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AN ESTIMATE FOR THE GAUSSIAN CURVATURE OF MINIMAL SURFACES WITH RAMIFICATION

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AN ESTIMATE FOR THE GAUSSIAN CURVATURE
OF MINIMAL SURFACES WITH RAMIFICATION
PHAM DUC THOAN
Abstract. In this article, we give an estimate for the Gaussian curvature of minimal surfaces whose the Gauss map has more
branching by the method of A. Ros [6] and by orbifold theory.

Contents
1. Introduction
2. Auxiliary lemmas
3. The proof of Theorems
3.1. The proof of Theorem 1
3.2. The proof of Theorem 3
3.3. The proof of Theorem 5
References

1
6
9
9
11
15
17

1. Introduction
Value distribution poperties of the Gauss map was stydied earlier.
In 1988, H. Fujimoto ([2]) proved Nirenberg’s conjecture that if M is
a complete non-flat minimal surface in R3 , then its Gauss map can
omit at most 4 points, and the bound is sharp. After that, he ([4])
also extended that result for complete minimal surfaces in Rm in the
case the Gauss map was assumed non-degenerate. In which, the Gauss
2010 Mathematics Subject Classification. Primary 53A10; Secondary 53C42,


30D35, 32H30.
Key words and phrases. Minimal surface, Gauss map, Ramification, Value distribution theory, estimate curvature, orbifold.
1


2

PHAM DUC THOAN

map can omit at most m(m + 1)/2 hyperplanes in general position in
Pm−1 (C).
In 1993, M. Ru ([12]) refined these results by studying the Gauss
maps of minimal surfaces in Rm with ramification.
In the case m = 3, M. Ru proved:
Theorem A. Let M be a complete minimal surface in R3 . If there
are q (q > 4) distinct points a1 , · · · , aq ∈ P1 (C) such that the classical
Gauss map of M is ramified over aj with multiplicity at least mj for
q

each j and

(1− m1j ) > 4 then M is flat, or equivalently g is constant.

j=1

In the case, m = 4, H. Fujimoto ([5]) poved the following theorem:
Theorem B. Suppose that M is a complete non-flat minimal surface in R4 and g = (g1 , g2 ) is the classical Gauss map of M. Let
a11 , ..., a1q1 , a21 , ..., a2q2 be q1 + q2 (q1 , q2 > 2) distinct points in P1 (C).
(i) In the case gl ≡ constant (l = 1, 2), if gl is ramified over alj with
multiplicity at least mlj for each j (l = 1, 2) then

q1

1
) ≤ 2, or γ2 =
γ1 =
(1 −
m1j
j=1

q2

(1 −
j=1

1
) ≤ 2, or
m2j

1
1
+
≥ 1.
γ1 − 2 γ2 − 2
(ii) In the case where g1 or g2 is constant, say g2 ≡ constant, if g1
is ramified over a1j with multiplicity at least m1j for each j, we have
the following:
q1

(1 −


γ1 =
j=1

1
) ≤ 3.
m1j

Relate to this problem, G. Dethloff and P. H. Ha ([7]) showed that
the above theorems still hold when the Gauss map restrict on annular
end of M .
By estimate the Gaussian curvature of minimal surfaces we can get
the ”value distribution” properties of the Gauss map (see in [2], [3] and
[6] when m = 3). Using the method of A. Ros [6] and theory orbifold,
we will give an estimate for the Gaussian curvature of minimal surfaces


AN ESTIMATE FOR THE GAUSSIAN CURVATURE OF MINIMAL...

3

in R3 and R4 whose Gauss maps ramified over the set of distinct points.
Namely, we will prove the followings:
Theorem 1. Let M be a minimal surface in R3 and q (q > 4) distinct
points a1 , · · · , aq ∈ P1 (C) and A be an annular end of M which is
conformal to {z : 0 < 1/r < |z| < z}, where z is conformal coordinate.
Suppose that the classical Gauss map g of M is ramified over aj and
the restriction of g to A is ramified over aj with multiplicity at least
mj for each j such that
q


(1 −
j=1

1
) > 4.
mj

(1.1)

Then one has a curvature estimate corresponding to A i.e there exists
a constant C, depending on the set of ramified points and A, but not
the surfaces, such that
|K(p)|1/2 d(p) ≤ C,

(1.2)

where K(p) is the Gaussian curvature of the surface at p and d(p) is
the geodesis distance from p to the boundary of M .
Corollary 2. If the Gauss map on an annular end A of a minimal
surface in R3 assumes five values on the unit sphere only finitely often
with ramification, one has a curvature estimate corresponding to A.
Proof. By passing to a sub-annular end A1 of A, we can see that the
Gauss map will omit 5 values on A1 . This implies that the condition
(1.1) is satisfied. Thus, the Theorem 1 deduce the Corollary 2.
Theorem 3. Let M be a minimal surface in R4 and g = (g1 , g2 ) be the
classical Gauss map of M . Let {al1 , · · · , alql } (l = 1, 2) be the families
of distinct points in P1 (C). Suppose that gl (l = 1, 2) is ramified over
alj with multiplicity at least mlj for each j such that
q1


(1 −

γ1 =
j=1

1
)>2
m1j

(1.3)


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PHAM DUC THOAN
q2

(1 −

γ2 =
j=1

1
)>2
m2j

1
1
+
< 1.

γ1 − 2 γ2 − 2

(1.4)

(1.5)

Then one has a curvature estimate i.e there exists a constant C, depending on the set of ramified points, but not the surfaces, such that
inequality of type (1.2) holds.
Corollary 4. Let M be a minimal surface in R4 and g = (g1 , g2 ) be the
classical Gauss map of M where g1 or g2 is constant, say g2 ≡ constant.
Let {a1 , · · · , aq } be the families of distinct points in P1 (C). Suppose that
g1 is ramified over aj with multiplicity at least mj for each j such that
q

(1 −
j=1

1
) > 3.
mj

(1.6)

Then one has a curvature estimate.
Proof. Since g2 is constant, the condition (1.4) is satisfied. The condition (1.6) implies the conditions (1.3) and (1.5). Then the Theorem 3
deduce the Corollary 4.
In the higher dimension case, the result of M. Ru ([12]) can be stated
as follows:
Theorem C. Let M be a complete minimal surface in Rm . Suppose
that the (generalized) Gauss map G of M is k−nondegenerate (that

is G(M ) is contained in a k−dimensional linear subspace in Pm−1 (C),
but none of lower dimension), 1 ≤ k ≤ m − 1. Let {Hj }qj=1 be hyperplanes in general position in Pm−1 (C). If G is ramified over Hj with
multiplicity at least mj for each j and
q

(1 −
j=1

k
k
) > (k + 1)(m − − 1) + m
mj
2

then M is flat, or equivalently G is constant.
In particular, if there are q (q > m(m + 1)/2) hyperplanes {Hj }qj=1
in general position in Pm−1 (C) such that G is ramified over Hj with


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5

multiplicity at least mj for each j, and
q

(1 −
j=1

m−1

m(m + 1)
)>
mj
2

then M is flat, or equivalently G is constant.
In 1997, R. Osserman and M. Ru ([10]) generalized the proof of A.
Ros in [6] to minimal surfaces in Rm . They proved that if minimal
surfaces whose Gauss map omits more than m(m + 1)/2 hyperplanes
in general position then there exists a constant C, depending on the
set of omitted hyperplanes, but not the surfaces, such that inequality
of type (1.2) holds.
Recently, P. H. Ha ([8]) gave an improvement of the Theorem of M.
Ru. He proved the following theorem:
Theorem D. Let x : M → Rm be a minimal surface 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

1−
j=1

1
mj

>q−

m+2
q−1

+
.
m−1
2

Then one has a curvature estimate i.e there exists a constant C, depending on the set of hyperplanes {Hj }qj=1 , but not the surfaces, such
that inequality of type (1.2) holds.
A natural question is that how the type of the above theorem is
in the case of set of hyperplanes in N − subgeneral in Pm−1 (C). The
final purpose of this article is to give some affirmative answers for this
question. Namely, we will prove the followings:
Theorem 5. Let x : M → Rm be a minimal surface in Rm with
its Gauss map G : M → Pm−1 (C). Let {Hj }qj=1 be hyperplanes in
N −subgeneral position in Pm−1 (C). Suppose that G is ramified over
Hj with multiplicity at least mj for each j such that
q

1−
j=1

1
mj

>q−

q − (2N − n + 1) 2N − n + 1
+
,
n
2



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PHAM DUC THOAN

where n = m − 1. Then one has a curvature estimate.
In particular, in the case the set of hyperplanes are located in general
in Pm−1 (C), Theorem 5 immidately become to the Theorem D.
The main idea to prove the theorems is refine the original ideas of
A. Ros ([6]) and M. Ru ([10]). After that, we use arguments similar to
those used by P. H. Ha ([8]), A. Ros and M. Ru to finish the proofs.
2. Auxiliary lemmas
In this section, we recall some auxililary results of minimal surfaces
and geometric orbifold which will be used later. We first recall the
classical results of the Nevanlinna theory.
Theorem 6. ( [5, Cartan-Nochka’s theorem]) Let f : C → Pn (C) be
a linearly nondegenerate holomorphic mapping and {Hj }qj=1 be hyperplanes in N − subgeneral position in Pn (C). Then, we have
q
i=1

δ [n] (Hi , f ) ≤ 2N − n + 1.

As a particular case n = 1, we recover the following classical result:
Theorem 7. ([9, Nevanlinna]) Let f : C → P1 (C) be a non-constant
meromorphic function. Then
δ(a) ≤ 2.
a∈P1 (C)

Now, we recall the some results of the orbifold theory which was

introduced by F. Campana ([1]).
Proposition 8. ([1]) Let fn : (X, ∆) → (X , ∆ ) be a sequence of orbifold morphism. Assume that (fn ), regarded as a sequence of holomorphic maps from X to X converge locally uniformly to a holomorphic
map f : X → X . Then either f (X) ⊂ supp(∆ ) or f is an orbifold
morphism from (X, ∆) to (X , ∆ ).
Proposition 9. ([11]) Let ω be a hermitian metric on X compact.
Then (X, ∆) is hyperbolically (reps. classically hyperbolically) imbedded
in X iff there is a positive constant c such that f ∗ ω ≤ c · hP for all


AN ESTIMATE FOR THE GAUSSIAN CURVATURE OF MINIMAL...

7

orbifold (reps. classically orbifold) morphism f : D → (X, ∆), where
hP denotes the Poincar metric.
We need to prove following Proposition for curve orbifold:
Proposition 10. Let aj be q distinct points in P1 (C) and put ∆ =
q

1−
j=1

1
mj

aj with q > 2. If
q

1−


deg(∆) =
j=1

1
mj

>2

then (P1 (C), ∆) is hyperbolically imbedded in P1 (C), thus is also hyperbolic.
Proof. Suppose that (P1 (C), ∆) is not hyperbolically imbedded in P1 (C).
By Proposition 9, we can show that there exists a sequence of orbifold
morphisms fn : D → (P1 (C), ∆) such that lim ||fn = +∞||. Thanks to
Brody reparametrization, we obtain a sequence of orbifold morphisms
gn : D(0, rn ) → (P1 (C), ∆) with rn → +∞, converging to a holomorphic map f : C → P1 (C) which either a non-constant orbifold
morphism f : C → (P1 (C), ∆) or a non-constant holomorphic map
f : C → supp(∆). Since supp(∆) is discrete, the first case is not possible. The second case is not possible either by the result of Nevanlinna
(Theorem 7). Thus Proposition 10 is proved.
Now, we prove two Propositions which generalized the theorems of
E. Rousseau ([11, Theorem 5.3 and Theorem 5.1]) for higher dimension
orbifold.
Proposition 11. Let Hj be q hyperplanes located in N −subgeneral
position in Pn (C) and ∆ =

q
j=1

1−

1
mj


>q−

1
mj

q

1−

deg(∆) =
j=1

Hj with q > 2N . If
q − (2N − n + 1)
n

then every orbifold morphism f : C → (Pn (C), ∆) are constant.
Proof. Suppose that f is l−nondegenerate (1 ≤ l ≤ n), we may assume
that f (C) ⊂ Pl (C). Then Zj = Pl (C) ∩ Hj are q hyperplanes in Pl (C),


8

PHAM DUC THOAN

located in N −subgeneral position. By the First Main Theorem of
Nevanlinna theory we have
T (r, f ) ≥ N(H,f ) (r) + C,
where C is a constant. Since f ∗ Hj has multiplicity at least mj at every

point of f −1 (Hj ), we have
mj [l]
N(H,f ) (r) ≥
N
(r).
l (H,f )
Therefore
l
δ [l] (Hj , f ) ≥ 1 −
.
mj
Then
q
j=1

q
j=1

δ [l] (Hj , f ) ≥

1−

l
mj

= l deg(∆) − (l − 1)q.

By the Theorem 6, we deduce that
l deg(∆) − (l − 1)q ≤ 2N − l + 1.
Since condition q > 2N and the fact that l ≤ n, we have

q − (2N + 1)
q − (2N + 1)
−1≤q−
− 1.
l
n
That is a contradition. Thus, Proposition 11 is proved.
deg(∆) ≤ q −

Proposition 12. Let Hj be q hyperplanes located in N −subgeneral
q

position in Pn (C) and ∆ =

1−
j=1

q

1−

deg(∆) =
j=1

1
mj

1
mj


Hj with q > 2N . If

>q−

q − (2N − n + 1)
n

then (Pn (C), ∆) is hyperbolically imbedded in Pn (C), thus is also hyperbolic.
Proof. Suppose that (Pn (C), ∆) is not hyperbolically imbedded in Pn (C).
Similar to proof of Proposition 10, we obtain a sequence of orbifold
morphisms gn : D(0, rn ) → (Pn (C), ∆) with rn → +∞, converging to a
holomorphic map f : C → Pn (C) which either a non-constant orbifold
morphism f : C → (Pn (C), ∆) or a non-constant holomorphic map
f : C → supp(∆).


AN ESTIMATE FOR THE GAUSSIAN CURVATURE OF MINIMAL...

9

The first case does not happen by Proposition 11, so the second case
must happen. By Proposition 8, for each ∆j = 1 − m1j Hj , 1 ≤
j ≤ q that either g is an orbifold morphism from C to (Pn (C), ∆) or
f (C) ⊂ supp(∆j ). Thus, there exists a partition of {1, 2 · · · , q} = I ∪ J
such that for all j ∈ J, LI ⊂ Hj and f is an orbifold morphism from
C to (LI , ∆ ) where LI = ∩i∈I Hi and ∆ = j∈J 1 − m1j (Hj ∩ LI ).
Assume that card(I) = l and rank{Hi : i ∈ I} = k ≤ l, we have
dim LI = n−k. Then q−l hyperplanes Hj ∩LI are in (N −l)−subgeneral
position in LI .
The sequence gn : D(0, rn ) → (Pn (C), ∆) can be seen as a sequence of orbifold morphism gn : D(0, rn ) → (Pn (C), ∆J ) where ∆J =

1
j∈J 1 − mj Hj since ∆J ≤ ∆. Therefore, by Proposition 8, it converges to a map f which is either an orbifold morphism from C to
(Pn (C), ∆ ) or verifies f (C) ⊂ supp(∆ ).
Since the condition q > 2N , we get q − l > 2(N − l) and using again
Theorem 11,
(q − l) − (2(N − l) − (n − k) + 1)
n−k
for k = 1, · · · , n − 1. We have
deg(∆J ) ≤ (q − l) −

deg(∆) = deg(∆I ) + deg(∆J )
(q − l) − (2(N − l) − (n − k) + 1)
n−k
q − (2N − n + 1)
≤q−
.
n
This is a contradition. Thus, Proposition 12 is proved.
≤ l + (q − l) −

Proposition 13. ([8]) Let ω be a hermitian metric on X compact.
Assume that the orbifold (X, ∆) is hyperbolic and hyperbolically imbedded in X. Then the set of all orbifold morphisms f : D → (X, ∆) is
relatively compact in Hol(D, X), the set of all holomorphic of D into
X.
3. The proof of Theorems
3.1. The proof of Theorem 1.


10


PHAM DUC THOAN

Proof. We need the following Lemma of A. Ros [6, Lemma 6]:
Lemma 14. ([6]) Let x(v) : M → R3 be a sequence of conformal minimal immersion, {g v } ⊂ M(M ) the sequence of their Gauss maps and
Kv the Gauss curvature of x(v) . Suppose that {g v } converges to a meromorphic function g ∈ M(M ), the sequence {Kv } is uniformly bounded
and that {x(v) (p0 )} converges for some point p0 ∈ M . Then we have
the following possibilities:
(i) g is constant map, or
(ii) a subsequence {Kv } of {Kv } converges to zero, or
(iii) a subsequence {x(v ) } of {x(v) } converges to a conformal minimal
immersion x : M → R3 whose Gauss map is g.
Now the proof of Theorem 1.
We shall prove the Theorem 1 by reduction to absurdity. Suppose
that the Theorem 1 is not true. We will constract a non-flat complete minimal surface whose classical Gauss map is ramified a set
of distinct points. Then there exists a sequence of (non complete)
minimal surfaces x(v) : Mv → R3 and points pv ∈ Mv such that
|Kv (pv )|1/2 dv (pv ) → ∞, and the classical Gauss map g v of x(v) is ramified over a fixed set of q distinct points aj in P1 (C) and the restriction
g v to annular end A is ramified over aj with multiplicity at least mj
for each j.
The arguments of R. Osserman and M. Ru in [10, pp. 590-591] show
that we can choose the surfaces Mv satisfying condition
Kv (pv ) = −1, −4 ≤ Kv ≤ 0 on Mv for all v and dv (pv ) → ∞.

(3.7)

By translations of R3 we can assume that x(v) (pv ) = 0 and Mv is
simply connected, by taking its universal covering, if necessary. By the
uniformization theorem, we can see that Mv is conformally equivalent
to either the unit disk D or the complex plane C, and we can suppose
that pv maps onto 0 for each v.

If Mv is conformally equivalent to C, g v is the meromorphic function
on C, thus is holomorphic function into P1 (C) which ramified over aj


AN ESTIMATE FOR THE GAUSSIAN CURVATURE OF MINIMAL...

11

with multiplicity at least m∗j ≥ 2. By assumption q > 4 and we have
q

1−
j=1

1
m∗j

> 2.

(3.8)

This implies that g v is constant by the result of Nevanlinna (Theorem
7), so Kv ≡ 0, which contradicts to the condition that Kv (0) = −1.
Thus, we have constructed a sequence of minimal surfaces, x(v) :
D → R3 , satisfying (3.7). By Proposition 10, the orbifold (P1 (C), ∆)
∆ = qj=1 1 − m1j aj is hyperbolic and hyperbolically imbedded in
P1 (C). Therefore, we obtain a subsequence of classical Gauss maps g v
of x(v) exists- without of loss generality we assume that g v : D → P1 (C)
converges uniformly on every compact subset of D to a map g : D →
P1 (C).

Again, the arguments of R. Osserman and M. Ru in [10, pp. 591-592]
or of A. Ros [6, pp. 247-248] implies that g is non-constant. Moreover,
by Lemma 14 and by arguments of R. Osserman and M. Ru in [10, pp.
591-592], there exists a subsequence {x(v ) } of {x(v) } which converges to
a complete minimal immersion x : D → R3 and whose classical Gauss
map is g. Since g v : D → (P1 (C), ∆) are the orbifold morphisms,
g : D → (P1 (C), ∆) is the orbifold morphism or g(D) ⊂ supp(∆)
from Proposition 8. Since supp(∆) is discrete, the second possibility
is not happen. By taking sub-annular end if necessary, from Theorem
A for ramification of the Gauss map on annular end (see [7]), the first
possibility is not happen either. This is a contradition. Thus Theorem
1 is proved.
3.2. The proof of Theorem 3.
Proof. We first recall some notations on the Gauss map of minimal
surfaces in R4 . Let x = (x1 , x2 , x3 , x4 ) : M → R4 be a non-flat complete
minimal surface in R4 . By definition, we may regard the classical Gauss
map g as a pair of meromorphic functions g = (g1 , g2 ) on M to P1 (C) ×
P1 (C). We call (φdz, g1 , g2 ) the Weierstrass data. We know that the
zeros of φdz of order k coincide exactly with the poles of g1 or g2 of


12

PHAM DUC THOAN

order k. Now the Gauss curvature K of M is given by
K=−

2
2

|φ| (1 + |g1 |2 )(1 + |g2 |2 )

|g1 |2
|g2 |2
+
(1 + |g1 |2 )2 (1 + |g2 |2 )2

. (3.9)

We need the following lemma:
Lemma 15. ([6]) Let F ⊂ M(D) be a family of meromorphic maps
defined on the unit disc. Then F is relatively compact if and only
8|f |2
2
if the family {|∇F |e =
: F ∈ F, f is associated of F } is
(1 + |f |2 )2
uniformly bounded on compact subset of D.
We will prove the following Lemma which is similar to Lemma 14:
Lemma 16. Let x(v) : M → R4 be a sequence of conformal minimal
immersion, {g v = (g1v , g2v )} ⊂ M(M ) the sequence of their Gauss maps
and Kv the Gauss curvature of x(v) . Suppose that {g v } converges to a
meromorphic function g = (g1 , g2 ) ∈ M(M ), the sequence {Kv } is
uniformly bounded and that {x(v) (p0 )} converges for some point p0 ∈
M . Then we have the following possibilities:
(i) g is constant map, or
(ii) a subsequence {Kv } of {Kv } converges to zero, or
(iii) a subsequence {xv } of {x(v) } converges to a conformal minimal
immersion x : M → R4 whose classical Gauss map is g.
Proof. Suppose that g is non-constant and that −1 ≤ Kv in M for each

v ∈ N. Let p ∈ M be a point and (Up , z) a complex local coordinate
centered at p. Let g1v , g2v and φv be the maps given by the Weierstrass
representation of x(v) . We put
M1 = {p : gl (p) = ∞, l = 1, 2 and p isn’t a branch point of g1 or g2 }.
Take a point p ∈ M1 , we have g1 (p) = ∞ and g2 (p) = ∞. So g(p) ∈
C × C and g1 (p) = 0 or g2 (p) = 0. By choose Up and > 0 sufficiently
small, we have that g1 or g2 are holomorphic and without branch points
on Up . Thus,
2|g2 |2
2|g1 |2
2

2
,
or
≥ 2 2 , in Up .
(1 + |g1 |2 )3 (1 + |g2 |2 )
(1 + |g2 |2 )3 (1 + |g2 |2 )


AN ESTIMATE FOR THE GAUSSIAN CURVATURE OF MINIMAL...

13

As g v → g, for v large enough, we have
2|(g1v ) |2

(1 + |g1v |2 )3 (1 + |g2v |2 )

2


, or

2|(g2v ) |2

(1 + |g2v |2 )3 (1 + |g2v |2 )

2

, in Up .

Thus, in Up
|Kv | =

2
v
|φv |2 (1 + |g1 |2 )(1 + |g2v |2 )

|(g1v ) |2
|(g2v ) |2
+
(1 + |g1v |2 )2 (1 + |g2v |2 )2

2



|φv |2

.


By |Kv | ≤ 1, we have |φv | ≥ in Up for large v, and then {φv } relatively compact in M(Up ). Therefore the sequence of globally defined
holomorphic 1-forms {φv dz} is relatively compact on M1 , because it is
relatively compact in a neighborhood of each of their points. Note that
M \ M1 is a discrete set. Taking a subsequence if necessary, we can
assume that {φv dz} converges on M1 either to an nonzero holomorphic
1-form φdz or to infinity. We consider each case separately.
(a) The case {φv dz} converges to infinity on M1 . Let p is a point
branch of gl with gl (p) = ∞ (l = 1, 2). Hence in a small disk D(2 ) of
Up , gl are holomorphic and so glv are also holomorphic from Hurwitz
theorem with v large. Thus φv has not zeros on D(2 ) with v large and
converges to infinity on ∂D( ). From the maximum modulus principle,
we conclude that {φv } converges to infinity on D( ).
Now, suppose that g1 (p) = ∞ or g2 (p) = ∞, say g1 . Then in a small
disk D(2 ) of Up , g1 has neither zeros nor poles other than p. So g1v φv
is an holomorphic map without zeros for v large. As g1v φv converges
uniformly to infinity on ∂D( ), the maximum modulus principle implies
that g1v φv converges to infinity on D( ).
Therefore it follows that |φv |2 (1+|g1v |2 )(1+|g2v |2 ) converges to infinity
on Up for each p ∈ M. By hypothesis g v → g and Lemma 15, we
conclude finally, from (3.9), that {Kv } converges to the zero function
on M .
(b) The case {φv dz} converges to an holomorphic 1-form φdz on
M1 . Let p ∈ M \ M1 and D( ) be a small disc contained in Up , as
φv → φ on ∂D( ), we see that {φv } is bounded on ∂D( ), then by the


14

PHAM DUC THOAN


maximum modulus principle, it is also bounded on D( ). Thus, {φv } is
relatively compact on D( ). We conclude easily that φdz extends in a
holomorphic way to M and that the global 1-forms φv dz converges to
φdz on M . Moreover, from Hurwitz theorem, we have that the zeros of
φdz occur precisely at the poles of g1 or g2 and the order of the zero is
coincide with the order of the poles. Then g1 φdz, g2 φdz and g1 g2 φdz are
holomorphic in M , g1v φv dz → g1 φdz, g2v φv dz and g1v g2v φv dz → g1 g2 φdz.
As {x(v) (p0 )} converges for some point p0 ∈ M , we conclude, from
(??), that the sequence of harmonic maps x(v) converges uniformly on
compact subsets of M to an conformal minimal immersion x : M → R4 ,
whose Weierstrass representation is given by (φdz, g1 , g2 ). In particular,
g = (g1 , g2 ) is the classical Gauss map of x.

Now we prove Theorem 3.
Similar to proof of Theorem 1, suppose that the Theorem 3 is not
true. We will constract a non-flat complete minimal surface whose
classical Gauss map is ramified a set of distinct points. Then there
exists a sequence of (non complete) minimal surfaces x(v) : Mv → R4
and points pv ∈ Mv such that |Kv (pv )|1/2 dv (pv ) → ∞, and the classical
Gauss map g (v) = (g1v , g2v ) of x(v) whose component maps glv (l = 1, 2)
are ramified over a fixed set of ql distinct points alj in P1 (C) with
multiplicity at least mlj for each j satisfying the hypothesis (1.3), (1.4)
and (1.5) of the theorem.
The arguments of R. Osserman and M. Ru in [10, pp. 590-591] show
that we can choose the surfaces Mv satisfying condition (3.7).
Similar to the proof of Theorem 1, we obtain a subsequence of classical Gauss maps g (v) of x(v) exists- without of loss generality we assume
that g (v) = (g1v , g2v ) : D → P1 (C) × P1 (C) converges uniformly on every
compact subset of D to a map g = (g1 , g2 ) : D → P1 (C) × P1 (C). Moreover, using Lemma 16, we can show that there exists a subsequence
{x(v ) } of {x(v) } which converges to a complete minimal immersion

x : D → R4 and whose classical Gauss map is g.


AN ESTIMATE FOR THE GAUSSIAN CURVATURE OF MINIMAL...

15

Since glv : D → (P1 (C), ∆) are the orbifold morphisms, g l : D →
(P1 (C), ∆) (l = 1, 2) is the orbifold morphism or g l (D) ⊂ supp(∆l )
from Proposition 8.
If g1 and g2 are the orbifold morphisms, (i) of Theorem B show a
contradition to one of the conditions (1.3), (1.4) and (1.5) of the theorem. Thus g 1 (D) ⊂ supp(∆1 ) or g 2 (D) ⊂ supp(∆2 ). Since supp(∆l )
is discrete, it follows that g 1 or g 2 is constant, says g 2 is. By g is nonconstant, g1 is also non-constant either. But the conditions (1.3) and
(1.5) deduce that
q1

1−
j=1

1
m1j

> 3.

Therefore (ii) of Theorem B implies that g1 is constant. This is a
contradition. Thus Theorem 3 is proved.
3.3. The proof of Theorem 5.
Proof. Similar to proof of Theorem 1, suppose that the Theorem 5
is not true. We will constract a non-flat complete minimal surface
whose (generalized) Gauss map is ramified a set of hyperplanes located N −subgeneral position. The there exists a sequence of (non

complete) minimal surfaces x(v) : Mv → Rm and points pv ∈ Mv
such that |Kv (pv )|1/2 dv (pv ) → ∞, and such that the Gauss map G(v)
of x(v) is ramified over a fixed set of q hyperplanes {Hj }qj=1 located
N −subgeneral position in Pm−1 (C) with multiplicity at least mj , for
each j.
The arguments of R. Osserman and M. Ru in [10, pp. 590-591] show
that we can choose the surfaces Mv satisfying (3.7). In addition, since
assumption
q

1−
j=1

1
mj

>q−

q − (2N − n + 1)
n

and by Propositions 11, 12, we deduce that Mv can not conformally
equivalent to C.


16

PHAM DUC THOAN

By Proposition 13 and using the same arguments of Theorem 1, we

can show that there exists a subsequence {x(v ) } of {x(v) } which converges to a complete minimal immersion x : D → Rm and whose Gauss
map is G. Since G(v) : D → (Pn (C), ∆) are the orbifold morphisms
then G : D → (Pn (C), ∆) is the orbifold morphism or G(D) ⊂ supp(∆)
from Proposition 8.
If the first case is happen, we suppose that G is k−nondegenerate
(k ≤ n), so G(D) ⊂ Pk (C). Then Hj ∩ Pk (C) are q hyperplanes located
N −subgeneral position in Pk (C). By a slight refinement of Theorem C,
we have
q

k · deg(∆) − kq + q =

1−
j=1

k
mj

k
≤ (k + 1)(N − ) + N + 1
2

1 k q − 2N − 1
− −
.
2 2
k
By assumption of the Theorem 5, we have
⇒ deg(∆) ≤ q + N −


deg(∆) > q + N −

1 n q − 2N − 1
− −
.
2 2
n

Therefore
(n − k)

q − 2N − 1 1

kn
2

< 0.

This implies
k < n and

2(q − 2N − 1)
< 1.
kn

Note that we have 2(q − 2N − 1) > 2nN − n2 − n ≥ n(n − 1) from the
assumption of the Theorem. Thus, n − 1 < k < n. This a contradition,
so the second case has must be happen. Using the same arguments of
the proof in Proposition 12, there exists a partition of {1, 2, · · · , q} =
I ∪ J such that for all j ∈ J, LI ⊂ Hj and G is an orbifold morphism

from D to (LI , ∆ ) where LI = ∩i∈I Hi and ∆ = j∈J 1 − m1j (Hj ∩
LI ). Assume that card(I) = l and rank{Hi : i ∈ I} = k ≤ l, we have
dim LI = n−k. Then q−l hyperplanes Hj ∩LI are in (N −l)−subgeneral
position in LI and k ≤ l ≤ N − (n − k). We consider G(D) ⊂ Pt (C)


AN ESTIMATE FOR THE GAUSSIAN CURVATURE OF MINIMAL...

17

with t ≤ n − k ≤ N − l. Again, by a slight refinement of Theorem C,
we have
1−
j∈J

t
mj

t
≤ (t + 1)(N − l − ) + N − l + 1.
2

Deduce
deg(∆) ≤

1−
j∈J

≤q+N −


1
mj

+l

1 t q − 2N − 1 l(1 + t)
− −

.
2 2
t
t

Therefore
2(q − 2N − 1)

l(1 + t)
n−t
− (n − t) +
< 0.
tn
t

By 2(q − 2N − 1) > n(n − 1), we have
(n − t)

n − 1 − t l(1 + t)
+
< 0.
t

t

This is a contradition. Thus, Theorem 5 is proved.
Acknowledgements. This work was completed during a stay of
the first author at the Vietnam Institute for Advanced Study in Mathematics (VIASM).
References
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18

PHAM DUC THOAN

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Pham Duc Thoan Department of Information Technology
National University of Civil Engineering
55 Giai Phong str., Hanoi, Vietnam
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