.6) From (2.5) and (2.6) we have a recipe that let us obtain affine maximal immersions from either harmonic vector fields or holomorphic curves. But when the expressions in (2.1) and (2.4) fail to be Riemannian metrics, it appears a natural set of singularities of ψ that motivates the study of affine maximal surfaces with some admissible singularities. Definition 2.1. Let Σ be a Riemann surface. We say that a map ψ : Σ −→ R3 is an affine maximal map if there exists a harmonic vector field N : Σ −→ R3 such that 3

Figure 1: Some rotational affine maximal maps.

det[N, ∗dN, dN ] does not vanish identically and ψ is given as in (2.5) (see Figure 1 for some rotational examples). We shall say that N is the affine co-normal of the affine maximal map. The singular set Sψ of ψ is the set of points where det[N, ∗dN, dN ] vanishes. It is clear that Σ \ Sψ is dense in Σ. Consider ψ : Σ → R3 an affine maximal map with affine co-normal N : Σ → R3 . From now on, I will be an interval and γ(s) : I → Σ a regular analytic curve. Let us take α = ψ ◦ γ and U = N ◦ γ. By the Inverse Function Theorem it is not difficult to prove (see [ACMi]) that there exists a conformal parameter z := s + i t, with s ∈ I. Thus, we can parametrize a piece of the affine maximal map by ψ : Ω ⊂ C → R3 with I ⊂ Ω, ψ(s, 0) = α(s) and N (s, 0) = U (s). By denoting η(s) = Nt (s, 0) and using the Identity Principle and (2.5), we have that the map ψ can be recovered as Z z ψ = α(s0 ) − N ∧ ∗dN, z ∈ Ω ⊂ C, s0 ∈ I, (2.7) s0

where the affine co-normal N is given by   Z z N (z) = Re U (z) − i η(ζ)dζ ,

z ∈ Ω ⊂ C,

s0 ∈ I,

(2.8)

s0

being U (z) and η(z) the holomorphic extensions of U and η to a neighborhood Ω of I. Moreover, from (2.5) we have that along α η ∧ U = −α0 ,

(2.9)

where by prime we indicate derivative with respect to s. Remark 1. It is clear from (2.7) and (2.8) that the analytic maps η, U and α determine uniquely an affine maximal map containing α and such that, along α, its affine co-normal is U and (2.9) holds. 4

If η ≡ 0 on I, then α0 ≡ 0 on I and ψ is recovered as in (2.7) where, in this case, N (z) = Re U (z), z ∈ Ω. If η 6= 0 on I, then from (2.9), 1 α0 ∧ η, hη, ηi hU 0 , α0 i 0 0 0 U ∧ η = λ2 α + 0 0 α ∧ η, hα , α i U = λ1 η −

(2.10) (2.11)

for some regular analytic functions λ1 , λ2 : I → R. By deriving (2.10) and using (2.11) we get that λ1 and λ2 satisfy det[α00 , α0 , η] hη, ηi hη 0 , ηi 0 0 0 0 hα , α i + hα00 , α0 i. = λ1 det[η , η, α ] − hη, ηi

hU 0 , α0 i = λ1 hη 0 , α0 i + hα0 , α0 iλ2

(2.12) (2.13)

All these ingredients let us solve the following geometric Cauchy problem for affine maximal maps: Let α, η : I −→ R3 be analytic maps satisfying hα0 , ηi = 0 on I. Find all affine maximal maps ψ : Σ −→ R3 , or equivalently their harmonic affine co-normal N , for which there exists a regular analytic curve Γ ⊂ Σ so that ψ(Γ) = α and η ∧ N (Γ) = −α0 . Any pair of maps α, η satisfying the above conditions will be called a pair of initial data. Remark 2. The above geometric affine Cauchy problem extends those studied in [ACG] and [AMaM2] for affine maximal surfaces. Observe that in this case the affine normal of an affine maximal map may not be well-defined on its singular set. Theorem 2.2. Let α, η be a pair of initial data. If α0 ∧ η 6= 0 on I, then for any regular analytic function λ1 : I → R such that λ1 hη 0 , α0 i +

det[α00 , α0 , η] 6= 0 hη, ηi

on I,

there exists a unique solution ψ to the geometric Cauchy problem for affine maximal maps such that 1 U := λ1 η − α0 ∧ η hη, ηi is the affine co-normal along α. The map ψ can be explicitly constructed in a neighborhood of α as in (2.7), where N is given by   Z z 1 0 N (z) = Re λ1 (z) η(z) − α (z) ∧ η(z) − i η(ζ)dζ , z ∈ Ω, hη, ηi(z) s0 5

η(z), λ1 (z) and α(z) being holomorphic extensions of η, λ1 and α, respectively. Moreover, ψ is an affine maximal surface in a neighborhood of α and the affine metric g along α satisfies det[α00 , α0 , η] . g(α0 , α0 ) = −λ1 hη 0 , α0 i − hη, ηi Proof. It follows easily from the previous considerations following the same scheme as in [AMaM2, Section 3]. Next we analyze the interesting case of affine maximal maps with a given curve in its singular set. More precisely we want to solve the following problem Given an analytic map α : I → R3 , find all affine maximal maps ψ : Σ → R3 for which there exists a regular analytic curve Γ in the singular set of ψ so that ψ(Γ) = α. In this case, we say that α is a singular curve of ψ. Let α : I → R3 be a singular curve of ψ. If α0 6= 0 on I, then from (2.9), (2.10), (2.11), (2.12) and (2.13) we obtain 0 = hU, α0 i,

0 = hU, α00 i,

0 = hη, α0 i.

(2.14)

Theorem 2.3. Let α : I → R3 be an analytic curve with non-vanishing curvature on I. Then, for any regular analytic functions λ, φ : I → R, λ > 0, there is a unique affine maximal map ψ having α as a singular curve and such that ψ is given by (2.7), where its affine co-normal N is given by (2.8), and U , η by U = λα0 ∧ α00 , η = φα0 ∧ α00 −

(2.15) λ|α0

1 (α0 ∧ α00 ) ∧ α0 . ∧ α00 |2

(2.16)

Conversely, every affine maximal map with α as a singular curve can be locally constructed in this way. Proof. If α is a singular curve with non-vanishing curvature on I of an affine maximal map ψ, then from (2.9), (2.14) and by straightforward computations we see that there exist regular analytic functions λ, φ : I → R, λ > 0, such that U and η are given as in (2.15) and (2.16), respectively. On the other hand, if U and η are given by (2.15) and (2.16), we can define ψ as in (2.7) where N is given by (2.8). Now, since N is harmonic and using (2.15) and (2.16), we obtain that, along α, det[N, Ns , Nt ]t = det[U, η 0 , η] − det[U, U 0 , U 00 ] 1 = − − λ3 det[α0 , α00 , α000 ]2 < 0 λ on I, where s + i t is a conformal parameter around I. Consequently, det[N, dN, ∗dN ] does not vanish identically and near {(s, 0) | s ∈ I} the singular set of ψ is a regular curve, that is, the points of α are the unique singular points in a neighborhood of it. 6

Theorem 2.4. There are no affine maximal maps containing a straight line. Proof. Let α be an straight line with direction vector ~v contained in an affine maximal map ψ with co-normal N . Then, from (2.9), we have that hU, ~v i = hη, ~v i = 0 on I. Consequently, from the Identity Principle and (2.8), det[N, ∗dN, dN ] vanishes in a neighborhood of α which contradicts that ψ is an affine maximal map. Theorem 2.5. Assume that α is constant in I and let β : I → R3 be an analytic regular curve satisfying det[β, β 0 , β 00 ] > 0 on I. Then, 1. There exists a unique affine maximal map ψ with singular set α and such that η = 0 and U = β on I. 2. There exists a unique affine map ψ with singular set α and such that η = β and U = 0 on I. 3. For any positive analytic function λ : I → R, there exists a unique affine maximal map ψ with singular set α and such that η = β and U = λβ on I. Conversely, any affine maximal map with α as a singular curve can be constructed, locally, as in either 1., 2. or 3. for some analytic curve β. Proof. If η ≡ 0, taking U = β we can define ψ as in (2.7), where N is given as in (2.8). It is clear that, along α, det[N, Ns , Nt ]t = − det[β, β 0 , β 00 ] 6= 0, where s + i t is a conformal parameter around I. Thus, ψ is an affine maximal map such that its singular set in a neighborhood of α is α. Otherwise, we can assume η 6= 0 on I. In this case, we have that either U ≡ 0 on I, or we can assume that U = λη, λ > 0. In both cases and reasoning similarly, if we take η = β, we construct by using (2.7) and (2.8) affine maximal maps containing α in its singular set. Conversely, it is clear from (2.9) that we can take U and η as either in 1., 2. or 3.

3

Isolated singularities

In this section we overview an interesting kind of singularities which do not take place in the Euclidean situation, that is, in minimal surfaces theory. Let D be a topological disk containing the origin ~0 and D? = D\{~0}. Definition 3.1. If a continuous map ψ : D → R3 is not regular at the origin, but it is an affine maximal immersion on D? , we will call ~0 (or ψ(~0)) an isolated singularity of ψ. We will say that ~0 is an embedded isolated singularity if ψ is embedded on a neighborhood U ? of ~0. Let ψ : D? → R3 be an affine maximal immersion around an isolated singularity ψ(~0). If we take on D? the structure of Riemann surface induced by the affine metric g, then D? is conformally a punctured disk or an annulus. It is not a restriction to assume that ψ(~0) = (0, 0, 0). (3.1) 7

3.1

The conformal structure of an annulus.

In this situation there exists a conformal parametrization ψ : ∆− /2πZ → R3 , where = {z = s + i t | −  < t < 0}, such that ψ extends continuously to {Im(z) = 0} and ψ(s, 0) = p, for all s ∈ R. Note that Theorem 2.5 gives information about when a curve of singular points degenerates into an isolated singularity. In fact, it provides a tool for constructing a large family of affine maximal maps with isolated singularities such that the affine co-normal is well-defined on the singular curve. More precisely, for each 2π-periodic analytic regular curve β : R → R3 satisfying the condition ∆−

det[β, β 0 , β 00 ] 6= 0 on R, and for each positive 2π-periodic regular analytic function λ : R → R+ , we can consider the family F of holomorphic curves given by the following holomorphic extensions: Gβ (z) = β(z), Z z β0 F (z) = − i β(ζ)dζ, 0 Z z βλ F (z) = λ(z)β(z) − i β(ζ)dζ,

(3.2)

0

where z lies in a neighbourhood ∆ = {z = s + i t | −  < t < } of R, for  > 0 small enough. Then, for any F ∈ F we have that Z z e ψ(z) = N ∧ ∗dN, (3.3) 0

being N = Re(F ), is an affine maximal map with {Im(z) = 0} as the singular set of ψe e 0) = (0, 0, 0), s ∈ R. in ∆ and ψ(s, Moreover, ψe induces a well-defined affine maximal map ψ : ∆ /2πZ → R3 whose affine co-normal is induced by N and such that ψ|∆− is an affine maximal immersion around the isolated singularity p. Among the examples of this large family of affine maximal surfaces with isolated singularities, one can investigate which are embedded. Actually, if ψ is one of these examples constructed from a holomorphic curve F ∈ F, then, by Theorem 11 in [GMi], we can assume that, up to an equiaffine transformation, ψ is the continuous vertical graph of a function φ : U ? = U \ {~0} → R on some punctured disk. From the standard affine maximal surfaces theory, (see [LSZ]), φ is a locally strongly convex solution of

−3/4 ρ = det ∇2 φ ,

φyy ρxx − 2φxy ρxy + φxx ρyy = 0,

in U ? .

(3.4)

In particular, we have that φ has a bounded gradient ∇φ = (φx , φy ) at least in a neighborhood of the origin (see [Be]), and φ has a continuous extension to U that we will also denote by φ. 8

Let us consider the Legendre transformation Lψ of ψ = (ψ1 , ψ2 , ψ3 ) (see [LSZ, pp. 89])   N1 N2 N1 N2 ψ ψ ψ ψ L = (L1 , L2 , L3 ) = − , , ψ1 + ψ2 + ψ3 N3 N3 N3 N3 = (φx , φy , xφx + yφy − φ) . (3.5) It is not difficult to check that Lψ is also a locally strongly convex immersion with Euclidean normal 1 1 (−ψ nL = p (−x, −y, 1). (3.6) 1 , −ψ2 , 1) = p 1 + x2 + y 2 1 + ψ12 + ψ22 Since the limit of Lψ3 when (x, y) tends to (0, 0) is 0, then (Lψ1 , Lψ2 ) tends to a curve Γ which is limit of the convex sections Lψ3 = ε when ε tends to 0. From the convexity of ψ, Lψ (U ? ) can be parametrized as the graph of a convex function φL on a domain V in the exterior of Γ, such that Γ is a boundary component of the closure of V. It is clear from (3.5) and (3.6) that φL and its gradient ∇φL extend continuously to Γ. Actually, φL ≡ 0 and ∇φL ≡ 0 on Γ. Theorem 3.2. Let β = (β1 , β2 , β3 ) : R → R3 be a 2π-periodic regular analytic curve satisfying that γ = (β1 /β3 , β2 /β3 ) is a 2π-periodic analytic parametrization of a convex Jordan curve. Let N be the real part of a holomorphic curve F ∈ F and ψ : ∆− → R3 the affine maximal immersion given by (3.3). Then ψ is, around the isolated singularity, the graph of a solution φ of (3.4) such that 1. N is the affine co-normal vector field of the graph of φ. 2. φ extends continuously at the origin and it has the underlying conformal structure of an annulus. 3. φ is not C 1 at the origin and ∇φ tends to the convex Jordan curve γ at the origin. Proof. From (3.2), (3.5) and (3.6) we obtain lim Lψ (s, t) = Γ(s) = (γ(s), 0), t→0

lim nL (s, t) = (0, 0, 1). t→0

Consequently, for r, ε > 0 small enough, Lψ (∆r− ) lies in the half space R3+ = {x3 > 0} and γ ε = Lψ (∆r− ) ∩ {x3 = ε} are regular convex Jordan curves tending to Γ(R) when ε tends to 0. Hence, Lψ (∆r− ) ∩ {0 < x3 < ε} is globally the graph of a convex function φL (y1 , y2 ), (y1 , y2 ) ∈ ΩL , for some domain ΩL in the exterior of Γ(R) which contains Γ(R) in its boundary. But then, via the Legendre transform of (y1 , y2 , φL ) we conclude that ψ is also a graph of a solution φ(x1 , x2 ) of (3.4), where (x1 , x2 ) ∈ U ? for some neighborhood U of the origin in R2 . Remark 3. The particular case of embedded isolated singularities conformal to an annulus and such that its affine co-normal has a constant coordinate along its singular curve has been considered in [AMaM]. Remark 4. If the curve γ is not a simple curve in R2 , then the isolated singularity is not embedded. 9

3.2

The conformal structure of a punctured disk

Let D? = {(u, v) ∈ R2 | 0 < u2 + v 2 < 1} be the punctured unit disk and a a positive real number. Consider the harmonic vector field N a : D? → R3 given by   √ a 2 2 (u, v) ∈ D? . N (u, v) = u, v, −a log u + v − a , √ Then, det[N a , dN a , ∗dN a ] = −a log( u2 + v 2 )(du2 +dv 2 ) and N a is the affine co-normal of an affine maximal immersion f? → R3 , ψa : D f? is the conformal universal covering of D? . Actually, ψ a is given by (2.7), where D well-defined in D? . In fact,   √ √ u2 + v 2 a 2 2 2 2 ψ (u, v) = au log u + v , av log u + v , , (u, v) ∈ D? . 2 Consequently, ψ a is a rotational embedding which extends continuously to the origin. In order to extend the above example, we consider the family N of harmonic vector fields N = (N1 , N2 , N3 ) ∈ N which admit, in a punctured disk Dε? of radius ε > 0, a series development of the form N (u, v) = N (R cos(t), R sin(t)) = (Rp cos(pt), Rp sin(pt), A3 0 + a log(R)) ! X X X + Rm A1m (t), Rm A2m (t), Rm A3m (t) , m≥p+1

m≥q+1

(3.7)

m≥k

where Ajm (t) = ajm cos(mt) + bjm sin(mt), j = 1, 2, 3. From (3.7), we have that [N, Nu , Nv ] = R2p−2 (−ap log(R)+o(u, v)) for some bounded function o(u, v) in Dε . Thus, there exists ε0 , 0 < ε0 < ε, such that N is the affine cof?0 → R3 given as in (3.3) on the conformal normal of an affine maximal surface ψ : D ε f?0 of D?0 . The immersion ψ is, in fact, well-defined in D?0 and, after universal covering D ε ε ε a straightforward computation, we obtain that it can be written as ψ = (−aRp log(R) cos(pt), −aRp log(R) sin(pt), R2p /2) +(o(p), o(p), o(2p + 1)) in Dε?0 , where we denote by o(l), l ∈ Z, l ≥ 0, a function depending on R and t given as o(l) = Rl f (R, t) for a certain function f (R, t) bounded in a neighborhood of the origin. Hence, if we put ψ = (ψ1 , ψ2 , ψ3 ) we have that ψ1 + i ψ2 = aRp log(R)ei pt + o(p) moves around the origin p times. Consequently, ψ is embedded if and only if p = 1. Actually, every affine maximal surface conformally equivalent to a puncture disk with an embedded isolated singularity is, up to an equiaffine transformation, one of these examples (see [AMaM]). 10

4

Global results and open problems

The recent solution to the affine Bersntein problem [TrWa2] and the Calabi conjecture [LiJi] has motivated its extension to affine maximal surfaces with singularities. In this section we review some global results concerning to affine maximal graphs with isolated singularities and complete affine maximal maps. We also pose some unsolved problems.

4.1

Entire affine maximal graphs with a finite number of isolated singularities

Let us consider the equation φyy ωxx − 2φxy ωxy + φxx ωyy = 0,



−3/4 ω = det ∇2 φ ,

in R2 \ A,

(4.1)

where A is either empty or a finite number of points in R2 . If A = ∅, Trudinger and Wang [TrWa, TrWa2] proved that a solution of (4.1) must be a quadratic polinomial. When A = 6 ∅, we know the answer only for trivial solutions of (4.1), that is, solutions of the equation det(∇2 φ) := φxx φyy − φ2xy = 1 in R2 \ A. (4.2) More precisely (see [GMaMi] for details), let Ω ⊂ C be a bounded domain whose boundary consists of n analytic Jordan curves C1 , . . . , Cn and z0 ∈ Ω. We denote by P (resp. Q) the unique, up to imaginary (resp. real) additive constants, holomorphic function in Ω \ {z0 } with a pole of residue 1 at z0 whose real part (resp. imaginary part) is constant along each Cj , j = 1, . . . , n. If we take N = (−2 Re(Q), −2 Im(P ), 1) then the map ψ = (ψ1 , ψ2 , ψ3 ) : Ω \ {z0 } → R3 given by (3.3) is a well-defined improper affine sphere (i.e. an affine maximal surface with a constant affine normal vector field), see [FMaM], and (ψ1 , ψ2 ) : Ω \ {z0 } → R2 \ {p1 , . . . , pn } is a global homeomorphism, where pj = (ψ1 , ψ2 )(Cj ), j = 1, . . . , n. Actually, ψ is the graph of a solution of (4.2) with A = {p1 , . . . , pn } as set of isolated singularities. These examples give all the solutions of (4.2) up to equiaffine transformations. Actually one can prove Theorem 4.1 ([GMaMi]). Any solution φ of (4.2) is one of the examples described above. In particular, there exists a bijective correspondence between the conformal equivalence classes of one punctured bounded domain in C of connectivity n − 1 and the space of solutions of (4.2), module equiaffine transformations. Concerning to nontrivial solutions of (4.1) we know the existence of rotational examples given, up to an equiaffine transformation, by φ(x, y) =

R2 , 2

x = aR log(R) cos(t),

y = aR log(R) sin(t), 11

R > 1, t ∈ [0, 2π[,

for some a ∈ R. They are affine maximal graphs with only an isolated singularity (see Figure 1). Two questions arise naturally whose answers remain unknown until now: • Is there nontrivial and non rotational solutions of (4.1) with only an isolated singularity? • What about non trivial solutions of (4.1) with more than one isolated singularities?

4.2

Embeddeness and Completeness

Although the affine metric is not well-defined on the singular set, we can define completeness of affine maximal maps in a similar way as Kokubu, Umehara and Yamada have done for other kind of surfaces with singularities, [KUY]. Definition 4.2. Let ψ : Σ → R3 be an affine maximal map. We say that ψ is complete if there exist compacts sets K, K 0 , K ⊂ K 0 ⊂ Σ, and a symmetric 2-tensor T with compact support in Σ such that if µ : Σ → [0, 1] is a differentiable function satisfying that µ(p) = 0 for all p ∈ K and µ(q) = 1 for all q ∈ Σ \ K 0 , then ge = (1 − µ)T + µ|g| is a complete metric on Σ. By using a Theorem of Hubber (see [H, Th. 13]), we can prove Theorem 4.3 ([AMaM3]). Let ψ : Σ → R3 be a complete affine maximal map with affine co-normal N . Then Σ is conformally equivalent to the complement of a finite point set {p1 , . . . , pn } in a compact Riemann surface Σ. The points {p1 , . . . , pn } will be called ends of ψ. We will say that an end p is regular if dN + i ∗dN extends meromorphically to p; otherwise the end will be called irregular. Remark 5. The ends of a complete improper affine map (see [Ma]) are always regular. A regular end is particularly interesting because around it we can assume that it comes as in (3.3) from a harmonic map N : D? → R3 well-defined on the unit punctured disk D? and such that dN + i ∗dN extends meromorphically to the origin. But in this case N admits a series development as follows ! X X X m m m N = R A1m (t), R A2m (t), R A3m (t) + log(R)(a1 , a2 , a3 ), m≥p

m≥q

m≥k

for some a1 , a2 , a3 ∈ R and p, q, k ∈ Z, where Ajm (t) = ajm cos(mt) + bjm sin(mt),

j = 1, 2, 3,

A1p 6= 0, A2q 6= 0. By using this expression and studying when det[N, dN, ∗dN ] does not vanish on D? , one has 12

Theorem 4.4 ([AMaM3]). Let ψ : D? → R3 be an affine maximal regular complete end with affine co-normal N . Then, ψ is embedded if and only if the holomorphic curve dN + i ∗dN has at most a pole of order 2 at the origin. In this case ψ has at the origin the same behavior as a revolution example. About global characterizations we only know that the only affine complete affine maximal surface is the elliptic paraboloid ([LiJi, TrWa3]). Regarding complete affine maximal maps, the family is plenty of global examples (see for instance Figure 1), and it should be very interesting to have characterizations of then. Here we pose the following questions whose answers may be not difficult of finding: • Is the elliptic paraboloid the unique complete affine maximal map with only one end which is regular and embedded? • Is it possible to characterize complete affine maximal maps with only two ends which are embedded and regular?

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