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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.
NON-INTEGRATED DEFECT RELATIONS FOR THE GAUSS 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 relations in the sense of Fujimoto for the Gauss maps of complete minimal surfaces with finite total curvature in R3 , R4 . 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. 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 S 2 . 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 2 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 improvements 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 )1/2 and, for each α = (a0 : a1 ) ∈ P1 (C) with |a0 |2 + |a1 |2 = 1, we define Fα := a1 f0 − a0 f1 . Definition 1. We define the non-integrated defect (or the S−defect) of α for f by δfS (α) := 1 − inf{η ≥ 0; η satisfies condition (∗)S }. Here, condition (∗)S means that there exists a [−∞, ∞)−valued continuous subhamornic function u (≡ −∞) on M satisfying the following conditions: (C1) eu ≤ ||f ||η , (C2) for each ξ ∈ f −1 (α), there exists the limit lim(u(z) − log |z − ξ|) ∈ [−∞, ∞), z→ξ where z is holomorphic local coordinate around ξ. NON-INTEGRATED DEFECT RELATIONS 3 Definition 2. We define the H−defect of α for f by δfH (α) := 1 − inf{η ≥ 0; η satisfies condition (∗)H }. Here, condition (∗)H means that there exists a [−∞, ∞)−valued continuous 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 ; Fα has no zero of order less than m}. 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 non-intergrated defect by showing a relation between the nonintergrated defect and the defect (as in Nevanlinna theory) of a nonconstant holomorphic map of ∆R into P1 (C). We now recall the following. δfS (α)the 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 1 least m, then δfS (α) ≥ δfH (α) ≥ δfO (α) ≥ 1 − . In particular, if m f −1 (α) = ∅, then δfO (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 4 PHAM HOANG HA AND NGUYEN VAN TRAO Gauss map. For arbitrary q distinct points α1 , ..., αq in P1 (C), then q δgS (αj ) ≤ 4. j=1 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 = (g 1 , g 2 ) (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 = (g 1 , g 2 ) is the Gauss map of M. Let α11 , ..., α1q1 , α21 , ..., α2q2 be q1 + q2 (q1 , q2 > 2) distinct points in P1 (C). (i) In the case g l ≡ constant (l = 1, 2), then q1 q2 S 1j S 2j j=1 δg 1 (α ) ≤ 2, or j=1 δg 2 (α ) ≤ 2, or 1 q1 S 1j j=1 δg 1 (α ) −2 + 1 q2 S 2j j=1 δg 2 (α ) 2 −2 ≥ 1. (ii) In the case where one of g 1 and g is constant, say g 2 ≡ constant, we have the following q1 δgS1 (α1j ) ≤ 3. j=1 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 2 improved 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 , NON-INTEGRATED DEFECT RELATIONS 5 namely, a metric on M with isolated singularities which is locally written 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 := −ddc log 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) := M 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 biholomorphic with the unit disc in C, then M has the complete conformal metric with constant curvature −1 which is called the Poincar´e metric 2 of M and denoted by dσM . 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 ) := f0 f1 − f1 f0 . Let αj = (aj0 : aj1 )(1 ≤ j ≤ q) be q distinct points in P1 (C), with |aj0 |2 + |aj1 |2 = 1, we define Fj := aj1 f0 − aj0 f1 . We now consider [−∞, ∞)−valued continuous subhamornic functions 6 PHAM HOANG HA AND NGUYEN VAN TRAO uj (≡ −∞) on M and nonnegative numbers ηj (1 ≤ j ≤ q) satisfying the conditions: (D0) γ := q − 2 − qj=1 ηj > 0, (D1) euj ≤ ||f ||ηj for j = 1, ..., q, (D2) for each ξ ∈ f −1 (αj ) (1 ≤ j ≤ q), there exists the limit lim(uj (z) − log |z − ξ|) ∈ [−∞, ∞). 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 C||f ||2q−4 |W (f0 , f1 )|2 ≥ q . Πj=1 |Fj |2 log2 (µ||f ||2 /|Fj |2 ) ||f || q Πj=1 log(µ||f ||2 /|Fj |2 ) Lemma 4. (see [8]) There exist positive constants C and µ(> 1) depending only on αj and ηj (1 ≤ j ≤ q) which satisfy that if we set q C||f ||γ e j=1 uj |W (f0 , f1 )| v := q Πj=1 |Fj | log(µ||f ||2 /|Fj |2 ) on M − A and v := 0 on M ∩ A (where A := {z ∈ M ; Πqj=1 Fj = 0}), then v is continuous on M and satisfies the condition ∆ log v ≥ v 2 in the sense of distribution. Set Ωf = ddc log ||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 q ||f ||γ e j=1 uj |W (f0 , f1 )| λ := q Πj=1 |Fj | log(µ||f ||2 /|Fj |2 ) on M − A, where A := {z ∈ M ; Πqj=1 Fj = 0}, then ddc log λ2 ≥ (γ − )Ωf . Proof. We have q c 2 c dd log λ = (γ − )Ωf + dd log ≥ (γ − )Ωf + ddc log ||f || e j=1 uj Πqj=1 log(µ||f ||2 /|Fj |2 ) Πqj=1 ||f || . log(µ||f ||2 /|Fj |2 ) NON-INTEGRATED DEFECT RELATIONS 7 Using Lemma 4, we get ddc log 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 φ3 √ , g= 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/g02 . Now, for given q distinct points α1 , ..., αq in P1 (C) we assume that q δgS (αj ) > 4. (4.1) j=1 From (4.1), by Definition 1, there exist constants ηj ≥ 0(1 ≤ j ≤ q) such that q γ =q−2− ηj > 2 j=1 and continuous subhamornic functions uj (1 ≤ j ≤ q) on M satisfying conditions (D1) and (D2). We set a new metric q 2 dτ := |h|||g||γ e j=1 uj |W (g0 , g1 )| Πqj=1 |Gj | log(µ||g||2 /|Gj |2 ) 2 |dz|2 = λ2 |dz|2 , where Gj = aj1 g0 − aj0 g1 . It is easily to see that dτ 2 is well-defined on M. By Lemma 4, we see that dτ 2 is continuous and has strictly negative curvature on M. Now, because M has the continuous pseudo-metric dτ 2 which has 8 PHAM HOANG HA AND NGUYEN VAN TRAO strictly negative curvature on M and there is no continuous pseudometric with strictly negative curvature on a Riemann surface whose universal covering surface is biholomorphic to C, we give that the universal 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 2 , dτ 2 ≤ C0 dσM 2 where dσM denotes the Poincar´e metric on M . Now, for each al , we take a neighborhood Ul of al which is biholomorphic to ∆∗ = {z; 0 < |z| < 1} , where z(al ) = 0. The Poincar´e metric on domain ∆∗ is given by 2 dσ∆ ∗ = 4|dz|2 . |z|2 log2 |z|2 By using the distance decreasing property of the Poincar´e metric, we have |dz|2 2 dτ ≤ Cl 2 2 2 |z| log |z| with some Cl > 0 . This implies that, for a neighborhood Ul∗ of al which is relatively compact in Ul , we have Ωdτ 2 < +∞. Ul∗ Since M is compact, we have Ωdτ 2 ≤ M Ωdτ 2 + M −∪l Ul∗ Ωdτ 2 < +∞. (4.2) Ul∗ l 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 ddc log λ2 ≥ (γ − )Ωg by Lemma 5, we can find a subharmonic function v such that λ2 |dz|2 = ev ||g||2(γ− ) |ω|2 2 = ev+(γ− −2) log ||g|| ||g||4 |ω|2 = ew ds2 . NON-INTEGRATED DEFECT RELATIONS 9 So dτ 2 = ew ds2 , where w is a subharmonic function. Here, we can apply the result of Yau in [21] to see ew Ωds2 = +∞, M 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 minimal 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 biholomorphic to P1 (C) × P1 (C). By suitable identifications we may regard g as a pair of meromorphic functions g = (g 1 , g 2 ) on M. Set φi := ∂xi /dz for i = 1, ..., 4. Then, g 1 and g 2 are given by √ √ φ3 + −1φ4 2 −φ3 + −1φ4 1 √ √ g = , g = φ1 − −1φ2 φ1 − −1φ2 and the metric on M induced from R4 is given by ds2 = |φ|2 (1 + |g 1 |2 )(1 + |g 2 |2 )|dz|2 , (see [11]), √ where φ := φ1 − −1φ2 . Take reduced representations g l = (g0l : g1l ) on M and set ||g l || = (|g0l |2 + |g1l |2 )1/2 for l = 1, 2. Then we can rewrite ds2 = |h|2 ||g 1 ||2 ||g 2 ||2 |dz|2 (4.1), where h := φ/(g01 g02 ). We firstly study the case g l ≡ constant, for l = 1, 2. Assume that 10 PHAM HOANG HA AND NGUYEN VAN TRAO q1 S 1j j=1 δg 1 (α ) > 2, q2 S 2j j=1 δg 2 (α ) 1 q1 S 1j j=1 δg 1 (α ) > 2, and 1 + −2 q2 S 2j j=1 δg 2 (α ) −2 < 1. (5.3) From (5.3), by Definition 1, there exist constants ηlj ≥ 0(1 ≤ j ≤ q; l = 1, 2) such that q γl = ql − 2 − ηlj > 0, j=1 1 1 + < 1, γ1 γ2 and continuous subhamornic functions ulj (1 ≤ j ≤ q; l = 1, 2) on M satisfying conditions (D1) and (D2). Choose δ0 (> 0) such that γl − ql δ0 > 0 for all l = 1, 2, and 1 1 + = 1. γ1 − q1 δ0 γ2 − q2 δ0 If we set pl := 1/(γl − ql δ0 ), (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 q dτ22 := Πl=1,2 |h|||g l ||γl e j=1 uj |W (g0l , g1l )| Πqj=1 |Glj | log(µ||g l ||2 /|Glj |2 ) 2pl |dz|2 = Πl=1,2 λ2l |dz|2 , lj l l where Glj := alj 0 g1 − a1 g0 (l = 1, 2). It is also easy to see that dτ22 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 M Ωdτ22 ≤ M −∪l Ul∗ Ωdτ22 + l Ul∗ Ωdτ22 < +∞. (5.5) NON-INTEGRATED DEFECT RELATIONS 11 On the other hand, we now take a nowhere zero holomorphic form ω on M. Since ddc log λ2l ≥ pl (γl − l )Ωgl (l = 1, 2) by Lemma 5, we can find two subharmonic functions v1 , v2 such that Πl=1,2 λ2l |dz|2 = (Πl=1,2 evl ||g l ||2pl (γl − l ) )|ω|2 l 2 = (Πl=1,2 evl +(pl (γl − l )−1) log ||g || ||g l ||2 )|ω|2 = ew ds2 . So dτ22 = ew ds2 , where w is a subharmonic function by (5.4). Here, we apply again the result of Yau in [21] to get ew Ωds2 = +∞, M because of the minimality of M with respect to the metric ds2 . This contradicts the assertion (5.5). The proof of Theorem 2 is completed. We finally consider the case where g 2 ≡ constant and g 1 ≡ constant. 1 Suppose that qj=1 δgS1 (α1j ) > 3. By definition, there exist constants η1j ≥ 0(1 ≤ j ≤ q) such that q1 γ3 = q1 − 2 − η1j > 1 j=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 q1 dτ32 := |h|||g 1 ||γ3 e j=1 u1j |W (g01 , g11 )| 1 Πqj=1 |G1j | log(µ||g 1 ||2 /|G1j |2 ) 2 |dz|2 = λ23 |dz|2 . By exactly the same arguments as in the proof of Theorem 1, we get Theorem 2(ii). 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. 12 PHAM HOANG HA AND NGUYEN VAN TRAO References [1] L. V. Ahlfors, An extension of Schwarz’s lemma, Trans. Amer. Math. Soc., 43 (1938), 359-364. [2] C. C. Chen, On the image of the generalized Gauss map of a complete minimal surface in R4 , Pacific J. Math. 102 (1982), 9-14. [3] G. Dethloff and P. H. Ha, Ramification of the Gauss map of complete minimal surfaces in R3 and R4 on annular ends, Ann. Fac. Sci. Toulouse Math., 23 (2014), no. 4, 829-846. [4] R. S. Earp and M. Rosenberg, On values of the Gauss maps of complete minimal surfaces in R3 , Comment. Math. Helv. 63 (1988), 579-586. [5] Y. Fang, On the Gauss map of complete minimal surfaces with finite total curvature, Indiana Univ. Math. J. 42 (1993),1389-1411. [6] H. Fujimoto, Value distribution of the Gauss maps of complete minimal surfaces in Rm , J. Math. Soc. Japan, 35 (1983),no. 4, 663-681. [7] H. Fujimoto, On the number of exceptional values of the Gauss maps of minimal surfaces, J. Math. Soc. Japan, 40 (1988), 235-247. [8] H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces, J. Differential Geom., 29 (1989), 245-262. [9] H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces III, Nagoya Math. J., 124 (1991), 13-40. [10] H. Fujimoto, Unicity theorems for the Gauss maps of complete minimal surfaces II, Kodai Math. J, 16 (1993), 335-354. [11] H. Fujimoto, Value Distribution Theory of the Gauss map of Minimal Surfaces in Rm , Aspect of Math., Vol. E21, Vieweg, Wiesbaden (1993). [12] P. H. Ha, An estimate for the Gaussian curvature of minimal surfaces in Rm whose Gauss map is ramified over a set of hyperplanes, Differential Geom. Appl. 32 (2014), 130-138. [13] P. H. Ha and N. H. Trang, Modified defect relations of the Gauss map of complete minimal surfaces on annular ends, arXiv:1408.3075. [14] Y. Kawakami, The Gauss map of pseudo-algebraic minimal surfaces in R4 , Math. Nachr. 282 (2009), 211-218. [15] Y. Kawakami, R. Kobayashi, and R. Miyaoka, The Gauss map of pseudoalgebraic minimal surfaces, Forum Math., 20 (2008), 1055-1069. [16] X. Mo and R. Osserman, On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto’s theorem, J. Differential Geom. 31 (1990), 343-355. [17] R. Osserman, On complete minimal surfaces, Arch. Rational Mech. Anal., 13 (1963), 392-404. NON-INTEGRATED DEFECT RELATIONS 13 [18] R. Osserman, Global properties of minimal surfaces in E 3 and E n , Ann. of Math., 80 (1964), 340-364. [19] M. Ru, Gauss map of minimal surfaces with ramification, Trans. Amer. Math. Soc., 339 (1993), 751-764. [20] F. Xavier, The Gauss map of a complete non-flat minimal surface cannot omit 7 points of the sphere, Ann. of Math., 113 (1981), 211-214; Erratum: Ann. of Math., 115 (1982), 667. [21] S. T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 659-670. Department of Mathematics, Hanoi National University of Education, 136 XuanThuy str., Hanoi, Vietnam E-mail address: ha.ph@hnue.edu.vn, traonv@hnue.edu.vn [...]... distribution of the Gauss maps of complete minimal surfaces in Rm , J Math Soc Japan, 35 (1983),no 4, 663-681 [7] H Fujimoto, On the number of exceptional values of the Gauss maps of minimal surfaces, J Math Soc Japan, 40 (1988), 235-247 [8] H Fujimoto, Modified defect relations for the Gauss map of minimal surfaces, J Differential Geom., 29 (1989), 245-262 [9] H Fujimoto, Modified defect relations for the Gauss. .. Gauss map of minimal surfaces III, Nagoya Math J., 124 (1991), 13-40 [10] H Fujimoto, Unicity theorems for the Gauss maps of complete minimal surfaces II, Kodai Math J, 16 (1993), 335-354 [11] H Fujimoto, Value Distribution Theory of the Gauss map of Minimal Surfaces in Rm , Aspect of Math., Vol E21, Vieweg, Wiesbaden (1993) [12] P H Ha, An estimate for the Gaussian curvature of minimal surfaces in... (1982), 9-14 [3] G Dethloff and P H Ha, Ramification of the Gauss map of complete minimal surfaces in R3 and R4 on annular ends, Ann Fac Sci Toulouse Math., 23 (2014), no 4, 829-846 [4] R S Earp and M Rosenberg, On values of the Gauss maps of complete minimal surfaces in R3 , Comment Math Helv 63 (1988), 579-586 [5] Y Fang, On the Gauss map of complete minimal surfaces with finite total curvature, Indiana... pseudoalgebraic minimal surfaces, Forum Math., 20 (2008), 1055-1069 [16] X Mo and R Osserman, On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto’s theorem, J Differential Geom 31 (1990), 343-355 [17] R Osserman, On complete minimal surfaces, Arch Rational Mech Anal., 13 (1963), 392-404 NON- INTEGRATED DEFECT RELATIONS 13 [18] R Osserman, Global properties of minimal surfaces. .. Ann of Math., 80 (1964), 340-364 [19] M Ru, Gauss map of minimal surfaces with ramification, Trans Amer Math Soc., 339 (1993), 751-764 [20] F Xavier, The Gauss map of a complete non- flat minimal surface cannot omit 7 points of the sphere, Ann of Math., 113 (1981), 211-214; Erratum: Ann of Math., 115 (1982), 667 [21] S T Yau, Some function-theoretic properties of complete Riemannian manifolds and their... whose Gauss map is ramified over a set of hyperplanes, Differential Geom Appl 32 (2014), 130-138 [13] P H Ha and N H Trang, Modified defect relations of the Gauss map of complete minimal surfaces on annular ends, arXiv:1408.3075 [14] Y Kawakami, The Gauss map of pseudo-algebraic minimal surfaces in R4 , Math Nachr 282 (2009), 211-218 [15] Y Kawakami, R Kobayashi, and R Miyaoka, The Gauss map of pseudoalgebraic... subharmonic function by (5.4) Here, we apply again the result of Yau in [21] to get ew Ωds2 = +∞, M because of the minimality of M with respect to the metric ds2 This contradicts the assertion (5.5) The proof of Theorem 2 is completed We finally consider the case where g 2 ≡ constant and g 1 ≡ constant 1 Suppose that qj=1 δgS1 (α1j ) > 3 By definition, there exist constants η1j ≥ 0(1 ≤ j ≤ q) such that... (D2) Choose the positive real number 3 > 0 such that γ3 − 3 > 1 Set q1 dτ32 := |h|||g 1 ||γ3 e j=1 u1j |W (g01 , g11 )| 1 Πqj=1 |G1j | log(µ||g 1 ||2 /|G1j |2 ) 2 |dz|2 = λ23 |dz|2 By exactly the same arguments as in the proof of Theorem 1, we get Theorem 2(ii) Acknowledgements This work was completed during a stay of the first named author at the Vietnam Institute for Advanced Study in Mathematics... 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 12 PHAM HOANG HA AND NGUYEN VAN TRAO References [1] L V Ahlfors, An extension of Schwarz’s lemma, Trans Amer Math Soc., 43 (1938), 359-364 [2] C C Chen, On the image of the generalized Gauss map of a complete minimal surface in R4 , Pacific.. .NON- INTEGRATED DEFECT RELATIONS 11 On the other hand, we now take a nowhere zero holomorphic form ω on M Since ddc log λ2l ≥ pl (γl − l )Ωgl (l = 1, 2) by Lemma 5, we can find two subharmonic functions v1 , v2 such that Πl=1,2 λ2l |dz|2 = (Πl=1,2 ... 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... 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