tóm tắt luận án tiếng anh vấn đề duy nhất của ánh xạ phân hình vào không gian xạ ảnh và tính rẽ nhánh của ánh xạ gauss của mặt cực tiểu đầy

´ UNIVERSITE DE BRETAGNE OCCIDENTALE And HANOI NATIONAL UNIVERSITY OF EDUCATION PHAM HOANG HA On unicity problems of meromorphic mappings of Cn into PN (C) and the ramification of the Gauss maps of complete minimal surfaces Summery of Doctoral Thesis in Mathematics Supervisors: Professor GERD DETHLOFF and Professor DO DUC THAI Hanoi, May 3, 2013 i Introduction Motivation of the thesis Unicity problems of meromorphic mappings under a conditions on the inverse images of divisors were studied firstly by R Nevanlinna in 1925 He showed that for two nonconstant meromorphic functions f and g on the complex plane C, if they have the same inverse images for five distinct values then f ≡ g In 1975, H Fujimoto generalized Nevanlinna’s results to the case of meromorphic mappings of Cn into PN (C) He showed that for two linearly nondegenerate meromorphic mappings f and g of C into PN (C), if they have the same inverse images counted with multiplicities for 3N + hyperplanes in general position in PN (C) then f ≡ g and there exists a projective linear transformation L of PN (C) onto itself such that g = L.f if they have the same inverse images counted with multiplicities for 3N + hyperplanes in general position in PN (C) After that, this problem has been studied intensively by a number of mathematicans as H Fujimoto, W Stoll, L Smiley, M Ru, G Dethloff T V Tan, D D Thai - S D Quang and so on Here we introduce the necessary notations to state the results Let f be a nonconstant meromorphic mapping of Cn into PN (C) and H a hyperplane in PN (C) Let k be a positive integer or k = ∞ Denote by ν(f,H) the map of Cn into Z whose value ν(f,H) (a) (a ∈ Cn ) is the intersection multiplicity of the image of f and H at f (a) For every z ∈ Cn , we set ν(f,H),≤k (z) = ν(f,H) (z) if ν(f,H) (z) > k, if ν(f,H) (z) ≤ k, ν(f,H),>k (z) = ν(f,H) (z) if ν(f,H) (z) > k, if ν(f,H) (z) ≤ k Take a meromorphic mapping f of Cn into PN (C) which is linearly nondegenerate over C, a positive integer d, a positive integer k or k = ∞ and q hyperplanes H1 , , Hq in PN (C) located in general position with dim{z ∈ Cn : ν(f,Hi ), k (z) > and ν(f,Hj ), k (z) > 0} ≤ n − (1 ≤ i < j ≤ q), and consider the set F(f, {Hj }q , k, d) of all meromorphic maps g : Cn → PN (C) j=1 satisfying the conditions (a) g is linearly nondegenerate over C, (b) (ν(f,Hj ),≤k , d) = (ν(g,Hj ),≤k , d) (1 ≤ j ≤ q), (c) f (z) = g(z) on q j=1 {z ∈ Cn : ν(f,Hj ),≤k (z) > 0} When k = ∞, for brevity denote F(f, {Hj }q , ∞, d) by F(f, {Hj }q , d) Denote j=1 j=1 by S the cardinality of the set S The unicity problem of meromorphic mappings means that one gives an estimate for the cardinality of the set F(f, {Hj }q , k, d) Some natural questions arise and we j=1 state the followings Question The number of hyperplanes (fixed targets) in PN (C) which are used In particular, how about q? Question How about the truncated multiplicities (d and k) ? Question Whether the fixed targets (hyperplanes) can be generalized to moving targets (moving hyperplanes) or hypersurfaces? On the question and 2, we list here some known results as Smiley Tan F(f, {Hi }3N +2 , 1) = 1, Thai-Quang i=1 [2.75N ] F(f, {Hi }i=1 F(f, {Hi }3N +1 , 1) = 1, N ≥ 2, Dethloffi=1 , 1) = for N ≥ N0 (where the number N0 can be explicitly calculated) and Chen-Yan F(f, {Hi }2N +3 , 1) = i=1 When q < 2N + 3, there are some results which were given by Tan and Quang Those results lead us to the question What can we say about the unicity theorems with truncated multiplicities in the case where q ≤ 2N + 2? The first purpose of this thesis is to study these problems Firstly, we will give a new aspect for the unicity problem with q = 2N + 2, and we also study the unicity theorems with ramification of truncations The second purpose of this thesis is to give some answers relative to the question Our results are following the results of Ru, Dethloff-Tan, Thai-Quang On the other hand, there are many interesting unicity theorems for meromorphic functions on C given by certain conditions of derivations We would like to study the unicity problems of such type in several complex variables for fixed and moving targets Parallel to the development of Nevanlinna theory, the value distribution theory of the Gauss map of minimal surfaces immersed in Rm was studied by many mathematicans as R Osserman, S S Chern, F Xavier, H Fujimoto, S J Kao, M Ru and others Let M now be a non-flat minimal surface in R3 , or more precisely, a connected oriented minimal surface in R3 By definition, the Gauss map G of M is the map which maps each point p ∈ M to the unit normal vector G(p) ∈ S of M at p Instead of G, we study the map g := π ◦ G : M → C := C ∪ {∞}(= P1 (C)) for the stereographic projection π of S onto P1 (C) By associating a holomorphic local √ coordinate z = u + −1v with each positive isothermal coordinate system (u, v), M is considered as an open Riemann surface with a conformal metric ds2 and by the assumption of minimality of M, g is a meromorphic function on M After that, we can generalize to the definition of Gauss map of minimal surfaces in Rm So there are many analogous results between the Gauss maps and meromorphic mappings of C into PN (C) One of them is the small Picard theorem In 1965, R Osserman showed that the complement of the image of the Gauss map of a nonflat complete minimal surface immersed in R3 is of logarithmic capacity zero in P1 (C) In 1981, a remarkable improvement was given by F Xavier that the Gauss map of a nonflat complete minimal surface immersed in R3 can omit at most six points in P1 (C) In 1988, H Fujimoto reduced the number six to four and this bound is sharp: In fact, we can see that the Gauss map of Scherk’s surface omits four points in P1 (C) In 1991, S J Kao showed that the Gauss map of an end of a non-flat complete minimal surface in R3 that is conformally an annulus {z|0 < 1/r < |z| < r} must also assume every value, with at most exceptions In 2007, Jin-Ru generalized Kao’s results for the case m > On the other hand, in 1993, M Ru studied the Gauss map of minimal surface in Rm with ramification That are generalizations of the above-mentioned results A natural question is that how about the Gauss map of minimal surfaces on annular ends with ramification The last purpose of this thesis is to answer to this question for the case m = 3, We refer to the work of Dethloff-Ha-Thoan for the case m > Aim of study The aim of study is to study the unicity problems for meromorphic mappings of Cn into PN (C) with fixed hyperplanes, moving hyperplanes and truncated multiplicities Besides, this thesis also studies the Gauss map of minimal surfaces in R3 , R4 on annular ends with ramification Object and scope of study As in motivation of the thesis above, the objects of the thesis are studying the unicity problems for meromorphic mappings of Cm into Pn (C) and the ramification of the Gauss map of minimal surfaces in R3 , R4 In this thesis, the main purpose is improving the recent known results Method of study In order to solve the problems of the thesis, we use the study methods and techniques of Complex Analysis, Nevanlinna theory, Riemann surfaces, Differential Geometry and we introduce some new techniques The results and significance of the thesis The thesis includes chapters In chapter 1, we study the unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets In particular, we give a new unicity theorem for the above-mentioned first purpose of this thesis After that we study the unicity theorems with ramification of truncations which is an improvement of Thai-Quang’s results At the end of this chapter we give a unicity theorem of meromorphic mappings with a conditions on derivations In chapter 2, we study the unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables sharing few moving targets In particular, we improve strongly the results of Dethloff- Tan before Beside that, we also give a unicity theorem of meromorphic mappings for moving targets with a conditions on derivations In chapter 3, we recall the Gauss map of minimal surfaces in Rm and we study the ramification of the Gauss map on annular ends in minimal surfaces in R3 , R4 Structure of the thesis The structure of this thesis includes an introduction, the references and chapters which are based on previous results These three chapters are based on four articles (two of them were published and the others are submitted) Chapter 1: Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets Chapter 2: Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables sharing small identical sets Chapter 3: Value distribution of the Gauss map of minimal surfaces on annular ends Chapter Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets The unicity theorems with truncated multiplicities of meromorphic mappings of Cn into the complex projective space PN (C) sharing a finite set of fixed hyperplanes in PN (C) has been studied intensively by H Fujimoto, L Smiley, S Ji, M Ru, D.D Thai, G Dethloff, T.V Tan, S.D Quang, Z Chen, Q Yan and others The unicity problem has grown into a huge theory We report here briefly the unicity problems with multiplicities of meromorphic mappings Theorem A.(Smiley) If q ≥ 3N + then F(f, {Hi }q , 1) = i=1 Theorem B.(Thai-Quang) If N ≥ then F(f, {Hi }3N +1 , 1) = i=1 Theorem C.(Dethloff-Tan)There exists a positive integer N0 (which can be explicitly calculated) such that F(f, {Hi }q , 1) = for N ≥ N0 and q = [2.75N ] i=1 Theorem D.(Chen-Yan) If N ≥ then F(f, {Hi }2N +3 , 1) = i=1 Theorem E.(Tan) For each mapping g ∈ F(f, {Hi }2N +2 , N + 1), there exist a i=1 constant α ∈ C and a pair (i, j) with ≤ i < j ≤ q, such that (Hi , g) (Hi , f ) =α (Hj , f ) (Hj , g) Theorem F (Quang) Let f1 and f2 be two linearly nondegenerate meromorphic mappings of Cn into PN (C) (N ≥ 2) and let H1 , , H2N +2 be hyperplanes in PN (C) located in general position such that dim{z ∈ Cn : ν(f1 ,Hi ) (z) > and ν(f1 ,Hj ) (z) > 0} ≤ n − for every ≤ i < j ≤ 2N + Assume that the following conditions are satisfied (a) min{ν(f1 ,Hj ),≤N , 1} = min{ν(f2 ,Hj ),≤N , 1} (1 ≤ j ≤ 2N + 2), (b) f1 (z) = f2 (z) on 2N +2 j=1 {z ∈ Cn : ν(f1 ,Hj ) (z) > 0}, (c) min{ν(f1 ,Hj ),≥N , 1} = min{ν(f2 ,Hj ),≥N , 1} (1 ≤ j ≤ 2N + 2), Then f1 ≡ f2 F(f, {Hi }2N +2 , 1) ≤ i=1 Theorem G (Quang) If N ≥ then In the first part of this chapter, we would like to study the unicity theorems for the case q ≤ 2N + In particular, we shall prove Theorem 1.2 (Ha-Quang) which gives a new aspect of them in the first part of this chapter In 2006, Thai-Quang showed that Theorem H (Thai-Quang) (a) If N = 1, then F(f, {Hi }3N +1 , k, 2) ≤ for i=1 k ≥ 15 N −1 24 (c) If N ≥ 4, then F(f, {Hi }3N , k, 2) ≤ for k > 3N + + i=1 N −3 60 3N (d) If N ≥ 6, then F(f, {Hi }i=1−1 , k, 2) ≤ for k > 3N + 11 + N −5 The second part of this chapter studies the unicity problems of meromorphic map(b) If N ≥ 2, then F(f, {Hi }3N +1 , k, 2) ≤ for k ≥ 3N + + i=1 ping with ramification of truncations We are going to improve Theorem G by Theorem 1.3 (Ha) In particular, we ramify truncations ki for each hyperplanes Hi (1 ≤ i ≤ q), and we then give its corollaries As far as we know, there are many interesting unicity theorems for meromorphic functions on C given by the certain conditions of derivations We will give a unicity theorem of such type in several complex variables for fixed targets That is a unicity theorem with truncated multiplicities in the case where N + ≤ q < 2N + We will prove Theorem 1.4 (Ha-Quang) in the last part of this chapter 1.1 Basic notions and auxiliary results from Nevanlinna theory In this section, we recall some notions and auxiliary results from Nevanlinna theory We introduce the definition of the divisors on Cn , the counting functions of the divisors, the characteristic function, the proximity function After that, we recall some results which play essential roles in Nevanlinna theory as the first main theorem, the second main theorem for hyperplanes, the logarithmic derivative lemma We also introduce some lemmas or propositions which are used for the proof of main results in this chapter 1.1.19 Lemma Suppose that Φα (F0 , , FM ) ≡ with |α| ≤ M (M − 1) If ν ([d]) := {νF0 ,≤k0 , d} = {νF1 ,≤k1 , d} = · · · = {νFM ,≤kM , d} for some d ≥ |α|, then νΦα (z0 ) ≥ {ν ([d]) (z0 ), d−|α|} for every z0 ∈ {z : νF0 ,≤k0 (z) > 0} \ A, where A is an analytic subset of codimension ≥ 1.1.20 Lemma Suppose that the assumptions in Lemma 1.1.19 are satisfied If F0 = · · · = FM ≡ 0, ∞ on an analytic subset H of pure dimension n−1, then νΦα (z0 ) ≥ M, ∀ z0 ∈ H Let f : Cn → PN (C) be a linearly nondegenerate meromorphic 1.1.21 Lemma mapping Let H1 , H2 , , Hq be q hyperplanes in PN (C) located in general position Assume that kj ≥ N − (1 ≤ j ≤ q) Then q q−N −1− j=1 N T (r, f ) ≤ kj + q 1− j=1 N (N ) N(f,Hj ),≤kj (r) + o(T (r, f )) kj + 1.1.22 Lemma Assume that there exists Φα = Φα (Fcj0 , , Fcj0 M ) ≡ for some M (M − 1) c ∈ C, |α| ≤ , ≥ |α| and the assumptions in Lemma 1.1.19 are satisfied Then, for each ≤ i ≤ M, the following holds: M (2−|α|) N(f i ,Hj ),≤kij (r)+M 0 (1) N(f i ,Hj ),≤kij (r) ≤ N (r, ν ) ≤ T (r)+ j=j0 l=0 ( M (M −1) ) (r)+o(T (r)) ),>klj0 N(f l ,H2j Φα 1.2 A unicity theorem with truncated multiplicities of meromorphic mappings in several complex variables sharing 2N + hyperplanes Theorem 1.2 (Ha-Quang) Let f and f be two linearly nondegenerate meromorphic mappings of Cn into PN (C) (N ≥ 2) and let H1 , , H2N +2 be hyperplanes in PN (C) located in general position such that dim{z ∈ Cn : ν(f ,Hi ) (z) > and ν(f ,Hj ) (z) > 0} ≤ n − for every ≤ i < j ≤ 2N + Let m be a positive integer such that m> 2N + N +1 2N + N +1 −2 Assume that the following conditions are satisfied (a) min{ν(f ,Hj ) , 1} = min{ν(f ,Hj ) , 1} (1 ≤ j ≤ 2N + 2), 2N +2 j=1 {z (b) f (z) = f (z) on ∈ Cn : ν(f ,Hj ) (z) > 0}, (c) min{ν(f ,Hj ) (z), ν(f ,Hj ) (z)} > N or ν(f ,Hj ) (z) ≡ ν(f ,Hj ) (z) (mod m) for all z ∈ (f , Hj )−1 (0) (1 ≤ j ≤ 2N + 2) Then f ≡ f 1.3 A unicity theorem for meromorphic mapping sharing few fixed targets with ramification of truncations Theorem 1.3 (Ha) Let f , f , f : Cn −→ PN (C) be three meromorphic mappings and let {Hi }q be hyperplanes in general position Let d, k, k1i , k2i , k3i be the integers i=1 with ≤ k1i , k2i , k3i ≤ ∞ (1 ≤ i ≤ q) We set M = max{kji }, m = min{kji } (1 ≤ j ≤ 3, ≤ i ≤ q), k = max{ {i ∈ {1, · · · , q} | kji = m} | ≤ j ≤ 3} Define by d = if M = m and d = min{kji − m > | ≤ j ≤ 3; ≤ i ≤ q} if M = m Assume that the following conditions are satisfied (i) dim{z ∈ Cn : ν(f j ,Hi ),≤kji > and ν(f j ,Hl ),≤kjl > 0} ≤ n − (1 ≤ j ≤ 3; ≤ i < l ≤ q) (ii) min(ν(f j ,Hi ),≤kji , 2) = (ν(f t ,Hi ),≤kti , 2) (iii) f ≡ f j on q α=1 {z (1 ≤ j < t ≤ 3; ≤ i ≤ q) ∈ Cn : ν(f ,Hα ),≤k1α (z) > 0} (1 ≤ j ≤ 3) Then f ≡ f or f ≡ f or f ≡ f if one of the following conditions is satisfied 16 1) N ≥ 2, 3N − ≤ q ≤ 3N + 1, m > 3N + + and 3(N − 1) 2N k 2N (q − k) 3N + N + − (2q − 5N − 3) > m+1 m+d+1 M +1 6(4 − k) 6k 24 − 6k 12 3(2k + 1) + + + and ν(f j ,Hl ),≤kl > 0} ≤ n − ( ≤ i < l ≤ 3N + 1) (ii) min(ν(f j ,Hi ),≤ki , 2) = (ν(f t ,Hi ),≤ki , 2) j (iii) f ≡ f on 3N +1 α=1 {z (1 ≤ j < t ≤ 3; ≤ i ≤ 3N + 1) n ∈ C : ν(f ,Hα ),≤kα (z) > 0} (1 ≤ j ≤ 3) Then f ≡ f or f ≡ f or f ≡ f if one of the following conditions is satisfied 14 a) N ≥ 2, kj = k1 + for every ≤ j ≤ 3N + and k1 > 3N + + 3(N − 1) 16 b) N ≥ 2, kj = k1 + for every ≤ j ≤ 3N + and k1 > 3N + + 3(N − 1) *) When k = and M = m + d, by using the proof for the case of Theorem 1.3, we have the following Corollary Let f , f , f : Cn −→ P1 (C) be three meromorphic functions and let {Hi }4 be hyperplanes in general position Let ki (1 ≤ i ≤ 4) be the positive integers i=1 satisfying the following conditions (i) dim{z ∈ Cn : ν(f j ,Hi ),≤ki > and ν(f j ,Hl ),≤kl > 0} ≤ n − ( ≤ j ≤ 3; ≤ i < l ≤ 4) (ii) min(ν(f j ,Hi ),≤ki , 2) = (ν(f t ,Hi ),≤ki , 2) (iii) f ≡ f j on α=1 {z (1 ≤ j < t ≤ 3; ≤ i ≤ 4) ∈ Cn : ν(f ,Hα ),≤kα (z) > 0} (1 ≤ j ≤ 3) Assume that one of the following conditions is satisfied a) k1 = 9, k2 = k3 = k4 = 66 b) k1 = 10, k2 = k3 = k4 = 36 10 c) k1 = 11, k2 = k3 = k4 = 26 d) k1 = 12, k2 = k3 = k4 = 21 e) k1 = 13, k2 = k3 = k4 = 18 f ) k1 = 14, k2 = k3 = k4 = 16 Then f ≡ f or f ≡ f or f ≡ f 1.4 A unicity theorem for meromorphic mapping sharing few fixed targets with a conditions on derivations Take a meromorphic mapping f of Cn into PN (C) which is linearly nondegenerate over C, a positive integer d, a positive integer k or k = ∞ and q hyperplanes H1 , , Hq in PN (C) located in general position with dim{z ∈ Cn : ν(f,Hi ) (z) > and ν(f,Hj ) (z) > 0} ≤ n − (1 ≤ i < j ≤ q), and consider the set G(f, {Hj }q , k, d) of all meromorphic maps g : Cn → PN (C) j=1 satisfying the conditions (a) g is linearly nondegenerate over C, (b) min{ν(f,Hj ),≤k , d} = min{ν(g,Hj ),≤k , d} (1 ≤ j ≤ q), (c) Let f = (f0 : · · · : fN ) and g = (g0 : · · · : gN ) be reduced representations of f and g, respectively Then, for each ν(f,Hi ), k (z) j N and for each ω ∈ q i=1 {z ∈ Cn : > 0}, the following two conditions are satisfied: (i) If fj (ω) = then gj (ω) = 0, (ii) gi fi (ω) = Dα (ω) for each n-tuple α = fj gj (α1 , , αn ) of nonnegative integers with |α| = α1 + + αn d and for each |α| ∂ i = j, where Dα = α1 ∂ z1 ∂ αn zn If fj (ω)gj (ω) = then Dα Remark that the condition (c) does not depend on the choice of reduced representations The last part of this chapter proves the following Theorem 1.4 (Ha-Quang) If N ≥ and d 3dN − 2N + 2N d − 2N d2 for each k > − 2(d − 1)N + d − 2d2 11 N −1, then G(f, {Hi }3N +2−2d , k, d) = i=1 Chapter Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables sharing small identical sets The unicity theorems with truncated multiplicities of meromorphic mappings of Cn into the complex projective space PN (C) sharing a finite set of fixed (or moving) hyperplanes in PN (C) have received much attention in the last few decades, and they are related to many problems in Nevanlinna theory and hyperbolic complex analysis For moving targets and truncated multiplicites, the following results are best and due to Dethloff-Tan They proved the following Theorem of Dethloff-Tan Let f, g : Cn −→ PN (C) (N ≥ 2) be two nonconstant meromorphic mappings, and let {aj }3N +1 be ”small” (with respect to f ) meromorphic j=1 mappings of Cn into PN (C) in general position such that (f, ) ≡ 0, (g, ) ≡ (1 3N + 1) and f is linearly nondegenerate over R({aj }3N +1 ) Set M = 3N (N + j=1 i 1) 2N +2 N +1 2N +2 N +1 −1 +N (3N +4) Assume that the following conditions are satisfied (i) dim{z ∈ Cn : ν(f,ai ), (1 i N + 3, M (z) j > and ν(f,aj ), j∈D {z > 0} n−2 3N + 1) (ii) min{ν(f,ai ) , M } = min{ν(g,ai ) , M } ((1 (iii) f (z) = g(z) on M (z) ∈ Cn : ν(f,aj ), of {1, · · · , 3N + 1} with D = N + 12 i M (z) 3N + 1) > 0}, where D is an arbitrary subset Then f ≡ g We would like to emphasize here that the assumption D = N + in the abovementioned theorem is essential in their proofs It seems to us that some key techniques in their proofs could not be used for D < N + The first main purpose of the present chapter is to give a unicity theorem with truncated multiplicities of meromorphic mappings in several complex variables sharing N + moving targets In particular, we prove Theorem 2.2 (Ha-Quang-Thai) It is an improvement of the above-mentioned theorem of Dethloff-Tan In this chapter, we also would like to study the unicity problems of meromorphic mappings in several complex variables for moving targets with conditions on derivations We will prove Theorem 2.3 (Ha-Quang-Thai) in the last part of this chapter 2.1 Preliminaries Let a1 , , aq (q ≥ N + 1) be q meromorphic mappings of Cn into PN (C) with reduced representations aj = (aj0 : · · · : ajN ) (1 j general position if det(ajk l ) ≡ for any q) We say that a1 , , aq are located in j0 < j1 < < jN q We also say that a1 , , aq are located in pointwise general position if the hyperplanes a1 (z), , aq (z) are in general position as a set of fixed hyperplanes at every point z ∈ Cn q Let Mn be the field of all meromorphic functions on Cn Denote by R aj j=1 ⊂ ajk q Mn the smallest subfield which contains C and all with ajl ≡ Define R aj j=1 ⊂ ajl Mn to be the smallest subfield which contains all h ∈ Mn with hk ∈ R aj q j=1 for some positive integer k Let f be a meromorphic mapping of Cn into PN (C) with reduced representation f = (f0 : · · · : fN ) We say that f is linearly nondegenerate over R if f0 , , fN are linearly independent over R aj q j=1 (R aj aj q j=1 q j=1 R aj q j=1 , respectively) Let f , a be two meromorphic mappings of Cn into PN (C) with reduced representaN tions f = (f0 : · · · : fN ), a = (a0 : · · · : aN ) respectively Put (f, a) = fi We say i=0 that a is ”small” with respect to f if T (r, a) = o(T (r, f )) as r → ∞ After that, we recall some results which play essential roles in Nevanlinna theory as the first main theorem, the second main theorem for moving targets, the logarithmic derivative lemma We also introduce some lemmas or propositions which are used for 13 the proof of main results in this chapter 2.2 A unicity theorem with truncated multiplicities of meromorphic mappings in several complex variables sharing few moving targets In this section, we prove the following Theorem 2.2 (Ha-Quang-Thai) Let k be a positive integer or k = ∞ and d be a positive integer or d = ∞ such that the following is satisfied + d+1 k+1 2N + N +1 2N + −2 < N +1 N +2 2N + − N (N + 2)(N (N + 2) + 1) k+1 Let f, g : Cn → PN (C) (N ≥ 2) be two nonconstant meromorphic mappings, and let {aj }3N +1 be ”small” (with respect to f ) meromorphic mappings of Cn into PN (C) in j=1 general position such that dim{z ∈ Cn : ν(f,ai ), j k (z)ν(f,aj ), k (z) > 0} n − (1 i< 3N + 1) Assume that f, g are linearly nondegenerate over R({aj }3N +1 ) and the following are j=1 satisfied (i) (ν(f,Hj ), k , d) (ii) f (z) = g(z) on = (ν(g,Hj ), j∈D {z k , d) (1 ∈ Cn : ν(f,aj ), j 3N + 1) N (N +2) (z) > 0}, where D be an arbitrary subset of {1, · · · , 3N + 1} with D = N + Then f ≡ g 2.3 A unicity theorem for meromorphic mapping with a conditions on derivations In the present section, we will prove the following Theorem 2.3 (Ha-Quang-Thai) Let f, g : Cn → PN (C) be two meromorphic mappings, and k be a positive integer with k > 2N +12N +6N −1 Let {at }N +2 be ”small” t=1 (with respect to f ) meromorphic mappings of Cn into PN (C) in general position such that dim{z ∈ Cn : ν(f,as ), k (z)ν(f,at ), k (z) > 0} n − (1 s 0}, the following two conditions are satisfied: (a) If fj (ω) = then gj (ω) = 0, (b) fi gi (ω) = Dα (ω) for each n-tuple α = fj gj (α1 , , αn ) of nonnegative integers with |α| = α1 + + αn 2N and for each |α| ∂ i = j, where Dα = α1 ∂ z1 ∂ αn zn If fj (ω)gj (ω) = then Dα Then f ≡ g Remark that the condition (ii) in Theorem 2.3 does not depend on the choice of reduced representations 15 Chapter Value distribution of the Gauss map of minimal surfaces on annular ends Let M be a non-flat minimal surface in R3 , or more precisely, a connected oriented minimal surface in R3 By definition, the Gauss map G of M is the map which maps each point p ∈ M to the unit normal vector G(p) ∈ S of M at p Instead of G, we study the map g := π ◦ G : M → C := C ∪ {∞}(= P1 (C)) for the stereographic projec√ tion π of S onto P1 (C) By associating a holomorphic local coordinate z = u + −1v with each positive isothermal coordinate system (u, v), M is considered as an open Riemann surface with a conformal metric ds2 and by the assumption of minimality of M, g is a meromorphic function on M In 1988, H Fujimoto proved Nirenberg’s conjecture that if M is a complete non-flat minimal surface in R3 , then its Gauss map can omit at most points, and the bound is sharp In 1991, S J Kao showed that the Gauss map of an end of a non-flat complete minimal surface in R3 that is conformally an annulus {z|0 < 1/r < |z| < r} must also assume every value, with at most exceptions On the other hand, in 1993, M Ru studied the Gauss map of minimal surface in Rm with ramification In this chapter, we shall study the Gauss map of minimal surfaces in R3 , R4 on annular ends with ramification In particular, we prove Theorem 3.4.5, Theorem 3.4.6 (Dethloff-Ha) We would like to refer the case Rm (m > 4) to Dethloff-Ha-Thoan 16 3.1 Minimal surface in Rm We recall some basic facts in differential geometry In particular, we recall the definitions of minimal surfaces, curvature of a Riemann surface, the conditions for a surface to be a minimal surface 3.2 The Gauss map of minimal surfaces In this section, we recall some notions on the Gauss map of minimal surfaces and the relations of the Gauss map with the minimality properties of the surfaces 3.3 Meromorphic functions with ramification Let f be a nonconstant holomorphic map of a disc ∆R := {z ∈ C; |z| < R} into P1 (C), where < R < ∞ Take a reduced representation f = (f0 : f1 ) on ∆R and define ||f || := (|f0 |2 + |f1 |2 )1/2 , W (f0 , f1 ) := f0 f1 − f1 f0 Let aj (1 ≤ j ≤ q) be q distinct points in P1 (C) We may assume aj = (aj : aj ) with |aj |2 + |aj |2 = 1(1 ≤ j ≤ q), and set Fj := aj f1 − aj f0 (1 ≤ j ≤ q) 3.3.1 Definition One says that the meromorphic function f is ramified over a point a = (a0 : a1 ) ∈ P1 (C)with multiplicity at least e if all the zeros of the function F := a0 f1 − a1 f0 have orders at least e If the image of f omits a, one will say that f is ramified over a with multiplicity ∞ 3.3.2 Proposition (Fujimoto) For each depending only on a1 , · · · , aq and on ∆ log Πq j=1 > there exist positive constants C1 and µ respectively such that ||f || log(µ||f ||2 /|Fj |2 ) 3.3.3 Lemma Suppose that q − − ≥ C1 ||f ||2q−4 |W (f0 , f1 )|2 Πq |Fj |2 log2 (µ||f ||2 /|Fj |2 ) j=1 q j=1 mj > and f is ramified over aj with multiplicity at least mj for each j(1 ≤ j ≤ q) Then there exist positive constants C and µ(> 1) depending only on aj and mj (1 ≤ j ≤ q) which satisfy that if we set q j=1 mj q−2− v := C||f || 1− m Πq |Fj | j=1 j |W (f0 , f1 )| log(µ||f ||2 /|Fj |2 ) 17 on ∆R − {F1 Fq = 0} and v = on ∆R ∩ {F1 Fq = 0}, then v is continuous on ∆R and satisfies the condition ∆ log v ≥ v in the sense of distribution 3.3.4 Lemma (Generalied Schwarz’ Lemma) Let v be a nonnegative real-valued continuous subhamornic function on ∆R If v satisfies the inequality ∆ log v ≥ v in the sense of distribution, then v(z) ≤ R2 2R − |z|2 q j=1 mj 3.3.5 Lemma For every δ with q − − > qδ > and f is ramified over aj with multiplicity at least mj for each j(1 ≤ j ≤ q), there exists a positive constant C0 such that q−2− ||f || q j=1 mj −qδ |W (f0 , f1 )| 1− m j Πq |Fj | j=1 −δ ≤ C0 R2 2R − |z|2 3.3.6 Proposition Let f : C → P1 (C) be a holomorphic map For arbitrary distinct points a1 , , aq ∈ P1 (C) and f is ramified over aj with multiplicity at least mj for each j, (1 ≤ j ≤ q) satisfying q (1 − j=1 ) > mj Then f is constant 3.4 The Gauss map of minimal surfaces with ramification 3.4.1 Definition One says that g of an open Riemann surface A into Pm−1 (C) is ramified over a hyperplane H = {(w0 : · · · : wm−1 ) ∈ Pm−1 (C) : a0 w0 + +am−1 wm−1 = 0}with multiplicity at least e if all the zeros of the function (g, H) := a0 g0 + +am−1 gm−1 have orders at least e, where g = (g0 : : gm−1 ) If the image of g omits H, one will say that g is ramified over H with multiplicity ∞ 3.4.2 Theorem (Ru) For any complete minimal surface M immersed in Rm and assume that the Gauss map g of M is k−nondegenerate (that is g(M ) is contained in a k−dimensional linear subspace of Pm−1 (C), but none of lower dimension ), ≤ k ≤ m − Let {Hj }q be hyperplanes in general position in Pm−1 (C) If g is ramified over j=1 Hi with multiplicity at least mi for each i and q (1 − j=1 k k ) > (k + 1)(m − − 1) + m mj 18 then M is flat, or equivalently, g is constant 3.4.3 Theorem (Ru) Let M be a non-flat complete minimal surface in R3 If there are q (q > 4) distinct points a1 , , aq ∈ P1 (C) such that the Gauss map of M is ramified over aj with multiplicity at least mj for each j, then q j=1 (1 − ) mj ≤ 3.4.4 Corollary The Gauss map g assumes every value on unit sphere with the possible exception of four values 3.4.5 Theorem (Dethloff-Ha) Let M be a non-flat complete minimal surface in R3 and let A be an annular end of M which is conformal to {z| < 1/r < |z| < r}, where z is the conformal coordinate If there are q (q > 4) distinct points a1 , , aq ∈ P1 (C) such that the Gauss map of M is ramified over aj with multiplicity at least mj for each q j=1 (1 j on A, then − ) mj ≤ 3.4.6 Theorem (Dethloff-Ha) Suppose that M is a complete non-flat minimal surface in R4 and g = (g , g ) is the Gauss map of M Let A be an annular end of M which is conformal to {z|0 < 1/r < |z| < r}, where z is the conformal coordinate Let a11 , , a1q1 , a21 , , a2q2 be q1 + q2 (q1 , q2 > 2) distinct points in P1 (C) (i) In the case g l ≡ constant (l = 1, 2), if g l is ramified over alj with multiplicity at least mlj for each j (l = 1, 2) on A, then γ1 = q1 j=1 (1 − ) m1j ≤ 2, or γ2 = q2 j=1 (1 − ) m2j ≤ 2, or 1 + ≥ γ1 − γ2 − (ii) In the case where one of g and g is constant, say g ≡ constant, if g is ramified over a1j with multiplicity at least m1j for each j, we have the following q1 (1 − γ1 = j=1 ) ≤ m1j 3.4.7 Corollary.(Kao) The Gauss map g of minimal surfaces in R3 on an annular end must assume every value on unit sphere with the possible exception of four values 19 CONCLUSION AND RECOMMENDATIONS Conclusions The main results of the thesis are the followings • Unicity theorems with truncated multiplicities of meromorphic mappings of Cn into PN (C) for fixed hyperplanes, truncated multiplicities and a small set of identity or with the conditions of derivations • Unicity theorems with truncated multiplicities of meromorphic mappings of Cn into PN (C) for moving targets, and a small set of identity or with the conditions of derivations • The ramification of the Gauss map of minimal surfaces in Rm (m = 3; 4) on an annular end Recommendations on further research What can we say about the unicity theorems of meromorphic mappings of Cn into PN (C) for q(< 2N + 2) hyperplanes? Can we give some unicity theorems of meromorphic mappings of Cn into PN (C) for moving targets with the conditions as in the case for hyperplanes Can we obtain on unicity theorems for the Gauss maps of minimal surfaces in m R ? However, due to limited time we could not get the results to these problems We hope that these problems will soon be resolved in the near future 20 ... be a minimal surface 3.2 The Gauss map of minimal surfaces In this section, we recall some notions on the Gauss map of minimal surfaces and the relations of the Gauss map with the minimality... image of the Gauss map of a nonflat complete minimal surface immersed in R3 is of logarithmic capacity zero in P1 (C) In 1981, a remarkable improvement was given by F Xavier that the Gauss map of... this bound is sharp: In fact, we can see that the Gauss map of Scherk’s surface omits four points in P1 (C) In 1991, S J Kao showed that the Gauss map of an end of a non-flat complete minimal