Differential Geometry and its Applications 32 (2014) 130–138

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Differential Geometry and its Applications

www.elsevier.com/locate/difgeo

An estimate for the Gaussian curvature of minimal surfaces inRm whose Gauss map is ramified over a set of hyperplanes

Pham Hoang Ha ∗

Department of Mathematics, Hanoi National University of Education, 136 XuanThuy str., Hanoi,Vietnam

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 October 2011Available online xxxxCommunicated by F. Pedit

MSC:primary 53A10, 14E22secondary 32H30, 32Q45

Keywords:Minimal surfaceGaussian curvatureRamificationOrbifoldsKobayashi hyperbolicity

In this article, using some new results on geometric orbifold we construct an estimatefor the Gaussian curvature of complete minimal surfaces in Rm. Thus, we get theramification of Gauss map of complete minimal surfaces.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Let x : M → Rm be a minimal surface, where M is a connected, oriented, real-dimension 2 manifold with-out boundary, and x = (x1, . . . , xm) is an immersion. Then M is a Riemann surface in the reduced structuredefined by local isothermal coordinate (u, v). The (generalized) Gauss map of the minimal surface M ,

g =(∂x1

∂z: · · · : ∂xm

∂z

): M → Qm−2(C) ⊂ Pm−1(C)

is holomorphic by the minimality of M , where z = u+ iv. The metric ds2 on M , induced from the standardmetric in Rm, is ds2 = 2

∑mj=1 |

∂xj

∂z |2dzdz, and the Gaussian curvature K is given by

* Current address: Laboratoire de Mathématiques, Université de Brest, 6, avenue Victor Le Gorgeu, CS 93837, 29238, BrestCedex 3, France.

E-mail address: phamhoangha23@gmail.com.

0926-2245/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.difgeo.2013.11.005

P.H. Ha / Differential Geometry and its Applications 32 (2014) 130–138 131

K = −4 |g̃ ∧ g̃′|2|g̃|6 = −4

∑j<k |gjg′k − gkg

′j |2

(∑m

j=1 |gj |2)3

where g̃ = (g1, . . . , gm), gj = ∂xj

∂z , 1 � j � m.It is a natural question to study the “value distribution” properties of the Gauss map g. H. Fujimoto

(see [4]) has shown that the Gauss map of a complete non-flat minimal surfaces can omit at mostm(m + 1)/2 hyperplanes in general position in Pm−1(C) under the assumption that g is non-degenerate.He also gave an example to show that m(m + 1)/2 is the best possible number when m is odd. After that,the “non-degenerate” assumption was removed by M. Ru (see [10]).

Moreover, by estimating the Gaussian curvature of minimal surfaces one can get the “value distribution”properties of the Gauss maps. This problem was studied, for instance, in [3,5] and [8] when m = 3. Later,Osserman and Ru generalized the proof of Ros in [8] to higher dimension. Namely, they proved the followingtheorem.

Theorem 1. (See [7, Theorem 1.1].) Let x : M → Rm be a minimal surface immersed in Rm. Suppose thatits Gauss map g omits more than m(m + 1)/2 hyperplanes in Pm−1(C), located in general position. Thenthere exists a constant C, depending on the set of omitted hyperplanes, but not the surface, such that

∣∣K(p)∣∣1/2d(p) � C (1)

where K(p) is the Gaussian curvature of the surface at p, and d(p) is the geodesic distance from p to theboundary of M .

In fact, given any point p on a complete surface satisfying the hypotheses, inequality (1) must hold withd(p) arbitrarily large, so that K(p) = 0. But a minimal surface in Rm with K ≡ 0 must lie on a plane andhence its Gauss map g is constant. So that Theorem 1 implies the “value distribution” properties of theGauss map of minimal surfaces in Rm.

On the other hand, Ru (see [11]) also gave generality of “value distribution” properties of the Gauss mapby using the same arguments of the proofs for the above-mentioned results of Fujimoto [4] and Ru [10].He studied the Gauss map with ramification. To give that result, we recall some definitions.

One says that g is ramified over a hyperplane H = {(w0 : · · · : wm−1) ∈ Pm−1(C) : a0w0 + · · · +am−1wm−1 = 0} with multiplicity at least e if all the zeros of the function (g,H) := a0g0 + · · ·+ am−1gm−1have orders at least e, where g = (g0 : . . . : gm−1). If the image of g omits H, one will say that g is ramifiedover H with multiplicity ∞. Ru stated the following.

Theorem 2. (See [11, Theorem 1].) For any complete minimal surface M immersed in Rm with its Gaussmap g. Let {Hj}qj=1 be hyperplanes in general position in Pm−1(C). If g is ramified over Hj with multiplicityat least mj for each j and

q∑j=1

(1 − m− 1

mj

)>

m(m + 1)2

then M is flat, or equivalently, g is constant.

In this article, we would like to study the Gauss map with ramification by estimating the Gaussiancurvature. To do this, we have to use some new results and new techniques on geometric orbifold.

Orbifolds were introduced by Campana in [1] as complex spaces endowed with an additional structure inthe form of a certain Q-Weil divisor. He also explained that they occur naturally in the study of fibrations forthe purpose of classification theory. The Q-Weil divisor encodes information about multiple fibers. Recently,

132 P.H. Ha / Differential Geometry and its Applications 32 (2014) 130–138

Campana and Winkelmann [2] and Rousseau [9] studied hyperbolicity aspects of orbifolds. They gave somenew results about this field. Here, we continue giving another one which will be needed for our main result.Then, we use the orbifold results and the arguments as in the proof of Osserman and Ru in [7] to improvethe omitted hyperplanes assumption in Theorem 1 by the ramification assumption.

In particular, the purpose of this paper is to prove the following.

Main Theorem. Let x : M → Rm be a minimal surface immersed in Rm with its Gauss map g : M →Pm−1(C). Let {Hj}qj=1 be hyperplanes in general position in Pm−1(C). Suppose that g is ramified over Hj

with multiplicity at least mj for each j and

q∑j=1

(1 − 1

mj

)> q − q − 1

m− 1 + m + 22 .

Then there exists a constant C, depending on the set of hyperplanes {Hj}qj=1, but not the surface, such that

∣∣K(p)∣∣1/2d(p) � C.

The Main theorem thus implies Theorem 1 and Theorem 2.

2. Some auxiliary results of minimal surfaces

In this section, we recall some auxiliary results of minimal surfaces which are used for proving the MainTheorem.

Theorem 3. (See [6, Theorem 1.2].) An inequality of the form (1) holds for all minimal surfaces in Rm

whose Gauss map omits a neighborhood of some hyperplane in Pm−1(C).

Theorem 4. (See [11, Theorem 1].) For any complete minimal surface M immersed in Rm and assume thatthe Gauss map g of M is k-nondegenerate (that is g(M) is contained in a k-dimensional linear subspace ofPm−1(C), but none of lower dimension), 1 � k � m− 1. Let {Hj}qj=1 be hyperplanes in general position inPm−1(C). If g is ramified over Hj with multiplicity at least mj for each j and

q∑j=1

(1 − k

mj

)> (k + 1)

(m− k

2 − 1)

+ m

then M is flat, or equivalently, g is constant.

For the aim of the proof of the Main theorem, we give the following proposition which is, in fact, a slightimprovement of Theorem 4.

Proposition 5. For any complete minimal surface M immersed in Rm. Assume that the Gauss map g of Mis k-nondegenerate and g(M) ⊂ Pk(C). Let {Hj}qj=1 be hyperplanes in Pm−1(C) such that {Hj ∩Pk(C)}qj=1are hyperplanes in n-subgeneral position in Pk(C) (k � n � m−1). If g is ramified over Hj with multiplicityat least mj for each j and

q∑j=1

(1 − k

mj

)> (k + 1)

(n− k

2

)+ n + 1

then M is flat, or equivalently, g is constant.

P.H. Ha / Differential Geometry and its Applications 32 (2014) 130–138 133

Proof. Using the same arguments as the proof of Ru in [11, pp. 758–762] we get Proposition 5. �Lemma 6. (See [7, Lemma 2.1].) Let Dr be the disk of radius r, 0 < r < 1, and let R be the hyperbolic radiusof Dr. Let ds2 = λ(z)2|dz|2 be any conformal metric on Dr with the property that the geodesic distancefrom z = 0 to |z| = r is greater than or equal to R. If the Gauss curvature K of the metric ds2 satisfies−1 � K � 0 then the distance of any point to the origin in the metric ds2 is greater than or equal to thehyperbolic distance.

Lemma 7. (See [7, Lemma 2.2].) Let ds2n be a sequence of conformal metrics on the unit disk D whose

curvatures satisfy −1 � Kn � 0. Suppose that D is a geodesic disk of radius Rn with respect to themetric ds2

n, where Rn → ∞, and that the metrics ds2n converge, uniformly on compact sets, to a metric ds2.

Then all distances to the origin with respect to ds2 are greater than or equal to the corresponding hyperbolicdistances in D. In particular, ds2 is complete.

Lemma 8. (See [7, Lemma 3.2].) Let x(n) = (x(n)1 , . . . , x

(n)m ) : M → Rm be a sequence of minimal immersions,

and g(n) : M → Qm−2(C) ⊂ Pm−1(C) the sequence of their Gauss maps. Suppose that {g(n)} convergesuniformly on every compact subset of M to a non-constant holomorphic map g : M → Qm−2(C) ⊂ Pm−1(C)and that there is some p0 ∈ M such that for each j, 1 � j � m, {x(n)

j (p0)} converges. Assume also that{|Kn|} is uniformly bounded, where Kn is the Gauss curvature of the minimal surface x(n). Then

(i) either a subsequence {Kn′} of {Kn} converges to zero or(ii) a subsequence {x(n′)} of {x(n)} converges to a minimal immersion, x : M → Rm, whose Gauss map is g.

3. Geometric orbifold

We always assume all complex spaces to be irreducible, reduced, normal, Hausdorff and paracompact.We now recall some notations and results of geometric orbifold as in [2] and [9].

Definition 1. Let (X,Δ) be an orbifold with Δ =∑

i(1− 1mi

)Zi where mi ∈ N∪ {∞} are multiplicities andwhere the Zi are distinct irreducible hypersurfaces.

A holomorphic map h from the unit disk D = {z ∈ C : |z| < 1} to X is a (non-classical) orbifoldmorphism from D to (X,Δ) if h(D) ⊂ supp(Δ) and if, moreover multx(h∗Zi) � mi for all i and x ∈ D

with h(x) ∈ supp(Zi), where supp(A) means the support of divisor A. If mi = ∞, we require h(D)∩Zi = ∅.The map h is called a “classical orbifold morphism” if the condition “multx(h∗Zi) � mi” is replaced by thecondition “multx(h∗Zi) is a multiple of mi”.

Definition 2. Let (X,Δ) and (X ′, Δ′) be orbifolds. Let Δ1 be union of all irreducible components of Δ

with multiplicity ∞. An orbifold morphism (resp. classical orbifold morphism) from (X,Δ) to (X ′, Δ′) is aholomorphic map f : X \Δ1 → X ′ such that:

1) f(X) ⊂ supp(Δ′).2) For every orbifold morphism (resp. every classical orbifold morphism) in the sense of Definition 1,

g : D → (X,Δ) with g(D) ⊂ f−1(supp(Δ′)) the composed map fog : D → X ′ defines an orbifold morphism(resp. a classical orbifold morphism) from D to (X ′, Δ′).

For the aim of this article, we only consider the non-classical versions. We refer to [1,2] and [9] for moredetails and others.

Remark 1. If f : (X,Δ) → (X ′, Δ′) is an orbifold morphism and Δ′′ is a Q+-Weil divisor on X ′ withΔ′′ � Δ′, then f is an orbifold morphism to (X ′, Δ′′), too.

134 P.H. Ha / Differential Geometry and its Applications 32 (2014) 130–138

Definition 3. The orbifold Kobayashi pseudo-distance d(X,Δ) on (X,Δ) is the largest pseudo-distance onX \Δ1 such that

g∗d(X,Δ) � dP

for every orbifold morphism g : D → (X,Δ), where dP denotes the Poincaré distance on D.

Definition 4. An orbifold (X,Δ) is hyperbolic if d(X,Δ) is a distance on X \Δ1.

Definition 5. We say that (X,Δ) is hyperbolically imbedded in X if for any two sequences of points(pn), (qn) ⊂ X \Δ1 converging to two points p, q ∈ X, then

d(X,Δ)(pn, qn) → 0(n → ∞) ⇒ p = q.

Proposition 9. (See [9].) Let ω be a Hermitian metric on X compact. The orbifold (X,Δ) is hyperbolicallyimbedded in X iff there is a positive constant c such that

f∗ω � c hP

for all orbifold morphisms f : D → (X,Δ), where hP denotes the Poincaré metric.

Proposition 10. Let ω be a Hermitian metric on X compact. Assume that the orbifold (X,Δ) is hyperbolicand hyperbolically imbedded in X then the set of all orbifold morphisms f : D → (X,Δ) is relatively compactin Hol(D,X), the set of all holomorphic maps of D into X.

Proof. Using Proposition 9, one get a positive constant c such that

f∗ω � c hP

for all orbifold morphisms f : D → (X,Δ), where hP denotes the Poincaré metric.Let δ be the distance function on X induced by 1

cω. Then

f∗δ � δP

for all orbifold morphisms f : D → (X,Δ), where δP denotes the Poincaré distance on D. That impliesδ � d(X,Δ).

On the other hand, we also have the claim that, for each x ∈ D, the set of {f(x)} for all orbifold morphismf : D → (X,Δ) is relatively compact in X by the compactness of X.

Thus, we now see that the set of all orbifold morphisms f : D → (X,Δ) is relatively compact inHol(D,X). �Theorem 11. (See [9, Theorem 5.3].) Let H1, . . . , Hq be q hyperplanes in general position in Pn(C) withq > 2n. Let Δ =

∑qj=1(1 − 1

mj)Hj with

deg(Δ) > q − q − 1n

+ 1.

Then every orbifold morphism f : C → (Pn(C), Δ) is constant.

P.H. Ha / Differential Geometry and its Applications 32 (2014) 130–138 135

Theorem 12. (See [9, Theorem 5.1].) Let H1, . . . , Hq be q hyperplanes in general position in Pn(C) withq > 2n. Let Δ =

∑qj=1(1 − 1

mj)Hj with

deg(Δ) > q − q − 1n

+ 1.

Then (Pn(C), Δ) is hyperbolic and hyperbolically imbedded in Pn(C).

Proposition 13. (See [2, Proposition 7].) Let fn : (X,Δ) → (X ′, Δ′) be a sequence of orbifold morphisms.Assume that (fn), regarded as a sequence of holomorphic maps from X to X ′ converge locally uniformly toa holomorphic map f : X → X ′.

Then either f(X) ⊂ supp(Δ′) or f is an orbifold morphism from (X,Δ) to (X ′, Δ′).

4. Proof of the Main Theorem

We shall prove the Main Theorem by reduction to absurdity.Suppose the Main theorem is not true. We will construct a non-flat complete minimal surface whose Gauss

map is ramified a set of hyperplanes in general position, thus getting a contradiction with Proposition 5.So suppose the conclusion of the Main theorem is not true; then there is a sequence of (non complete)minimal surfaces x(n) : Mn → Rm and points pn ∈ Mn such that |Kn(pn)|1/2dn(pn) → ∞, and such thatthe Gauss map g(n) of x(n) is ramified a fixed set of q hyperplanes {Hj}qj=1 in general position in Pm−1(C).

First, we use the same arguments of Osserman and Ru in [7, pp. 590–591] to claim that the surfaces Mn

can be chosen so that

Kn(pn) = −1, −4 � Kn � 0 on Mn for all n and dn(pn) → ∞. (2)

We now prove the claim. Without loss of generality, we can assume that Mn is a geodesic disk centeredat pn. Let M ′

n = {p ∈ Mn : dn(p, pn) � dn(pn)/2}. Then Kn is uniformly bounded on M ′n and d′n(p) =

distance of p to the boundary of M ′n tends to zero as p → ∂M ′

n. Hence |Kn(p)|(d′n(p))2 has a maximum ata point p′n interior to M ′

n. Therefore

∣∣Kn

(p′n

)∣∣(d′n(p′n))2 �∣∣Kn(pn)

∣∣(d′n(pn))2 = 1

4∣∣Kn(pn)

∣∣(dn(p))2 → ∞.

So we can replace the Mn by the M ′n, with |Kn(p′n)|(d′n(p′n))2 → ∞. We rescale M ′

n to make Kn(p′n) = −1.By the invariance under scaling of the quantity K(p)(d(p))2, we shall have d′n(p′n) → ∞; here, withoutcausing confusion, we use the same notation d′n to denote the geodesic distance with respect to the rescaledmetric. Again we can assume that M ′

n is a geodesic disc centered at p′n, and let

M ′′n =

{p ∈ M ′

n

∣∣∣ dn(p, p′n) < d′n(p′n)2

}.

Then p ∈ M ′′n implies that d′n(p) � d′n(p′n) − dn(p, p′n) � d′n(p′n)

2 and

∣∣Kn(p)∣∣ (d′n(p′n))2

4 �∣∣Kn(p)

∣∣(d′n(p))2 �

∣∣Kn

(p′n

)∣∣(d′n(p′n))2 =(d′n

(p′n

))2.

Therefore |Kn(p)| � 4 on M ′′n . Furthermore, d′′n(p′n) = d(p′n, ∂M ′′

n ) = d′n(p′n)/2 → ∞. This proves the claim.

136 P.H. Ha / Differential Geometry and its Applications 32 (2014) 130–138

By translations of Rm we can assume that x(n)(pn) = 0. We can also assume that Mn is simply connected,by taking its universal covering, if necessary. By the uniformization theorem, Mn is conformally equivalentto either the unit disc D or the complex plane C, and we can suppose that pn maps onto 0 for each n.

If Mn is conformally equivalent to C, g(n) is the orbifold morphism of C into (Pm−1(C), Δ), where Δ =∑qj=1(1 − 1

mj)Hj will be denoted throughout this section. By assumption, we have

degΔ =q∑

j=1

(1 − 1

mj

)> q − q − 1

m− 1 + m + 22 > q − q − 1

m− 1 + 1.

This implies, by Theorem 11, that g(n) is constant, so Kn ≡ 0, which contradicts the condition that|Kn(0)| = 1.

So we have constructed a sequence of minimal surfaces, x(n) : D → Rm, satisfying (2). Since, by The-orem 12, (Pm−1(C), Δ) is hyperbolic and hyperbolically imbedded, then using Proposition 10, we geta subsequence of Gauss maps g(n) of x(n) exists – without of loss generality we assume g(n) itself – suchthat g(n) : D → Pm−1(C) converges uniformly on every compact subset of D to a map g : D → Pm−1(C).

We now use again the same arguments of Osserman and Ru [7, pp. 591–592] to claim that g is non-constant. Suppose not, i.e., g is a constant map, and g maps the disk D onto a single point P . Let H be anyhyperplane not containing the point P , and let U , V be disjoint neighborhoods of H and P respectively.Let C be the constant in Theorem 3 such that

∣∣K(p)∣∣1/2d(p) � C

for any minimal surface S in Rm whose Gauss map omits the neighborhood U of H, where p is a point of Sand d(p) is the geodesic distance of p to the boundary of S. Choose r < 1 such that the hyperbolic distanceR of z = 0 to |z| = r satisfies R > C. Since g(n) converges uniformly to g on |z| � r, the image of |z| = r liesin the neighborhood V of p for sufficiently large n, say n � n0. It follows that for n � n0, the image of thedisk |z| � r under g(n) omits the neighborhood U of H and we may therefore apply the above inequality toconclude

∣∣Kn(0)∣∣1/2dn(r) � C

where dn(r) is the geodesic distance from the origin to the boundary of the surface x(n) : Dr → Rm.But |Kn(0)| = 1 for all n, and hence dn(r) � C for n � n0.

On the other hand, we shall get a lower bound for dn(r) from Lemma 6. The surface x(n) : {|z| < 1} → Rm

is a geodesic disk of radius Rn. If we reparametrize by w = rnz where |w| = rn has hyperbolic radius Rn,then the circle |z| = r corresponds to |w| = rnr, and by Lemma 6, the distance in the surface metric fromthe origin to any point on the circle |z| = r, or equivalently, |w| = rnr, is greater than or equal to thehyperbolic distance from 0 to |w| = rnr. But as n → ∞, Rn → ∞ and rn → 1, so that the hyperbolic radiusof |w| = rnr tends to the hyperbolic radius of |w| = r, which is R. Since by assumption R > C we havefor n sufficiently large that the surface distance from z = 0 to |z| = r is greater than C, contradicting theearlier bound dn(r) � C. Thus we conclude that the limit function g can not be constant.

Therefore the hypotheses of Lemma 8 are satisfied. Since |Kn(0)| = 1, the possibility (i) of Lemma 8cannot happen. Thus, there exists a subsequence {x(n′)} of {x(n)} which converges to a minimal immersionx : D → Rm and whose Gauss map is g. By (2) and by Lemma 7, x is complete. Since g(n) are orbifoldmorphism of D into (Pm−1(C), Δ) then g is an orbifold morphism of D into (Pm−1(C), Δ) or g(D) ⊂Supp(Δ) from Proposition 13.

The first case gives g constant thanks to Proposition 5 with k = n = m− 1.

P.H. Ha / Differential Geometry and its Applications 32 (2014) 130–138 137

In the second case, by Remark 1 and Proposition 13, for each Δ′j = (1 − 1

mj)Hj , 1 � j � q, we show

that either g is an orbifold morphism of D into (Pm−1(C), Δ′j) or g(D) ⊂ supp(Δ′

j). So g is ramified overHj with multiplicity at least mj or the image of g is contained in Hj for each j. Then, there is a partitionof indices {1, 2, . . . , q} = I ∪ J such that g is an orbifold morphism from D into (LI , Δ

′′), where

LI =⋂i∈I

Hi and Δ′′ =∑i∈J

(1 − 1

mj

)Hj ∩ LI :=

∑i∈J

(1 − 1

mj

)H ′

j .

We may assume that card(I) = l, 1 � l � m− 1, then dimLI = m− 1 − l and g is ramified over H ′j with

multiplicity at least mj for each j ∈ J .Suppose that g is k-nondegenerate, we may assume g(D) ⊂ Pk(C), then k � m − 1 − l and {H ′′

j =H ′

j ∩ Pk(C)}j∈J are q − l hyperplanes in (m− 1 − l)-subgeneral position in Pk(C).Using Proposition 5, we get

∑j∈J

(1 − k

mj

)� (k + 1)

(m− l − 1 − k

2

)+ m− l

⇒∑j∈J

(1 − 1

mj

)� k + 1

k

(m− l − 1 − k

2

)+ m− l

k+ q − l − q − l

k

⇒q∑

j=1

(1 − 1

mj

)� k + 1

k

(m− l − 1 − k

2

)+ m

k+ q − q

k.

Now, combining with the assumption of the Main Theorem, we have

q − q − 1m− 1 + m + 2

2 <k + 1k

(m− l − 1 − k

2

)+ m

k+ q − q

k

⇔ (q − 1)(m− 1 − k)k(m− 1) <

k + 1k

(m− l − 1 − k

2

)+ m− 1

k− m + 2

2 .

Noting that we have q > 1 + (m− 1)(m + 2)2 from the assumption of the Main Theorem. Thus, we get

(m + 2)(m− 1 − k)2k <

k + 1k

(m− l − 1 − k

2

)+ m− 1

k− m + 2

2

⇔ (m− k)2 − 3(m− k) + 2kl + 2l + 22k < 0

⇔ (m− k − 1)(m− k − 2) + 2kl + 2l2k < 0 (with m− k � l + 1 � 2).

That is a contradiction. Thus, the Main Theorem is proved.

Acknowledgements

We are grateful to Professors Gerd-Eberhard Dethloff and Do Duc Thai for many stimulating discussionsconcerning this material. The research is partially supported by a NAFOSTED grant of Vietnam.

References

[1] F. Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier 54 (2004) 499–665.

138 P.H. Ha / Differential Geometry and its Applications 32 (2014) 130–138

[2] F. Campana, J. Winkelmann, A Brody theorem for orbifolds, Manuscr. Math. 128 (2009) 195–212.[3] H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces, J. Differ. Geom. 29 (1989) 245–262.[4] H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces II, J. Differ. Geom. 31 (1990) 365–385.[5] H. Fujimoto, On the Gauss curvature of minimal surfaces, J. Math. Soc. Jpn. 44 (1992) 427–439.[6] R. Osserman, Global properties of minimal surfaces in E3 and En, Ann. Math. 80 (1964) 340–364.[7] R. Osserman, M. Ru, An estimate for the Gauss curvature on minimal surfaces in Rm whose Gauss map omits a set of

hyperplanes, J. Differ. Geom. 46 (1997) 578–593.[8] A. Ros, The Gauss map of minimal surfaces, in: Differential Geometry, Valencia, 2001, World Sci. Publ., River Edge, NJ,

2002, pp. 235–252.[9] E. Rousseau, Hyperbolicity of geometric orbifolds, Trans. Am. Math. Soc. 362 (2010) 3799–3826.

[10] M. Ru, On the Gauss map of minimal surfaces immersed in Rn, J. Differ. Geom. 34 (1991) 411–423.[11] M. Ru, Gauss map of minimal surfaces with ramification, Trans. Am. Math. Soc. 339 (1993) 751–764.

Further reading

[12] S.S. Chern, R. Osserman, Complete minimal surface in euclidean n-space, J. Anal. Math. 19 (1967) 15–34.[13] H. Fujimoto, Value Distribution Theory of the Gauss Map of Minimal Surfaces in Rm, Aspects of Mathematics, vol. E21,

Vieweg, Wiesbaden, 1993.[14] S. Kobayashi, Hyperbolic Complex Spaces, Springer, Berlin, 1998.