AN ESTIMATE FOR THE GAUSSIAN CURVATURE OF MINIMAL SURFACES WITH RAMIFICATION

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 theoryAbstract. 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

OF MINIMAL SURFACES WITH RAMIFICATION

PHAM DUC THOAN

Abstract In this article, we give an estimate for the

Gauss-ian 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 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 dis-tribution theory, estimate curvature, orbifold.

1

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 each j and

q

P

j=1

(1 −m1

j) > 4 then M is flat, or equivalently g is constant

In the case, m = 4, H Fujimoto ([5]) poved the following theorem: Theorem B Suppose that M is a complete non-flat minimal sur-face 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 6≡ constant (l = 1, 2), if gl is ramified over alj with multiplicity at least mlj for each j (l = 1, 2) then

γ1 =

q 1

X

j=1

(1 − 1

m1j) ≤ 2, or γ2 =

q 2

X j=1

(1 − 1

m2j) ≤ 2, or 1

γ1− 2+

1

γ2− 2 ≥ 1.

(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:

γ1 =

q 1

X j=1

(1 − 1

m1j) ≤ 3.

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

in R3 and R4whose 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 X j=1

(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

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 1deduce 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, · · · , alq l} (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

γ1 =

q 1

X j=1 (1 − 1

γ2 =

q 2

X j=1

(1 − 1

1

γ1− 2 +

1

Then one has a curvature estimate i.e there exists a constant C, de-pending 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 X j=1

(1 − 1

Then one has a curvature estimate

Proof Since g2 is constant, the condition (1.4) is satisfied The condi-tion (1.6) implies the conditions (1.3) and (1.5) Then the Theorem 3

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 hyper-planes in general position in Pm−1(C) If G is ramified over Hj with multiplicity at least mj for each j and

q X j=1

(1 − k

mj) > (k + 1)(m −

k

2 − 1) + m 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

multiplicity at least mj for each j, and

q X j=1

(1 − m − 1

mj ) >

m(m + 1) 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 X j=1



1 − 1

mj



> q − q − 1

m − 1 +

m + 2

2 . Then one has a curvature estimate i.e there exists a constant C, de-pending 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

X

j=1



1 − 1

mj



> q − q − (2N − n + 1)

2N − n + 1

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), Theorem5 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 hyper-planes in N − subgeneral position in Pn(C) Then, we have

Pq i=1δ[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

X a∈P 1 (C)

δ(a) ≤ 2

Now, we recall the some results of the orbifold theory which was introduced by F Campana ([1])

Proposition 8 ([1]) Let fn: (X, ∆) → (X0, ∆0) be a sequence of orb-ifold morphism Assume that (fn), regarded as a sequence of holomor-phic maps from X to X0 converge locally uniformly to a holomorphic map f : X → X0 Then either f (X) ⊂ supp(∆0) or f is an orbifold morphism from (X, ∆) to (X0, ∆0)

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

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

P

j=1



1 −m1

j



aj with q > 2 If

deg(∆) =

q P j=1



1 − 1

mj



> 2 then (P1(C), ∆) is hyperbolically imbedded in P1(C), thus is also hyper-bolic

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 ||fn0 = +∞|| Thanks to Brody reparametrization, we obtain a sequence of orbifold morphisms

gn : D(0, rn) → (P1(C), ∆) with rn → +∞, converging to a holo-morphic 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 possi-ble The second case is not possible either by the result of Nevanlinna (Theorem 7) Thus Proposition10 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 ∆ =Pqj=11 − m1

j



Hj with q > 2N If deg(∆) =

q P j=1



1 − m1

j



> q − 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),

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

N(H,f )(r) ≥ mj

l N

[l]

(H,f )(r)

Therefore

δ[l](Hj, f ) ≥ 1 − l

mj. Then

Pq

j=1δ[l](Hj, f ) ≥Pqj=11 −ml

j



= 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

deg(∆) ≤ q − q − (2N + 1)

l − 1 ≤ q − q − (2N + 1)

That is a contradition Thus, Proposition 11is proved  Proposition 12 Let Hj be q hyperplanes located in N −subgeneral position in Pn(C) and ∆ =

q P j=1



1 −m1

j



Hj with q > 2N If

deg(∆) =

q X j=1



1 − 1

mj



> q − q − (2N − n + 1)

n then (Pn(C), ∆) is hyperbolically imbedded in Pn(C), thus is also hy-perbolic

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(∆)

The first case does not happen by Proposition11, so the second case must happen By Proposition 8, for each ∆0j = 1 − m1

j



Hj, 1 ≤

j ≤ q that either g is an orbifold morphism from C to (Pn(C), ∆) or

f (C) ⊂ supp(∆0j) Thus, there exists a partition of {1, 2 · · · , q} = I ∪ J such that for all j ∈ J , LI 6⊂ Hj and f is an orbifold morphism from

C to (LI, ∆0) where LI = ∩i∈IHi and ∆0 =P

j∈J



1 −m1

j

 (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∩LIare in (N −l)−subgeneral position in LI

The sequence gn : D(0, rn) → (Pn(C), ∆) can be seen as a se-quence of orbifold morphism gn: D(0, rn) → (Pn(C), ∆J) where ∆J = P

j∈J



1 −m1

j



Hj since ∆J ≤ ∆ Therefore, by Proposition 8, it con-verges to a map f which is either an orbifold morphism from C to (Pn(C), ∆0) or verifies f (C) ⊂ supp(∆0)

Since the condition q > 2N , we get q − l > 2(N − l) and using again Theorem 11,

deg(∆J) ≤ (q − l) − (q − l) − (2(N − l) − (n − k) + 1)

n − k for k = 1, · · · , n − 1 We have

deg(∆) = deg(∆I) + deg(∆J)

≤ l + (q − l) − (q − l) − (2(N − l) − (n − k) + 1)

n − k

≤ q − q − (2N − n + 1)

This is a contradition Thus, Proposition 12is proved  Proposition 13 ([8]) Let ω be a hermitian metric on X compact Assume that the orbifold (X, ∆) is hyperbolic and hyperbolically imbed-ded 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

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 mini-mal immersion, {gv} ⊂ M(M ) the sequence of their Gauss maps and

Kv the Gauss curvature of x(v) Suppose that {gv} converges to a mero-morphic 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 {Kv0} of {Kv} converges to zero, or

(iii) a subsequence {x(v0)} 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 com-plete 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)|1/2dv(pv) → ∞, and the classical Gauss map gv of x(v) is rami-fied over a fixed set of q distinct points aj in P1(C) and the restriction

gv 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, gv is the meromorphic function

on C, thus is holomorphic function into P1(C) which ramified over aj

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

q X j=1



1 − 1

m∗j



This implies that gv 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), ∆)

∆ = Pqj=11 −m1

j



aj is hyperbolic and hyperbolically imbedded in

P1(C) Therefore, we obtain a subsequence of classical Gauss maps gv

of x(v) exists- without of loss generality we assume that gv : 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 Lemma14and by arguments of R Osserman and M Ru in [10, pp 591-592], there exists a subsequence {x(v0)} of {x(v)} which converges to

a complete minimal immersion x : D → R3 and whose classical Gauss map is g Since gv : 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

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 P 1(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