Vietnam Journal of Mathematics 34:1 (2006) 71–94 An Extension of Uniqueness Theorems for Meromorphic Mappings Gerd Dethloff 1 and Tran Van Tan 2 1 Universit´e de Bretagne Occidentale UFR Sciences et Tec hniques D´ep artement de Math´ematiques 6, avenue Le Gorgeu, BP 452 29275 Brest Cedex, Fr ance 2 Dept. of Math., Hanoi University of Education, 136 Xuan Thuy R oad Cau Giay, Hanoi, Vietnam Received February 22, 2005 Revised June 20, 2005 Abstract. In this paper, we give some results on the number of meromorphic map- pings of C m into CP n under a condition on the inverse images of hyperplanes in CP n . At the same time, we give an answer for an open question posed by H. Fujimoto in 1998. 1. Introduction In 1926, Nevanlinna 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,andthatg is a special type of a linear fractional tran- formation of f if they have the same inverse images, counted with multiplicities, forfourdistinctvalues. In 1975, Fujimoto [2] generalized Nevanlinna’s result to the case of mero- morphic mappings of C m into CP n . This problem continued to be studied by Smiley [9], Ji [5] and others. Let f be a meromorphic mapping of C m into CP n and H be a hyperplane in CP n such that imf  H. Denote by v (f,H) the map of C m into N 0 such that v (f,H) (a)(a ∈ C m ) is the intersection multiplicity of the image of f and H at f(a). Let k be a positive interger or +∞. We set 72 Gerd Dethloff and Tran Van Tan v k) (f,H) (a)=  0ifv (f,H) (a) >k, v (f,H) (a)ifv (f,H) (a)  k. Let f be a linearly nondegenerate meromorphic mapping of C m into CP n and {H j } q j=1 be q hyperplanes in general position with (a) dim  z : v k) (f,H i ) (z) > 0andv k) (f,H j ) (z) > 0   m −2 for all 1  i<j q. For each positive integer p,denotebyF k ({H j } q j=1 ,f,p) the set of all linearly nondegenerate meromorphic mappings g of C m into CP n such that: (b) min  v k) (g,H j ) ,p  =min  v k) (f,H j ) ,p  , (c) g = f on q  j=1  z : v k) (f,H j ) (z) > 0  . In [5], Ji proved the following Theorem J. [5] If q =3n +1 and k =+∞, then for three mappings f 1 ,f 2 ,f 3 ∈ F k  {H j } q j=1 ,f,1  , the mapping f 1 × f 2 × f 3 : C m −→ CP n × CP n × CP n is algebraically degenerate, namely, {(f 1 (z),f 2 (z),f 3 (z)), z ∈ C m } is contained in a proper algebraic subset of CP n × CP n × CP n . In 1929, Cartan declared that there are at most two meromorphic functions on C which have the same inverse images (ignoring multiplicities) for four dis- tinct values. However in 1988, Steinmetz [10] gave examples which showed that Cartan’s declaration is false. On the other hand, in 1998, Fujimoto [4] showed that Cartan’s declaration is true if we assume that meromorphic functions on C share four distinct values counted with multiplicities truncated by 2. He gave the following theorem Theorem F. [4] If q =3n +1 and k =+∞ then F k  {H j } q j=1 ,f,2  contains at most two mappings. He also proposed an open problem asking if the number q =3n+1 in Theorem F can be replaced by a smaller one. Inspired by this question, in this paper we will generalize the above results to the case where the number q =3n +1 is in fact replaced by a smaller one. We also obtain an improvement concerning truncating multiplicities. Denote by Ψ the Segre embedding of CP n × CP n into CP n 2 +2n which is defined by sending the ordered pair ((w 0 , , w n ), (v 0 , , v n )) to ( , w i v j , )(in lexicographic order). Let h : C m −→ CP n × CP n be a meromorphic mapping. Let (h 0 : : h n 2 +2n ) be a representation of Ψ ◦ h .Wesaythath is linearly degenerate (with the algebraic structure in CP n × CP n given by the Segre embedding) if h 0 , , h n 2 +2n are linearly dependent over C. Our main results are stated as follows: Theorem 1. There are at most two distinct mappings in F k  {H j } q j=1 ,f,p  in each of the following cases: An Extension of Uniqueness Theorems 73 i) 1  n  3,q =3n +1,p=2and 23n  k  +∞ ii) 4  n  6,q =3n, p =2and (6n − 1)n n −3  k  +∞ iii) n ≥ 7,q =3n −1,p=1and (6n −4)n n −6  k  +∞ Theorem 2. Assume that q =  5(n +1) 2  , (65n + 171)n  k  +∞,where [x]:=max{d ∈ N : d  x} for a positive constant x. Then one of the following assertions holds : i) #F k  {H j } q j=1 ,f,1   2. ii) For any f 1 ,f 2 ∈ F k  {H j } q j=1 ,f,1  , the mapping f 1 ×f 2 : C m −→ CP n ×CP n is linearly degenerate (with the algebraic structure in CP n × CP n given by the Segre embedding). We finally remark that we obtained similar uniqueness theorems with moving targets in [11], but only with a bigger number of targets and with much bigger truncations. 2. Preliminaries We set z := (|z 1 | 2 + ···+ |z m | 2 ) 1/2 for z =(z 1 , ,z m ) ∈ C m ,B(r):=  z : z <r  ,S(r):=  z :   z = r  ,d c := √ −1 4π ( ∂ − ∂),υ:= (dd c z 2 ) m−1 and σ := d c log z 2 ∧ (dd c log z 2 ) m−1 . Let F be a nonzero holomorphic function on C m . For an m-tuple α := (α 1 , ,α m ) of nonnegative integers, set |α| := α 1 + ···+ α m and D α F := ∂ |α| F ∂z α 1 1 ∂z α m m . Wedefinethemapv F : C m → N 0 by v F (z):=max  p : D α F (z) = 0 for all α with |α| <p  .Letk be a positive integer or +∞. Define the map v k) F of C m into N 0 by v k) F (z):=  0ifv F (z) >k, v F (z)ifv F (z)  k. Let ϕ be a nonzero meromorphic function on C m . Wedefinethemapv k) ϕ as follows. For each z ∈ C m , choose nonzero holomorphic functions F and G on a neighborhood U of z such that ϕ = F G on U and dim  F −1 (0) ∩G −1 (0)   m −2. Then put v k) ϕ (z):=v k) F (z). Set   v k) ϕ   :=  z : v k) ϕ (z) > 0  . Define N k) (r, v ϕ ):= r  1 n k) (t) t 2m−1 dt, (1 <r<+∞) where 74 Gerd Dethloff and Tran Van Tan n k) (t):=    v k) ϕ   ∩B(t) v k) ϕ υ for m ≥ 2, and n k) (t):=  |z|t v k) ϕ (z)form =1. Set N(r, v ϕ ):=N +∞) (r, v ϕ ). For l a positive integer or +∞,set N k) l (r, v ϕ ):= r  1 n k) l (t) t 2m−1 dt, (1 <r<+∞) where n k) l (t):=    v k) ϕ   ∩B(t) min  v k) ϕ ,l  υ for m ≥ 2andn k) l (t):=  |z|t min  v k) ϕ (z),l  for m =1. Set N(r, v ϕ ):=N +∞) 1 (r, v ϕ )andN k) (r, v ϕ ):= N k) 1 (r, v ϕ ). For a closed subset A of a purely (m−1)-dimensional analytic subset of C m , we define N(r, A):= r  1 n(t) t 2m−1 dt, (1 <r<+∞), where n(t):= ⎧ ⎨ ⎩  A∩B(t) υ for m ≥ 2, #(A ∩B(t)) for m =1. Let f : C m → CP n be a meromorphic mapping. For arbitrarily fixed homo- geneous coordinates (w 0 : ···: w n )onCP n , we take a reduced representation f =(f 0 : ··· : f n ), which means that each f i is a holomorphic function on C m and f(z)=(f 0 (z):··· : f n (z)) outside the analytic set {f 0 = ··· = f n =0} of codimension ≥ 2. Set f := (|f 0 | 2 + ···+ |f n | 2 ) 1/2 . The characteristic function of f is defined by T f (r):=  S(r) log f σ −  S(1) log fσ, r > 1. For a nonzero meromorphic function ϕ on C m , the characteristic function T ϕ (r) of ϕ is defined by considering ϕ as a meromorphic mapping of C m into CP 1 . Let H = {a 0 w 0 +···+a n w n =0} be a hyperplane in CP n such that imf  H. Set (f,H):=a 0 f 0 + ···+ a n f n . We define N k) f (r, H ):=N k) (r, v (f,H) )andN k) l,f (r, H ):=N k) l (r, v (f,H) ). Sometimes we write N k) f (r, H )forN k) 1,f (r, H ), N l,f (r, H )forN +∞) l,f (r, H )and N f (r, H )forN +∞) +∞,f (r, H ). An Extension of Uniqueness Theorems 75 Set ψ f (H):= f  |a 0 | 2 + ···+ |a n | 2  1/2 (f,H) . We define the proximity function by m f (r, H ):=  S(r) log |ψ f (H) |σ −  S(1) log |ψ f (H) |σ. For a nonzero meromorphic function ϕ, the proximity function is defined by m(r, ϕ):=  S(r) log + | ϕ |σ. We note that m(r, ϕ)=m ϕ (r, +∞)+O(1) ([4], p. 135). We state First and Second Main Theorems of Value Distribution Theory. First Main Theorem. Let f : C m → CP n be a meromorphic mapping and H a hyperplane in CP n such that im f  H. Then N f (r, H )+m f (r, H )=T f (r). For a nonzero meromorphic function ϕ we have N(r, v 1 ϕ )+m(r, ϕ)=T ϕ (r)+O(1). Second Main Theorem. Let f : C m → CP n be a linearly nondegenerate meromorphic m apping and H 1 , , H q be hyperplanes in general p o sition in CP n . Then (q − n −1)T f (r)  q  j=1 N n,f (r, H j )+o(T f (r)) except for a set E ⊂ (1, +∞) of finite Lebesgue measure. The following so-called logarithmic derivative lemma plays an essential role in Nevanlinna theory. Theorem 2.1. ([5], Lemma 3.1) Let ϕ be a non-constant meromorphic function on C m . Then for any i, 1  i  m, we have m  r, ∂ ∂z i ϕ ϕ  = o(T ϕ (r)) as r →∞,r/∈ E, where E ⊂ (1, +∞) of finite Lebesgue measure. Let F, G and H be nonzero meromorphic functions on C m . For each l, 1  l  m, we define the Cartan auxiliary function by Φ l (F, G, H):=F ·G · H ·       111 1 F 1 G 1 H ∂ ∂z l  1 F  ∂ ∂z l  1 G  ∂ ∂z l  1 H        . By [4] (Proposition 3.4) we have the following 76 Gerd Dethloff and Tran Van Tan Theorem 2.2. Let F, G, H be nonzero meromorphic functions on C m .Assume that Φ l (F, G, H) ≡ 0 and Φ l  1 F , 1 G , 1 H  ≡ 0 for all l, 1  l  m. Then one of the following assertions holds i) F = G or G = H or H = F . ii) F G , G H , H F are all constant. 3. Proof of the Theorems First of all, we need the following lemmas: Lemma 1. Let f 1 , , f d be line arly nondegenerate meromorphic mappings of C m into CP n and {H j } q j=1 be hyperplanes in CP n .Then there exists a dense subset C⊂ C n+1  {0} such that for any c =(c 0 , , c n ) ∈C, the hyperplane H c defined by c 0 ω 0 + ···+ c n ω n =0satisfies dim (f −1 i (H j ) ∩f −1 i (H c ))  m − 2 for all i ∈{1, ,d} and j ∈{1, , q}. Proof. We refer to [5], Lemma 5.1.  Let f 1 ,f 2 ,f 3 ∈ F k  {H j } q j=1 ,f,1  ,forq ≥ n +1. Set T (r):=T f 1 (r)+T f 2 (r)+T f 3 (r). For each c ∈C, set F j ic := (f i ,H j ) (f i ,H c ) for i ∈{1, 2, 3} and j ∈{1, ,q}. Lemma 2. Assume that there exist j 0 ∈{1, , q},c ∈C,l ∈{1, , m} and a closed subset A of a purely (m −1) -dimensional an alytic subset of C m satisfying 1) Φ l c := Φ l  F j 0 1c ,F j 0 2c ,F j 0 3c  ≡ 0, and 2) min  v k) (f 1 ,H j 0 ) ,p  =min  v k) (f 2 ,H j 0 ) ,p  =min  v k) (f 3 ,H j 0 ) ,p  on C m \A, where p is a positive integer. T hen 2 q  j=1,j=j 0 N k) f i (r, H j )+ N k) p−1,f i (r, H j 0 )  N(r, v Φ l c )+(p − 1)N(r, A)  k +2 k +1 T (r)+(p +2) N(r, A)+o(T (r)) for all i ∈{1, 2, 3}. Proof. Without loss of generality, we may assume that l =1. For an arbitrary point a ∈ C m \ A satisfying v k) (f 1 ,H j 0 ) (a) > 0, we have v k) (f i ,H j 0 ) (a) > 0 for all i ∈{1, 2, 3}.Wechoosea such that a/∈ 3  i=1 f −1 i (H c ). We distinguish two cases, which lead to equations (1) and (2). An Extension of Uniqueness Theorems 77 Case 1. If v (f 1 ,H j 0 ) (a) ≥ p,thenv (f i ,H j 0 ) (a) ≥ p, i ∈{1, 2, 3}. This means that a is a zero point of F j 0 ic with multiplicity ≥ p for i ∈{1, 2, 3}.Wehave Φ 1 c = F j 0 1c F j 0 3c ∂ ∂z 1  1 F j 0 3c  − F j 0 1c F j 0 2c ∂ ∂z 1  1 F j 0 2c  + F j 0 2c F j 0 1c ∂ ∂z 1  1 F j 0 1c  − F j 0 2c F j 0 3c ∂ ∂z 1  1 F j 0 3c  + F j 0 3c F j 0 2c ∂ ∂z 1  1 F j 0 2c  − F j 0 3c F j 0 1c ∂ ∂z 1  1 F j 0 1c  . On the other hand F j 0 1c F j 0 3c ∂ ∂z 1  1 F j 0 3c  = −F j 0 1c ∂ ∂z 1 F j 0 3c F j 0 3c ,soa is a zero point of F j 0 1c F j 0 3c ∂ ∂z 1  1 F j 0 3c  with multiplicity ≥ p − 1. By applying the same argument also to all other combinations of indices, we see that a is a zero point of Φ 1 c with multiplicity ≥ p − 1. (1) Case 2. If v (f 1 ,H j 0 ) (a)  p,thenp 0 := v (f 1 ,H j 0 ) (a)=v (f 2 ,H j 0 ) (a)=v (f 3 ,H j 0 ) (a)  p. There exists a neighborhood U of a such that v (f 1 ,H j 0 )  p on U. In- deed, there exists otherwise a sequence {a s } ∞ s=1 ⊂ C m , with lim s→∞ a s = a and v (f 1 ,H j 0 ) (a s ) ≥ p+1 for all s. By the definition, we have D β (f 1 ,H j 0 )(a s )=0for all |β| <p+1. So D β (f 1 ,H j 0 )(a) = lim s→∞ D β (f 1 ,H j 0 )(a s ) = 0 for all |β| <p+1. Thus v (f 1 ,H j 0 ) (a) ≥ p + 1. This is a contradiction. Hence v (f 1 ,H j 0 )  p on U. We can choose U such that U ∩A = ∅ , v (f i ,H j 0 )  p on U and (f i ,H c )has no zero point on U for all i ∈{1, 2, 3}. Then v F j 0 1c = v F j 0 2c = v F j 0 3c  p on U. So U ∩{F j 0 1c =0} = U ∩{F j 0 2c =0} = U ∩{F j 0 3c =0}.Choosea such that a is regular point of U ∩{F j 0 1c =0}. By shrinking U we may assume that there exists a holomorphic function h on U such that dh has no zero point and F j 0 ic = h p 0 u i on U, where u i (i =1, 2, 3) are nowhere vanishing holomorphic functions on U (note that v F j 0 1c (a)=v F j 0 2c (a)=v F j 0 3c (a)=p 0 ). We have Φ 1 c = u 1  u 3 ∂ ∂z 1 u 2 − u 2 ∂ ∂z 1 u 3  h p 0 u 2 u 3 + u 2  u 1 ∂ ∂z 1 u 3 − u 3 ∂ ∂z 1 u 1  h p 0 u 3 u 1 + u 3  u 2 ∂ ∂z 1 u 1 − u 1 ∂ ∂z 1 u 2  h p 0 u 1 u 2 . So, we have a is a zero point of Φ 1 c with mulitplicity ≥ p 0 .(2) By (1), (2) and our choice of a, there exists an analytic set M ⊂ C m with codimension ≥ 2 such that v Φ 1 c ≥ min{v (f 1 ,H j 0 ) , p −1} on  z : v k) (f 1 ,H j 0 ) (z) > 0  \ (M ∪A). (3) For each j ∈{1, ,q}\{j 0 },leta (depending on j) be an arbitrary point in C m such that v k) (f 1 ,H j ) (a) > 0 (if there exist any). Then v k) (f i ,H j ) (a) > 0 78 Gerd Dethloff and Tran Van Tan for all i ∈{1, 2, 3}, since f 1 ,f 2 ,f 3 ∈ F k  {H j } q j=1 ,f,1  .Wecanchoosea/∈ f −1 i (H c ) ∪ f −1 i (H j 0 ),i =1, 2, 3. Then there exists a neighborhood U of a such that v (f i ,H j )  k on U and (f i ,H j 0 ), (f i ,H c )(i =1, 2, 3 ) have no zero point on U.WehaveB := f −1 1 (H j ) ∩ U = f −1 2 (H j ) ∩ U = f −1 3 (H j ) ∩ U and 1 F j 0 1c = 1 F j 0 2c = 1 F j 0 3c on B. Choose a such that a is a regular point of B.By shrinking U, we may assume that there exists a holomorphic function h on U such that dh has no zero point and U ∩{h =0} = B.Then 1 F j 0 2c − 1 F j 0 1c = hϕ 2 and 1 F j 0 3c − 1 F j 0 1c = hϕ 3 on U where ϕ 2 ,ϕ 3 are holomorphic functions on U . Hence, we get Φ 1 c = F j 0 1c F j 0 2c F j 0 3c        10 0 1 F j 0 1c hϕ 2 hϕ 3 ∂ ∂z 1  1 F j 0 1c  ϕ 2 ∂ ∂z 1 h + h ∂ ∂z 1 ϕ 2 ϕ 3 ∂ ∂z 1 h + h ∂ ∂z 1 ϕ 3        = F j 0 1c F j 0 2c F j 0 3c h 2     ϕ 2 ϕ 3 ∂ ∂z 1 ϕ 2 ∂ ∂z 1 ϕ 3     . Therefore, a is a zero point of Φ 1 c with multiplicity ≥ 2. Thus, for each j ∈ {1, ,q}\{j 0 }, there exists an analytic set N ⊂ C m with codimension ≥ 2 such that v Φ 1 c ≥ 2on  z : v k) (f 1 ,H j ) (z) > 0  \ N. (4) By (3) and (4), we have 2 q  j=1,j=j 0 N k) f 1 (r, H j )+N k) p−1,f 1 (r, H j 0 )  N(r, v Φ 1 c )+(p − 1)N(r, A). Similarly, we have 2 q  j=1,j=j 0 N k) f i (r, H j )+N k) p−1,f i (r, H j 0 )  N(r, v Φ 1 c )+(p−1)N(r, A),i=1, 2, 3. (5) Let a be an arbitrary zero point of some F j 0 ic ,a /∈ f −1 i (H c ), say i =1. We have Φ 1 c =  F j 0 2c − F j 0 3c  F j 0 1c ∂ ∂z 1  1 F j 0 1c  +  F j 0 3c − F j 0 1c  F j 0 2c ∂ ∂z 1  1 F j 0 2c  +  F j 0 1c − F j 0 2c  F j 0 3c ∂ ∂z 1  1 F j 0 3c  . (6) So we have An Extension of Uniqueness Theorems 79 v 1 Φ 1 c (a)  1+max{v 1 F j 0 ic (a),i=2, 3}  1+ v 1 F j 0 2c (a)+v 1 F j 0 3c (a). Furthermore, if 0 <v F j 0 1c (a)  k  and, hence, v k) (f 1 ,H j 0 ) (a) > 0  and a/∈ A,then by (3) we may assume that v 1 Φ 1 c (a) = 0 (outside an analytic set of codimension ≥ 2). (7) Let a be an arbitrary pole of all F j 0 ic , i =1, 2, 3. By (6) we have v 1 Φ 1 c (a)  max{v 1 F j 0 ic (a),i=1, 2, 3}+1< 3  i=1 v 1 F j 0 ic (a)(8) It follows from (6) that a pole of Φ 1 c is a zero or a pole of some F j 0 ic . Thus, by (6), (7) and (8), we have N  r, v 1 Φ 1 c   3  i=1 N  r, v 1 F j 0 ic  + 3  i=1  N  r, v F j 0 ic  − N k)  r, v F j 0 ic   +3 N(r, A)  3  i=1 N  r, v 1 F j 0 ic  + 1 k +1 3  i=1 N  r, v F j 0 ic  +3 N(r, A)  3  i=1 N  r, v 1 F j 0 ic  + 1 k +1 3  i=1 T F j 0 ic (r)+3N(r, A)  3  i=1 N  r, v 1 F j 0 ic  + 1 k +1 T (r)+3 N(r, A)+O(1). (9) We have Φ 1 c = F j 0 1c  F j 0 3c ∂ ∂z 1  1 F j 0 3c  − F j 0 2c ∂ ∂z 1  1 F j 0 2c  + F j 0 2c  F j 0 1c ∂ ∂z 1  1 F j 0 1c  − F j 0 3c ∂ ∂z 1  1 F j 0 3c  + F j 0 3c  F j 0 2c ∂ ∂z 1  1 F j 0 2c  − F j 0 1c ∂ ∂z 1  1 F j 0 1c  so m(r, Φ 1 c )  3  i=1 m(r, F j 0 ic )+2 3  i=1 m  r, F j 0 ic ∂ ∂z 1  1 F j 0 ic  +0(1). By Theorem 2.1, we have m  r, F j 0 ic ∂ ∂z 1  1 F j 0 ic  = o  T F j 0 ic (r)  . Thus, we get m(r, Φ 1 c )  3  i=1 m(r, F j 0 ic )+o(T (r)), (10) 80 Gerd Dethloff and Tran Van Tan (note that T F j 0 ic (r)  T f i (r)+O(1)). By (9), (10) and by First Main Theorem, we have N(r, v Φ 1 c )  T Φ 1 c (r)+O(1) = N  r, v 1 Φ 1 c  + m(r, Φ 1 c )+O(1)  3  i=1  N  r, v 1 F j 0 ic  + m(r, F j 0 ic )  + 1 k +1 T (r)+3 N(r, A)+o(T (r))  3  i=1 T F j 0 ic (r)+ 1 k +1 T (r)+3 N(r, A)+o(T (r))  3  i=1 T f i (r)+ 1 k +1 T (r)+3 N(r, A)+o(T (r)) = k +2 k +1 T (r)+3 N(r, A)+o(T (r)). (11) By (5) and (11) we get Lemma 2.  The following lemma is a version of Second Main Theorem without taking account of multiplicities of order >kin the counting functions. Lemma 3. Let f be a linearly n ondegen erate meromorphic mapping of C m into CP n and {H j } q j=1 (q ≥ n +2) be hyperplanes in CP n in general p osition. Take a positive integer k with qn q−n−1  k  +∞. Then T f (r)  k (q − n −1)(k +1)−qn q  j=1 N k) n,f (r, H j )+o(T f (r))  nk (q − n −1)(k +1)−qn q  j=1 N k) f (r, H j )+o(T f (r)) for all r>1 except a set E of finite Lebesgue measure. Proof. By First and Second Main Theorems, we have (q − n −1)T f (r)  q  j=1 N n,f (r, H j )+o  T f (r)   k k +1 q  j=1 N k) n,f (r, H j )+ n k +1 q  j=1 N f (r, H j )+o  T f (r)  ≤ k k +1 q  j=1 N k) n,f (r, H j )+ qn k +1 T f (r)+o  T f (r)  ,r/∈ E, which impies that  q − n −1 − qn k +1  T f (r)  k k +1 q  j=1 N k) n,f (r, H j )+o  T f (r)  . [...]... + 1)2 k+1 n+ nq k An Extension of Uniqueness Theorems 93 This contradicts q = 5(n+1) , k ≥ (65n + 171)n Thus, we get that αij = 1 2 for all 1 i = j q − 1 For 1 s < v 3, denote by Lsv the set of all j ∈ {1, , q − 2} such that (fs ,Hj ) (fv ,Hj ) (fs ,Hq−1 ) = (fv ,Hq−1 ) By (28), we have L12 ∪ L23 ∪ L13 = {1, , q − 2} If there exists some Lsv = ∅, we may assume without loss of generality 5(n + 1)... author would like to thank Professor Do Duc Thai for valuable discussions, the Universit´ de Bretagne Occidentale for its hospitality and e support, and the PICS-CNRS For MathVietnam for its support References 1 H Cartan, Un nouveau th´or`me d’unicit´ relatif aux fonctions m´romorphes,C R e e e e Acad Sci Paris 188 (1929) 301–330 2 H Fujimoto, The uniqueness problem of meromorphic maps into the complex...An Extension of Uniqueness Theorems 81 Thus, we have k (q − n − 1)(k + 1) − qn Tf (r) nk (q − n − 1)(k + 1) − qn q k) Nn,f (r, Hj )+o(T f (r)) j=1 q k) N f (r, H j ) + o(T f (r)) j=1 Proof of Theorem 1 Assume that there exist three distinct mappings f1 , f2 , f3 ∈ Fk ({Hj }q , f, p) Denote by Q the set which contains all indices j ∈ {1, , q} j=1 j j j satisfying Φl F1c , F2c , F3c ≡ 0 for some... ) = for all j ∈ {1, , n}, so f1 ≡ f2 1, , n ∈ L12 Then (f1 , Hq−1 ) (f2 , Hq−1 ) (as in the proof of Theorem 1) This is a contradiction Thus, we have Lsv = ∅ for all 1 s < v 3 Then for any 1 s < v 3, (fv , Hj ) (fs ,Hj ) there exists j ∈ {1, , q −2} such that (fs ,Hq−1 ) = Hence, we finally (fv , Hq−1 ) get that fs × fv : Cm −→ CP n × CP n is linearly degenerate We thus have completed the proof of. .. (j = 1, , n) For 1 (fs ,Hj ) (fs ,H3n−1 ) This is a contradiction Thus, for any case we have that f1 , f2 , f3 can not be distinct Hence, the proof of Theorem 1 is complete Proof of Theorem 2 Assume that #Fk ({Hj }q , f, 1) ≥ 3 Take arbitrarily j=1 three distinct mappings f1 , f2 , f3 ∈ Fk ({Hj }q , f, 1) We have to prove that j=1 fs × fv : Cm −→ CP n × CP n is linearly degenerate for all 1 s < v... Second Main Theorems, we have 3 N r, vg2 −bj + o Tg2 (r) Tg2 (r) j=1 1 k+1 3 N r, vg2 −bj + o Tg2 (r) j=1 3 Tg (r) + o Tg2 (r) k+1 2 This contradicts k ≥ 23 Case 2 If n ≥ 2, for each 1 i = j exists a constant αij such that 3n − 1, by (17) and Theorem 2.2, there (f2 , Hj ) (f3 , Hj ) (f1 , Hj ) (f1 , Hj ) = αij or = αij (f2 , Hi ) (f1 , Hi ) (f3 , Hi ) (f1 , Hi ) An Extension of Uniqueness Theorems or... Variables, Translations of Mathematical Monographs, 80 American Methematical Society, Providence, RI, 1990 8 B Shiffman, Introduction to the Carlson-Griffiths Equidistribution Theory, Lecture Note in Math Vol 981, Springer–Verlag, 1983 9 L Smiley, Geometric conditions for unicity of holomorphic curves, Contemp Math 25 (1983) 149–154 10 N Steinmetz, A uniqueness theorem for three meromorphic functions,... Hc )|2 ) 2 σ + O(1) 1 T (fi ,H1 ) (r) = (fi ,Hc ) S(r) log fi σ + O(1) = Tfi (r) + O(1), S(r) q Since f1 = f2 on j=1 k) z : v(f1 ,Hj ) (z) > 0 and i = 1, 2, 3 An Extension of Uniqueness Theorems k) 83 k) dim z : v(f1 ,Hi ) (z) > 0 m − 2 for all i = j, and v(f1 ,Hj ) (z) > 0 we have q k) N f1 (r, Hj ) T (f1 ,H1 ) − (f2 ,H1 ) (r) + 0(1) N r, v (f1 ,H1 ) − (f2 ,H1 ) (f1 ,Hc ) j=1 (f2 ,Hc ) (f1 ,Hc )... #({1, , q} \ Q) ≥ 3n − 1 Without loss of generality, we may assume that 1, , 3n − 1 ∈ Q Then we have / j j j Φl F1c , F2c , F3c ≡ 0 for all c ∈ C, l ∈ {1, , m}, j ∈ {1, , 3n − 1} j j j On the other hand, C is dense in Cn+1 Hence, Φl F1c , F2c , F3c ≡ 0 for all c ∈ Cn+1 \ {0}, l ∈ {1, , m}, j ∈ {1, , 3n − 1} In particular (for Hc = Hi ) we have Φl for all 1 i=j (f1 , Hj ) (f2 , Hj ) (f3... B is included in an analytic set of codimension ≥ 2 So we have 3 (n − 1)N (r, B) k) k) nN fi (r, Hj1 ) − Nn,fi (r, Hj1 ) i=1 By (25) we have (5n + 2)n T (r) + o(T (r)), (n − 1)k N (r, B) where we note that n ≥ 2, since n is even It is clear that k) k) k) min v(f1 ,Hj ) , 2 = min v(f2 ,Hj ) , 2 = min v(f3 ,Hj ) , 2 1 1 1 on Cm \B(⊆ Cm B) An Extension of Uniqueness Theorems 91 By Lemma 2 (with A = . Vietnam Journal of Mathematics 34:1 (2006) 71–94 An Extension of Uniqueness Theorems for Meromorphic Mappings Gerd Dethloff 1 and Tran Van Tan 2 1 Universit´e. v (f,H) ). Sometimes we write N k) f (r, H )forN k) 1,f (r, H ), N l,f (r, H )forN +∞) l,f (r, H )and N f (r, H )forN +∞) +∞,f (r, H ). An Extension of Uniqueness Theorems 75 Set ψ f (H):= f  |a 0 | 2 +. . ii) F G , G H , H F are all constant. 3. Proof of the Theorems First of all, we need the following lemmas: Lemma 1. Let f 1 , , f d be line arly nondegenerate meromorphic mappings of C m into CP n and {H j } q j=1 be