In this article, we give the nonintegrated defect relations in the sense of Fujimoto for the Gauss maps of complete minimal surfaces with finite total curvature in R 3 , R 4 . These are some strict improvements of previous results for the ramifications and modified defect relations of the Gauss maps of complete minimal surfaces with finite total curvature.
Trang 1GAUSS MAPS OF COMPLETE MINIMAL SURFACES
WITH FINITE TOTAL CURVATURE
PHAM HOANG HA AND NGUYEN VAN TRAO
Abstract In this article, we give the non-integrated defect
re-lations in the sense of Fujimoto for the Gauss maps of complete
minimal surfaces with finite total curvature in R 3
, R 4 These are some strict improvements of previous results for the ramifications
and modified defect relations of the Gauss maps of complete
min-imal surfaces with finite total curvature.
1 Introduction Let M be a complete non-flat minimal surface in R3 and let g be the Gauss map of M The initial result which says that g cannot omit a set of positive logarithmic capacity was obtained by Osserman [17] in
1963 In 1981, Xavier [20] proved that g can at most omit 6 points in
S2 In 1988, Earp and Rosenberg [4] proved that, if additionally M is
of finite topology and infinite total curvature, then g takes every value infinitely many times with 6 exceptions In the same year, Fujimoto [7] proved that g can omit at most 4 points, and the bound is sharp In
1990, Mo and Osserman [16] generalized Fujimoto’s method to obtain the following result which puts the connection between total curvature and Gauss map in final form: The Gauss map of a complete minimal surface in R3 assumes every value infinitely often with at most four exceptions, unless the surface have finite total curvature After that many results related to this topic were given (see [5], [19], [15], [12] and [3] for examples)
2010 Mathematics Subject Classification Primary 53A10; Secondary 53C42, 30D35, 32A22.
Key words and phrases Minimal surface, Gauss map, Defect relation.
1
Trang 22 PHAM HOANG HA AND NGUYEN VAN TRAO
On the other hand, Fujimoto [8], [9] introduced the modified defects for the Gauss maps of complete minimal surfaces He then gave the im-provements of the above-mentioned results Recently, the first named author and Trang [13] also proved the modified defect relations of the Gauss maps of complete minimal surfaces in R3 and R4 on annular ends which are similar to the ones obtained by Fujimoto in [8] In this article, we intend to study the non-integrated defect relations for the Gauss maps of complete minimal surfaces with finite total curvature These are the strict improvements of all previous results of Fujimoto on the modified defect relations for the Gauss maps of complete minimal surfaces in R3 and R4 with finite total curvature Thus, they are the improvements of previous results on ramifications for the Gauss maps
of complete minimal surfaces with finite total curvature in R3 and R4
2 Statements of the main results
We now recall the knowledges of modified defects in [8]
Let M be an open Riemann surface and f a nonconstant holomorphic map of M into P1(C) Assume that f has reduced representation f = (f0 : f1) Set ||f || = (|f0|2+|f1|2)1/2and, for each α = (a0 : a1) ∈ P1(C) with |a0|2+ |a1|2 = 1, we define Fα := a1f0− a0f1
Definition 1 We define the non-integrated defect (or the S−defect)
of α for f by
δSf(α) := 1 − inf{η ≥ 0; η satisfies condition (∗)S}
Here, condition (∗)S means that there exists a [−∞, ∞)−valued con-tinuous subhamornic function u (6≡ −∞) on M satisfying the following conditions:
(C1) eu ≤ ||f ||η,
(C2) for each ξ ∈ f−1(α), there exists the limit
lim
z→ξ(u(z) − log |z − ξ|) ∈ [−∞, ∞), where z is holomorphic local coordinate around ξ
Trang 3Definition 2 We define the H−defect of α for f by
δHf (α) := 1 − inf{η ≥ 0; η satisfies condition (∗)H}
Here, condition (∗)H means that there exists a [−∞, ∞)−valued con-tinuous subhamornic function u on M which is hamornic on M −f−1(α) and satisfies the conditions (C1) and (C2)
Definition 3 We define the O−defect of α for f by
δfO(α) := 1 − inf{ 1
m; Fα has no zero of order less than m}. Remark 1 We always have 0 ≤ δfO(α) ≤ δfH(α) ≤ δfS(α) ≤ 1
Moreover, Fujimoto [6, page 672] also gave the reasons why he calls
δS
f(α)the intergrated defect by showing a relation between the intergrated defect and the defect (as in Nevanlinna theory) of a non-constant holomorphic map of ∆R into P1(C)
We now recall the following
Definition 4 f is called to be ramified over a point α ∈ P1(C) with multiplicity at least m if all the zeros of the function Fα have orders at least m If the image of f omits α, we will say that f is ramified over
α with multiplicity ∞
Remark 2 If f ramified over a point α ∈ P1(C) with multiplicity at least m, then δfS(α) ≥ δHf (α) ≥ δfO(α) ≥ 1 − 1
m In particular, if
f−1(α) = ∅, then δO
f(α) = 1
In this article, we would like to show the S−defect relations for the Gauss maps of minimal surfaces with finite total curvature in R3 which are similar to the H−defect relations for the Gauss maps of minimal surfaces obtained by Fujimoto in [8] Namely, we prove the following for the first purpose
Theorem 1 Let x = (x1, x2, x3) : M → R3 be a non-flat complete minimal surface with finite total curvature and g : M → P1(C) the
Trang 44 PHAM HOANG HA AND NGUYEN VAN TRAO
Gauss map For arbitrary q distinct points α1, , αq in P1(C), then
q
X
j=1
δgS(αj) ≤ 4
Moreover, we also would like to consider a complete minimal surfaces
M immersed in R4, this special case has been investigated by various authors (see, for example [2], [8], [14], [3] and [13]) Then, the Gauss map of M may be identified with a pair of meromorphic functions
g = (g1, g2) (we also refer the readers to [11] for more details) We shall prove the following theorem for the last purpose of this article Theorem 2 Suppose that M is a complete non-flat minimal surface
in R4 with finite total curvature and g = (g1, g2) is the Gauss map of
M Let α11, , α1q 1, α21, , α2q 2 be q1+ q2 (q1, q2 > 2) distinct points in
P1(C)
(i) In the case gl6≡ constant (l = 1, 2), then
Pq 1
j=1δS
g 1(α1j) ≤ 2, or Pq 2
j=1δS
g 2(α2j) ≤ 2, or 1
Pq 1
j=1δS
g 1(α1j) − 2+
1
Pq 2
j=1δS
g 2(α2j) − 2 ≥ 1
(ii) In the case where one of g1 and g2is constant, say g2 ≡ constant,
we have the following
q 1 X
j=1
δgS1(α1j) ≤ 3
Remark 3 For the case of the Gauss maps of minimal surfaces with finite total curvature, we can show that:
i) Theorem 1 improved strictly Theorem 1.3 in [6](by reducing the number 6 to the number 4) and Theorem I in [8](by changing the H− defect relations to the S− defect relations)
ii) Theorem 2improved strictly Theorem 6.3 in [6] and Theorem III in [8] (by changing the H− defect relations to the S− defect relations)
3 Preliminaries and auxiliary lemmas
In this section, we recall some auxiliary lemmas in [8], [9]
Let M be an open Riemann surface and ds2 a pseudo-metric on M ,
Trang 5namely, a metric on M with isolated singularities which is locally writ-ten as ds2 = λ2|dz|2 in terms of a nonnegative real-value function λ with mild singularities and a holomorphic local coordinate z We define the divisor of ds2 by νds := νλ for each local expression ds2 = λ2|dz|2, which is globally well-defined on M We say that ds2 is a continuous pseudo-metric if νds ≥ 0 everywhere
Definition 5 (see [9]) We define the Ricci form of ds2 by
Ricds2 := −ddclog λ2 for each local expression ds2 = λ2|dz|2
In some cases, a (1, 1)−form Ω on M is regarded as a current on M
by defining Ω(ϕ) :=RM ϕΩ for each ϕ ∈ D, where D denotes the space
of all C∞ differentiable functions on M with compact supports Definition 6 (see [9]) We say that a continuous pseudo-metric ds2
has strictly negative curvature on M if there is a positive constant C such that
−Ricds2 ≥ C · Ωds2, where Ωds2 denotes the area form for ds2, namely,
Ωds2 := λ2(√
−1/2)dz ∧ d¯z for each local expression ds2 = λ2|dz|2
As is well-known, if the universal covering surface of M is biholo-morphic with the unit disc in C, then M has the complete conformal metric with constant curvature −1 which is called the Poincar´e metric
of M and denoted by dσ2
M Let f be a nonconstant holomorphic map of open minimal surface
M into P1(C) Take a reduced representation f = (f0 : f1) on M and define
||f || := (|f0|2+ |f1|2)1/2, W (f0, f1) := f0f10 − f1f00
Let αj = (aj0 : aj1)(1 ≤ j ≤ q) be q distinct points in P1(C),
with |aj0|2+ |aj1|2 = 1, we define Fj := aj1f0 − aj0f1
We now consider [−∞, ∞)−valued continuous subhamornic functions
Trang 66 PHAM HOANG HA AND NGUYEN VAN TRAO
uj(6≡ −∞) on M and nonnegative numbers ηj(1 ≤ j ≤ q) satisfying the conditions:
(D0) γ := q − 2 −Pq
j=1ηj > 0, (D1) eu j ≤ ||f ||η j for j = 1, , q,
(D2) for each ξ ∈ f−1(αj) (1 ≤ j ≤ q), there exists the limit
lim
z→ξ(uj(z) − log |z − ξ|) ∈ [−∞, ∞)
Lemma 3 (see [8]) For each > 0 there exist positive constants C and µ depending only on α1, · · · , αq and on respectively such that
∆ log
||f ||
Πqj=1log(µ||f ||2/|Fj|2)
≥ C||f ||
2q−4|W (f0, f1)|2
Πqj=1|Fj|2log2(µ||f ||2/|Fj|2). Lemma 4 (see [8]) There exist positive constants C and µ(> 1) de-pending only on αj and ηj(1 ≤ j ≤ q) which satisfy that if we set
v := C||f ||
γePqj=1 u j|W (f0, f1)|
Πqj=1|Fj| log(µ||f ||2/|Fj|2)
on M − A and v := 0 on M ∩ A (where A := {z ∈ M ; Πqj=1Fj = 0}), then v is continuous on M and satisfies the condition ∆ log v ≥ v2 in the sense of distribution
Set Ωf = ddclog ||f ||2 We have the following
Lemma 5 For each > 0 such that γ − > 0, there exists a real number µ(> 1) depending only on αj and ηj(1 ≤ j ≤ q) which satisfy that if we set
λ := ||f ||γePqj=1 u j|W (f0, f1)|
Πqj=1|Fj| log(µ||f ||2/|Fj|2)
on M − A, where A := {z ∈ M ; Πqj=1Fj = 0}, then
ddclog λ2 ≥ (γ − )Ωf Proof We have
ddclog λ2 = (γ − )Ωf + ddclog
||f ||ePqj=1 u j
Πqj=1log(µ||f ||2/|Fj|2)
≥ (γ − )Ωf + ddclog
||f ||
Πqj=1log(µ||f ||2/|Fj|2)
Trang 7
Using Lemma 4, we get
ddclog λ2 ≥ (γ − )Ωf
4 The proof of the Theorem 1
Proof Let x = (x1, x2, x3) : M → R3 be a non-flat complete minimal surface and g : M → P1(C) the Gauss map Set φi := ∂xi/∂z (i =
1, 2, 3) and r := φ1−√−1φ2 Then, the Gauss map g : M → P1(C) is given by
φ1−√−1φ2
, and the metric on M induced from R3 is given by
ds2 = |r|2(1 + |g|2)2|dz|2 (see [11])
Take a reduced representation g = (g0 : g1) on M and set ||g|| = (|g0|2+
|g1|2)1/2 Then we can rewrite ds2 = |h|2||g||4|dz|2, where h := r/g2
0 Now, for given q distinct points α1, , αq in P1(C) we assume that
q
X
j=1
δSg(αj) > 4 (4.1)
From (4.1), by Definition 1, there exist constants ηj ≥ 0(1 ≤ j ≤ q) such that
γ = q − 2 −
q
X
j=1
ηj > 2 and continuous subhamornic functions uj(1 ≤ j ≤ q) on M satisfying conditions (D1) and (D2)
We set a new metric
dτ2 := |h|||g||γePqj=1 u j|W (g0, g1)|
Πqj=1|Gj| log(µ||g||2/|Gj|2)
2
|dz|2 = λ2|dz|2,
where Gj = aj1g0− aj0g1
It is easily to see that dτ2 is well-defined on M By Lemma 4, we see that dτ2 is continuous and has strictly negative curvature on M Now, because M has the continuous pseudo-metric dτ2 which has
Trang 88 PHAM HOANG HA AND NGUYEN VAN TRAO
strictly negative curvature on M and there is no continuous pseudo-metric with strictly negative curvature on a Riemann surface whose universal covering surface is biholomorphic to C, we give that the uni-versal covering surface of M is biholomorphic to the unit disc By the generalized Schwarz’s lemma [1], there exists a positive constant C0 such that
dτ2 ≤ C0dσ2M, where dσ2M denotes the Poincar´e metric on M
Now, for each al, we take a neighborhood Ul of al which is biholomor-phic to ∆∗ = {z; 0 < |z| < 1} , where z(al) = 0 The Poincar´e metric
on domain ∆∗ is given by
dσ2∆∗ = 4|dz|
2
|z|2log2|z|2
By using the distance decreasing property of the Poincar´e metric, we have
dτ2 ≤ Cl |dz|
2
|z|2log2|z|2
with some Cl > 0 This implies that, for a neighborhood Ul∗ of al which is relatively compact in Ul, we have
Z
Ul∗
Ωdτ2 < +∞
Since M is compact, we have
Z
M
Ωdτ2 ≤
Z
M −∪ l Ul∗
Ωdτ2 +X
l
Z
Ul∗
Ωdτ2 < +∞ (4.2)
On the other hand, we now take a nowhere zero holomorphic form
ω on M
Choose the positive real number > 0 such that γ − > 2 Then, since
ddclog λ2 ≥ (γ −)Ωg by Lemma5, we can find a subharmonic function
v such that
λ2|dz|2 = ev||g||2(γ−)|ω|2
= ev+(γ−−2) log ||g||2||g||4|ω|2
= ewds2
Trang 9So dτ2 = ewds2, where w is a subharmonic function Here, we can apply the result of Yau in [21] to see
Z
M
ewΩds2 = +∞, because of the minimality of M with respect to the metric ds2 This contradicts the assertion (4.2) The proof of Theorem 1 is completed
5 The proof of Theorem 2
Proof Let x = (x1, x2, x3, x4) : M → R4 be a non-flat complete mini-mal surface in R4 As is well-known, the set of all oriented 2-planes in
R4 is canonically identified with the quadric
Q2(C) := {(w1 : : w4)|w12+ + w42 = 0}
in P3(C) By definition, the Gauss map g : M → Q2(C) is the map which maps each point p of M to the point of Q2(C) corresponding to the oriented tangent plane of M at p The quadric Q2(C) is biholomor-phic to P1(C) × P1(C) By suitable identifications we may regard g as
a pair of meromorphic functions g = (g1, g2) on M
Set φi := ∂xi/dz for i = 1, , 4 Then, g1 and g2 are given by
g1 = φ3 +
√
−1φ4
φ1−√−1φ2
, g2 = −φ3+√
−1φ4
φ1−√−1φ2
and the metric on M induced from R4 is given by
ds2 = |φ|2(1 + |g1|2)(1 + |g2|2)|dz|2, (see [11]),
where φ := φ1−√−1φ2
Take reduced representations gl = (gl
0 : gl
1) on M and set ||gl|| = (|gl
0|2+ |gl
1|2)1/2 for l = 1, 2 Then we can rewrite
ds2 = |h|2||g1||2||g2||2|dz|2 (4.1), where h := φ/(g1
0g2
0)
We firstly study the case gl 6≡ constant, for l = 1, 2 Assume that
Trang 1010 PHAM HOANG HA AND NGUYEN VAN TRAO
Pq 1
j=1δSg1(α1j) > 2,Pq 2
j=1δSg2(α2j) > 2, and 1
Pq 1
j=1δS
g 1(α1j) − 2 +
1
Pq 2
j=1δS
g 2(α2j) − 2 < 1. (5.3) From (5.3), by Definition1, there exist constants ηlj ≥ 0(1 ≤ j ≤ q; l =
1, 2) such that
γl = ql− 2 −
q
X
j=1
ηlj > 0, 1
γ1 +
1
γ2 < 1, and continuous subhamornic functions ulj(1 ≤ j ≤ q; l = 1, 2) on M satisfying conditions (D1) and (D2)
Choose δ0(> 0) such that γl− qlδ0 > 0 for all l = 1, 2, and
1
γ1− q1δ0 +
1
γ2− q2δ0 = 1.
If we set
pl := 1/(γl− qlδ0), (l = 1, 2),
we thus have
p1 + p2 = 1
Taking the positive real numbers l> 0(l = 1, 2) such that
pl(γl− l) > 1, (l = 1, 2) (5.4)
We now set a new metric
dτ22 := Πl=1,2 |h|||gl||γ lePqj=1 u j|W (gl
0, gl
1)|
Πqj=1|Gl
j| log(µ||gl||2/|Gl
j|2)
2pl
|dz|2 = Πl=1,2λ2l|dz|2,
where Gl
j := alj0gl
1− alj1gl
0(l = 1, 2)
It is also easy to see that dτ2
2 is well-defined on M We now use Lemma
4 to see that dτ22 is continuous and has strictly negative curvature Now repeating the proof for the case (4.2) in the proof of Theorem
1,we also have
Z
M
Ωdτ2 ≤
Z
M −∪ l U∗
Ωdτ2 +X
l
Z
U∗
Ωdτ2 < +∞ (5.5)
Trang 11On the other hand, we now take a nowhere zero holomorphic form
ω on M Since ddclog λ2l ≥ pl(γl− l)Ωgl(l = 1, 2) by Lemma5, we can find two subharmonic functions v1, v2 such that
Πl=1,2λ2l|dz|2 = (Πl=1,2evl||gl||2p l (γ l −l))|ω|2
= (Πl=1,2evl +(p l (γ l − l )−1) log ||g l || 2
||gl||2)|ω|2
= ewds2
So dτ2
2 = ewds2, where w is a subharmonic function by (5.4) Here, we apply again the result of Yau in [21] to get
Z
M
ewΩds2 = +∞, because of the minimality of M with respect to the metric ds2 This contradicts the assertion (5.5) The proof of Theorem 2is completed
We finally consider the case where g2 ≡ constant and g1 6≡ constant Suppose that Pq 1
j=1δS
g 1(α1j) > 3 By definition, there exist constants
η1j ≥ 0(1 ≤ j ≤ q) such that
γ3 = q1− 2 −
q 1 X
j=1
η1j > 1
and continuous subhamornic functions u1j(1 ≤ j ≤ q1) on M satisfying conditions (D1) and (D2) Choose the positive real number 3 > 0 such that γ3− 3 > 1
Set
dτ32 := |h|||g1||γ 3ePq1j=1 u 1j|W (g1
0, g11)|
Πq1
j=1|G1
j| log(µ||g1||2/|G1
j|2)
2
|dz|2 = λ23|dz|2
By exactly the same arguments as in the proof of Theorem 1, we get
Acknowledgements This work was completed during a stay of the first named author at the Vietnam Institute for Advanced Study
in Mathematics (VIASM) He would like to thank this institution for financial support and hospitality We are also grateful to Professor Do Duc Thai for many stimulating discussions concerning this material