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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
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) 4 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 AN ESTIMATE FOR THE GAUSSIAN CURVATURE OF MINIMAL... 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 6 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 [1] F. Campana, J. Winkelmann, A Brody theorem for orbifold, Manuscripta math. 128 (2009), 195-212. [2] H. Fujimoto, On the Gauss curvature of the minimal surfaces, J. Math. Soc. Japan. 44 (1992), 427-439. [3] H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces, J. Differential Geometry. 29 (1989), 245-262. [4] H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces II, J. Differential Geometry. 31 (1990), 365-385. [5] H. Fujimoto, Value Distribution Theory of the Gauss map of minimal Surfaces in Rm , Aspect of Math. E21, Vieweg, Wiesbaden, 1993. [6] A. Ros, The Gauss map of minimal Surfaces, in: Differential Geometry, Valencia, 2001, World Sci. Publ., River Edge, NJ. (2002), .235-252. 18 PHAM DUC THOAN [7] 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), 829-846. [8] P. H. Ha, An estimate for the Gauss curvature of minimal surfaces in Rm whose Gauss map is ramified over a set of hyperplanes, J. Differ. Geom. 32 (2014), 130-138. [9] R. Nevanlinna, Analytic functions, Berlin-Heidelberg-New York, Springer (1970). [10] R. Osserman and M. Ru, An estimate for the Gauss curvature on minimal surfaces in Rm whose Gauss map omits a set of hyperplanes, J. Differential Geom. 46 (1997), 578-593. [11] E. Rousseau, Hyperbolicity of geometric orbifold, Trans. Am. Math. Soc. 362 (2010), 3799-3826. [12] M. Ru, Gauss map of minimal surfaces with ramification, Trans. Amer. Math. Soc. 339 (1993), 751-764. Pham Duc Thoan Department of Information Technology National University of Civil Engineering 55 Giai Phong str., Hanoi, Vietnam Email: thoanpd@nuce.edu.vn [...]... Ramification of the Gauss map of complete minimal surfaces in R3 and R4 on annular ends, Ann Fac Sci Toulouse Math 23 (2014), 829-846 [8] P H Ha, An estimate for the Gauss curvature of minimal surfaces in Rm whose Gauss map is ramified over a set of hyperplanes, J Differ Geom 32 (2014), 130-138 [9] R Nevanlinna, Analytic functions, Berlin-Heidelberg-New York, Springer (1970) [10] R Osserman and M Ru, An estimate. .. a stay of the first author at the Vietnam Institute for Advanced Study in Mathematics (VIASM) References [1] F Campana, J Winkelmann, A Brody theorem for orbifold, Manuscripta math 128 (2009), 195-212 [2] H Fujimoto, On the Gauss curvature of the minimal surfaces, J Math Soc Japan 44 (1992), 427-439 [3] H Fujimoto, Modified defect relations for the Gauss map of minimal surfaces, J Differential Geometry.. .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. .. 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... 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... relations for the Gauss map of minimal surfaces II, J Differential Geometry 31 (1990), 365-385 [5] H Fujimoto, Value Distribution Theory of the Gauss map of minimal Surfaces in Rm , Aspect of Math E21, Vieweg, Wiesbaden, 1993 [6] A Ros, The Gauss map of minimal Surfaces, in: Differential Geometry, Valencia, 2001, World Sci Publ., River Edge, NJ (2002), 235-252 18 PHAM DUC THOAN [7] G Dethloff and P H Ha, Ramification. .. (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... (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... 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... 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 ... [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. .. 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... relatively compact in Hol(D, X), the set of all holomorphic of D into X The proof of Theorems 3.1 The proof of Theorem 10 PHAM DUC THOAN Proof We need the following Lemma of A Ros [6, Lemma 6]: Lemma