International audienceIn this paper, we extend the uniqueness theorem for meromorphic mappings to the case where the family of hyperplanes depends on the meromorphic mapping and where the meromorphic mappings may be degenerate

Dethloff, Gerd

Duc Quang, Si

Tran Van, Tan

HAL-Université de Bretagne Occidentale

A Uniqueness Theorem for Meromorphic Mappings withTwo Families of HyperplanesGerd Dethloff, Tan Tran Van, Si Duc QuangTo cite this version:Gerd Dethloff, Tan Tran Van, Si Duc Quang. A Uniqueness Theorem for Meromorphic Map-pings with Two Families of Hyperplanes. Proc. AMS, 2012, 140 (1), pp.189-197. <hal-00531022>HAL Id: hal-00531022http://hal.univ-brest.fr/hal-00531022Submitted on 1 Nov 2010HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A UNIQUENESS THEOREM FOR MEROMORPHICMAPPINGS WITH TWO FAMILIES OF HYPERPLANESGerd Dethloff and Si Duc Quang and Tran Van TanAbstractIn this paper, we extend the uniqueness theorem for meromorphicmappings to the case where the family of hyperplanes depends on themeromorphic mapping and where the meromorphic mappings may bedegenerate.1 IntroductionThe uniqueness problem of meromorphic mappings under a condition on theinverse images of divisors was first studied by Nevanlinna [6]. He showedthat for two nonconstant meromorphic functions f and g on the complexplane C, if they have the same inverse images for five distinct values, thenf ≡ g. In 1975, Fujimoto [3] generalized Nevanlinna’s result to the case ofmeromorphic mappings of Cm into CP n. He showed that for two linearlynondegenerate meromorphic mappings f and g of Cm into CP n, if they havethe same inverse images counted with multiplicities for (3n+2) hyperplanesin general position in CP n, then f ≡ g.In 1983, Smiley [9] showed thatTheorem 1. Let f, g be linearly nondegenerate meromorphic mappings ofCm into CP n. Let {Hj}qj=1 (q ≥ 3n + 2) be hyperplanes in CP n in generalposition. Assume thata) f−1(Hj) = g−1(Hj) , for all 1 ≤ j ≤ q (as sets),b) dim(f−1(Hi) ∩ f−1(Hj)) ≤ m− 2 for all 1 ≤ i < j ≤ q ,c) f = g on⋃qj=1 f−1(Hj) .Then f ≡ g.Mathematics Subject Classification 2000: Primary 32H30; Secondary 32H04, 30D35.Key words : Meromorphic mappings, uniqueness theorems.Acknowledgement: This work was done during a stay of the third named authorat the Institut des Hautes E´tudes Scientifiques, France. He wishes to express hisgratitude to this institute.1In 2006 Thai-Quang [11] generalized this result of Smiley to the case whereq ≥ 3n + 1 and n ≥ 2. In 2009, Dethloff-Tan [2] showed that for everynonnegative integer c there exists a positive integer N(c) depending only onc such that Theorem 1 remains valid if q ≥ (3n + 2 − c) and n ≥ N(c).They also showed that the coefficient of n in the formula of q can be replacedby a number which is smaller than 3 for all n >> 0. Furthermore, theyestablished a uniqueness theorem for the case of 2n + 3 hyperplanes andmultiplicities are truncated by n. At the same time, they strongly generalizedmany uniqueness theorems of previous authors such as Fujimoto [4], Ji [5]and Stoll [10]. Recently, by using again the technique of Dethloff-Tan [2],Chen-Yan [1] showed that the assumption “multiplicities are truncated by n”in the result of Dethloff-Tan can be replaced by “multiplicities are truncatedby 1”. In [8], Quang examined the uniqueness problem for the case of 2n+2hyperplanes.We would like to note that so far, all results on the uniqueness prob-lem have still been restricted to the case where meromorphic mappings aresharing a common family of hyperplanes. The purpose of this paper is tointroduce a uniqueness theorem for the case where the family of hyperplanesdepends on the meromorphic mapping. We also will allow that the meromor-phic mappings may be degenerate. For this purpose we introduce some newtechniques which can also be used to obtain simpler proofs for many otheruniqueness theorems.We shall prove the following uniqueness theorem:Theorem 2. Let f, g be nonconstant meromorphic mappings of Cm intoCP n. Let {Hj}qj=1 and {Lj}qj=1 (q > 2n + 2) be families of hyperplanes inCP n in general position. Assume thata) f−1(Hj) = g−1(Lj) for all 1 ≤ j ≤ q ,b) dim(f−1(Hi) ∩ f−1(Hj)) ≤ m− 2 for all 1 ≤ i < j ≤ q ,c) (f,Hi)(g,Li)=(f,Hj)(g,Lj)on⋃qk=1 f−1(Hk) \(f−1(Hi) ∪ f−1(Hj))for all 1 ≤i < j ≤ q .Then the following assertions hold :i) dim〈Imf〉 = dim〈Img〉 Def.=: p,where for a subset X ⊂ CP n, we denote by 〈X〉 the smallest projective sub-space of CP n containing X.2ii) If(∗) q > 2n + 3− p+√(2n+ 3− p)2 + 8(p− 1)(2n− p+ 1)2(≥ 2n+ 2),then(f,H1)(g, L1)≡ · · · ≡ (f,Hq)(g, Lq).Furthermore, there exists a linear projective transformation L of CP n intoitself such that L(f) ≡ g and L(Hj ∩ 〈Imf〉) = Lj ∩ L(〈Imf〉) for all j ∈{1, . . . , q}.Remark. 1.) In Theorem 2 condition c) is well defined since, by conditiona), (f,Hi)(g,Li)is a (nonvanishing) holomorphic function outside f−1(Hi).2.) The condition (∗) is satisfied in the following cases:+) q ≥ 2n+ 3 and p ∈ {1, 2, n− 1, n}, n ∈ Z+.+) q ≥ 2n+ p+ 1 and p ∈ {2, 3, . . . , n}, n ∈ Z+.3.) If there exists a subset {j0, . . . , jn} ⊂ {1, . . . , q} such that Hji ≡ Ljifor all i ∈ {0, . . . , n}, then the proof of Theorem 2 implies that f ≡ g.4.) For the special case where f, g are linearly nondegenerate (i.e. p = n)and Hj ≡ Lj , from Theorem 2 we get again the results of Dethloff-Tan [2]and Chen-Yan [1].2 PreliminariesWe set ‖z‖ := (|z1|2 + · · ·+ |zm|2)1/2 for z = (z1, . . . , zm) ∈ Cm and defineB(r) :={z ∈ Cm : ‖z‖ < r}, S(r) := {z ∈ Cm : ‖z‖ = r}for all 0 < r <∞. Definedc :=√−14π(∂ − ∂), υ := (ddc‖z‖2)m−1σ := dclog‖z‖2 ∧ (ddclog‖z‖2)m−1.Let F be a nonzero holomorphic function on Cm. For each a ∈ Cm,expanding F as F =∑Pi(z−a) with homogeneous polynomials Pi of degreei around a, we defineνF (a) := min{i : Pi 6≡ 0}.3Let ϕ be a nonzero meromorphic function on Cm. We define the zero divisorνϕ as follows: For each z ∈ Cm, we choose nonzero holomorphic functions Fand G on a neighborhood U of z such that ϕ = F/G on U and dim(F−1(0)∩G−1(0))6 m− 2. Then we put νϕ(z) := νF (z).Let ν be a divisor in Cm and k be positive integer or +∞. Set |ν| :={z : ν(z) 6= 0} and ν [k](z) := min{ν(z), k}.The truncated counting function of ν is defined byN [k](r, ν) :=r∫1n[k](t)t2m−1dt (1 < r < +∞),wheren[k](t) =∫|ν|∩B(t)ν [k] · υ for m > 2,∑|z|6tν [k](z) for m = 1.We simply write N(r, ν) for N [+∞](r, ν).For a nonzero meromorphic function ϕ on Cm, we set N[k]ϕ (r) := N [k](r, νϕ)and Nϕ(r) := N[+∞](r, νϕ). We have the following Jensen’s formula:Nϕ(r)−N 1ϕ(r) =∫S(r)log|ϕ|σ −∫S(1)log|ϕ|σ.Let f : Cm −→ CP n be a meromorphic mapping. For an arbitrary fixedhomogeneous coordinate system (w0 : · · · : wn) in CP n, we take a reducedrepresentation f = (f0 : · · · : fn), which means that each fi is a holomorphicfunction on Cm and f(z) = (f0(z) : · · · : fn(z)) outside the analytic set{f0 = · · · = fn = 0} of codimension > 2. Set ‖f‖ =(|f0|2 + · · ·+ |fn|2)1/2.The characteristic function Tf(r) of f is defined byTf (r) :=∫S(r)log‖f‖σ −∫S(1)log‖f‖σ, 1 < r < +∞.For a meromorphic function ϕ on Cm, the characteristic function Tϕ(r) of ϕis defined by considering ϕ as a meromorphic mapping of Cm into CP 1.4We state the First Main Theorem and the Second Main Theorem in ValueDistribution Theory: For a hyperplane H : a0w0 + · · · + anwn = 0 in CP nwith Imf 6⊆ H , we put (f,H) = a0f0 + · · ·+ anfn, where (f0 : · · · : fn) is areduced representation of f .First Main Theorem. Let f be a meromorphic mapping of Cm into CP n,and H be a hyperplane in CP n such that (f,H) 6≡ 0. ThenN(f,H)(r) 6 Tf (r) +O(1) for all r > 1.Let n,N, q be positive integers with q ≥ 2N −n+1 and N ≥ n. We say thathyperplanes H1, . . . , Hq in CPn are in N -subgeneral position if ∩Ni=0Hji = ∅for every subset {j0, . . . , jN} ⊂ {1, . . . , q}.Cartan-Nochka Second Main Theorem ([7], Theorem 3.1). Let f be alinearly nondegenerate meromorphic mapping of Cm into CP n and H1, . . . , Hqhyperplanes in CP n in N-subgeneral position (q ≥ 2N − n+ 1). Then(q − 2N + n− 1)Tf(r) 6q∑j=1N[n](f,Hj)(r) + o(Tf(r))for all r except for a subset E of (1,+∞) of finite Lebesgue measure.3 Proof of Theorem 3We first remark that f−1(Hj) = g−1(Lj) 6= CP n for all j ∈ {1, . . . , q},and that therefore {Hj ∩ 〈Imf〉}qj=1 (respectively {Lj ∩ 〈Img〉}qj=1) are hy-perplanes in 〈Imf〉 (respectively 〈Img〉) in n−subgeneral position: Indeed,otherwise there exists t ∈ {1, . . . , q} such that f−1(Ht) = CP n. Then bythe assumption b) we have dimf−1(Hj) ≤ m− 2 for all j ∈ {1, . . . , q} \ {t}.Therefore, f−1(Hj) = ∅ for all j ∈ {1, . . . , q} \ {t}. Then 〈Imf〉 6⊂ Hj forall j ∈ {1, . . . , q} \ {t}. Thus, {Hj ∩ 〈Imf〉}qj=1j 6=tare hyperplanes in 〈Imf〉 inn-subgeneral position.By the Cartan-Nochka Second Main Theorem, we have(q − 2n+ dim〈Imf〉 − 2)Tf(r) ≤q∑j=1j 6=tN[dim〈Imf〉](f,Hj)(r) + o(Tf(r)) = o(Tf(r)).5This is a contradiction to the fact that q > 2n+ 2.Since {Hj}n+1j=1 and {Lj}n+1j=1 are families of hyperplanes in general position,f˜ :=((f,H1) : · · · : (f,Hn+1))and g˜ :=((g, L1) : · · · : (g, Ln+1))arereduced representations of meromorphic mappings f˜ and g˜ respectively ofCm into CP n. Furthermore, dim〈Imf〉 = dim〈Imf˜〉, dim〈Img〉 = dim〈Img˜〉,Tf˜(r) = Tf(r) +O(1) and Tg˜(r) = Tg(r) +O(1).By assumptions a) and c) we thatf˜ = g˜ on ∪qj=1 f−1(Hj). (3.1)We now prove thatdim〈Imf〉 = dim〈Img〉 Def.= p. (3.2)This is equivalent to prove that dim〈Imf˜〉 = dim〈Img˜〉. Therefore, it sufficesto show that for any hyperplane H in CP n then(H, f˜) ≡ 0 if and only if (H, g˜) ≡ 0.Suppose that the above assertion does not hold. Without loss of the gener-ality, we may assume that there exists a hyperplane H such that (H, f˜) 6≡ 0and (H, g˜) ≡ 0. Then by (3.1) we have(f˜ , H) = 0 on ∪qj=1 f−1(Hj). (3.3)By (3.3) and by the First Main Theorem and the Cartan-Nochka SecondMain Theorem we have(q − 2n+ dim〈Imf〉 − 1)Tf(r) +O(1) ≤q∑j=1N[dim〈Imf〉](f,Hj)(r) + o(Tf (r))≤ dim〈Imf〉q∑j=1N[1](f,Hj)(r) + o(Tf(r))(3.3)≤ dim〈Imf〉N(f˜ ,H)(r) + o(Tf(r))≤ dim〈Imf〉Tf˜(r) + o(Tf(r))= dim〈Imf〉Tf(r) + o(Tf (r)).This is a contradiction to the fact that q > 2n + 2. We complete the proofof (3.2).6Now we prove that(f,H1)(g, L1)≡ · · · ≡ (f,Hq)(g, Lq). (3.4)We distinguish the following two cases:Case 1: There exists a subset J := {j0, . . . , jn} ⊂ {1, . . . , q} such that(f,Hj0)(g, Lj0)≡ · · · ≡ (f,Hjn)(g, Ljn)Def.≡ u .We have Pole(u)∪Zero(u) ⊂ f−1(Hj0)∩f−1(Hj1), which is an analytic setof codimension at least 2 by assumption b). Hence, Pole(u) ∪ Zero(u) = ∅.Since Hj0, ..., Hjn are hyperplanes in general position, F :=((f,Hj0) :· · · : (f,Hjn))is the reduced representation of a meromorphic mapping F ofCm into CP n. Still by the same reason TF (r) = Tf(r) +O(1).Suppose that (3.4) does not hold. Then, there exists i0 ∈ {1, . . . , q} \{j0, . . . , jn} such that(f,Hi0)(g, Li0)6≡ u. (3.5)Since the families {Hj}qj=1 and {Lj}qj=1 are in general position, there existhyperplanes H i0 : a0ω0 + · · ·+ anωn = 0, Li0 : b0ω0 + · · ·+ bnωn = 0 in CP nsuch that (f,Hi0) ≡ (F,H i0), and (g, Li0) ≡ b0(g, Lj0) + · · · + bn(g, Ljn) ≡(F,Li0)u. Therefore, by (3.5) we have(F,H i0)(F, Li0)≡ (f,Hi0)u(g, Li0)6≡ 1.By assumption c) and since Pole(u) ∪ Zero(u) = ∅, we have u = (f,Hj0 )(g,Lj0 )=(f,Hi0 )(g,Li0)= u (F,Hi0 )(F,Li0)on(⋃qk=1 f−1(Hk))\(f−1(Hi0)∪f−1(Hj0)) and u = (f,Hj1 )(g,Lj1 ) =(f,Hi0 )(g,Li0)= u (F,Hi0 )(F,Li0)on(⋃qk=1 f−1(Hk))\(f−1(Hi0)∪f−1(Hj1)). Then (F,Hi0 )(F,Li0) = 1on(⋃qk=1 f−1(Hk)) \ f−1(Hi0).Therefore,q∑k=1,k 6=i0N[1](f,Hk)(r) ≤ N (F,Hi0)(F,Li0)−1(r)≤ T (F,Hi0)(F,Li0)(r) +O(1) ≤ TF (r) +O(1) = Tf(r) +O(1).7Therefore, by the Cartan-Nochka Second Main Theorem we haveTf (r) +O(1) ≥q∑k=1,k 6=i0N[1](f,Hk)(r) ≥q∑k=1,k 6=i01pN[p](f,Hk)(r)≥ q − 2n+ p− 2pTf (r)− o(Tf(r)).This implies that q ≤ 2n+ 2. This is a contradiction. Hence, we get (3.4) inthis case.Case 2: For any subset J ⊂ {1, . . . , q} with #J = n + 1, there exists apair i, j ∈ J such that(f,Hi)(g, Li)6≡ (f,Hj)(g, Lj).We introduce an equivalence relation on L := {1, · · · , q} as follows: i ∼ j ifand only ifdet(f,Hi) (f,Hj)(g, Li) (g, Lj) ≡ 0.Set {L1, · · · , Ls} = L/ ∼. It is clear that ♯Lk ≤ n for all k ∈ {1, · · · , s}.Without loss of generality, we may assume that Lk := {ik−1 + 1, · · · , ik}(k ∈ {1, · · · , s}) where 0 = i0 < · · · < is = q.We define the map σ : {1, · · · , q} → {1, · · · , q} byσ(i) ={i+ n if i+ n ≤ q,i+ n− q if i+ n > q.It is easy to see that σ is bijective and | σ(i)− i |≥ n (note that q > 2n+2).This implies that i and σ(i) belong to distinct sets of {L1, · · · , Ls}. Thisimplies that for all i ∈ {1, . . . , q},Pi := det(f,Hi) (f,Hσ(i))(g, Li) (g, Lσ(i)) 6≡ 0.By the assumption and by the definition of function Pi, we haveνPi ≥ min{ν(f,Hi), ν(g,Li)}+min{ν(f,Hσ(i)), ν(g,Lσ(i))}+q∑j=1j 6=i,σ(i)ν[1](f,Hj)(3.6)8outside an analytic set of codimension ≥ 2.On the other hand, since f−1(Hk) = g−1(Lk) we havemin{ν(f,Hk), ν(g,Lk)} ≥ min{ν(f,Hk), p}+min{ν(g,Lk), p} − pmin{ν(f,Hk), 1}= ν[p](f,Hk)+ ν[p](g,Lk)− pν [1](f,Hk)for k ∈ {i, σ(i)}.Therefore, by (3.6) we haveνPi ≥ ν [p](f,Hi) + ν[p](g,Li)+ ν[p](f,Hσ(i))+ ν[p](g,Lσ(i))− pν [1](f,Hi) − pν[1](f,Hσ(i))+q∑j=1j 6=i,σ(i)ν[1](f,Hj)outside an analytic set of codimension ≥ 2.Then for all i ∈ {1, . . . , q} we haveNPi(r) ≥ N [p](f,Hi)(r) +N[p](g,Li)(r) +N[p](f,Hσ(i))(r) +N[p](g,Lσ(i))(r)− pN [1](f,Hi)(r)− pN[1](f,Hσ(i))(r) +q∑j=1j 6=i,σ(i)N[1](f,Hj)(r). (3.7)On the other hand, by Jensen’s formulaNPi(r) =∫S(r)log |Pi|σ +O(1)≤∫S(r)log(|(f,Hi)|2 + |(f,Hσ(i))|2) 12σ+∫S(r)log(|(g, Li)|2 + |(g, Lσ(i))|2) 12σ +O(1)≤ Tf(r) + Tg(r) +O(1).Therefore, by (3.7) for all i ∈ {1, . . . , q} we haveN[p](f,Hi)(r) +N[p](g,Li)(r) +N[p](f,Hσ(i))(r) +N[p](g,Lσ(i))(r)− pN [1](f,Hi)(r)− pN[1](f,Hσ(i))(r) +q∑j=1j 6=i,σ(i)N[1](f,Hj)(r)≤ Tf(r) + Tg(r) +O(1). (3.8)9By summing-up of both sides of the above inequality for all i ∈ {1, . . . , q},we have2q∑j=1(N[p](f,Hj)(r) +N[p](g,Lj)(r))+ (q − 2p− 2)q∑j=1N[1](f,Hj)(r)≤ q(Tf (r) + Tg(r))+O(1). (3.9)Therefore, since f−1(Hj) = g−1(Lj) we have2q∑j=1(N[p](f,Hj)(r) +N[p](g,Lj)(r))+q − 2p− 22q∑j=1(N[1](f,Hj)(r) +N[1](g,Lj)(r))≤ q(Tf(r) + Tg(r))+O(1). (3.10)Then(2 +q − 2p− 22p) q∑j=1(N[p](f,Hj)(r) +N[p](g,Lj)(r)) ≤ q(Tf (r) + Tg(r))+O(1).(3.11)By (3.11) and by the Cartan-Nochka Second Main Theorem we have(q + 2p− 2)(q − 2n+ p− 1)2p(Tf (r) + Tg(r)) ≤ q(Tf (r) + Tg(r))+ o(Tf(r) + Tg(r)).It follows that (q+2p− 2)(q− 2n+ p− 1) ≤ 2pq. Then q2− (2n+3− p)q−2(p− 1)(2n+1− p) ≤ 0. This is a contradiction to condition (∗) of Theorem2. Thus we have completed the proof of (3.4).Assume that Hj : aj0ω0 + · · · + ajnωn = 0, Lj : bj0ω0 + · · · + bjnωn =0 (j = 1, . . . , q).SetA :=a10 . . . a1na20 . . . a2n.... . ....a(n+1)0 . . . a(n+1)n , B :=b10 . . . b1nb20 . . . b2n.... . ....b(n+1)0 . . . b(n+1)n , and L = B−1·A.By (3.4), we have A(f) ≡ B(g), so we get L(f) ≡ g.10Set H∗j = (aj0, . . . , ajn) ∈ Cn+1, L∗j = (bj0, . . . , bjn) ∈ Cn+1. We writeH∗j = αj1H∗1 + · · ·+ αj(n+1)H∗n+1 and L∗j = βj1L∗1 + · · ·+ βj(n+1)L∗n+1.By (3.4) we haveαj1(f,H1) + · · ·+ αj(n+1)(f,Hn+1)βj1(g, L1) + · · ·+ βj(n+1)(g, Ln+1) ≡(f,H1)(g, L1)≡ · · · ≡ (f,Hn+1)(g, Ln+1)for all j ∈ {1, . . . , q}.This implies that(αj1 − βj1)(f,H1) + · · ·+ (αj(n+1) − βj(n+1))(f,Hn+1) ≡ 0 (3.12)for all j ∈ {1, . . . , q}.On the other hand f : Cm −→ 〈Imf〉 is linearly nondegenerate and {Hj}n+1j=1are in general position in CP n. Thus, by (3.12) we have(αj1 − βj1)(ω,H1) + · · ·+ (αj(n+1) − βj(n+1))(ω,Hn+1) = 0 (3.13)for all ω ∈ 〈Imf〉 for all j ∈ {1, . . . , q}.Let hyperplanes αj : αj1ω0 + · · · + αj(n+1)ωn = 0 and βj : βj1ω0 + · · · +βj(n+1)ωn = 0 (j = 1, . . . , q).By (3.13) we have(A(ω), αj) = (A(ω), βj) (3.14)for all ω ∈ 〈Imf〉 and j ∈ {1, . . . , q}.For any j ∈ {1, . . . , q} and for any ω ∈ 〈Imf〉 we have(ω,Hj) = αj1(ω,H1) + · · ·+ αj(n+1)(ω,Hn+1)= (A(ω), αj)(3.14)= (A(ω), βj)= (B · L(ω), βj)= βj1(L(ω), L1) + · · ·+ βj(n+1)(L(ω), Ln+1)= (L(ω), Lj).This implies that L(〈Imf〉∩Hj) = Lj∩L(〈Imf〉) for all j ∈ {1, . . . , q}, whichcompletes the proof of Theorem 2. 211References[1] Z. Chen and Q. Yan, Uniqueness theorem of meromorphic maps intoPN(C) sharing 2N +3 hyperplanes regardless of multiplicities, Internat.J. Math. 20 (2009), 717–726.[2] G. Dethloff and T. V. Tan, Uniqueness theorems for meromorphic map-pings with few hyperplanes, Bulletin des Sciences Mathe´matiques, 133(2009), 501-514.[3] H. Fujimoto, The uniqueness problem of meromorphic maps into thecomplex projective space, Nagoya Math. J. 58 (1975), 1-23.[4] H. Fujimoto, Uniqueness problem with truncated multiplicities in valuedistribution theory, Nagoya Math. J. 152 (1998), 131-152.[5] S. Ji, Uniqueness problem without multiplicities in value distributiontheory, Pacific J. Math. 135 (1988), 323-348.[6] R. Nevanlinna, Einige Eindeutigkeitssa¨tze in der Theorie der mero-morphen Funktionen, Acta. Math. 48 (1926), 367-391.[7] J. Noguchi, A note on entire pseudo-holomorphic curves and the proofof Cartan-Nochka’s theorem, Kodai Math. J. 28 (2005), 336-346.[8] S. D. Quang, Unicity problem of meromorphic mappings sharing fewhyperplanes, preprint.[9] L. Smiley, Geometric conditions for unicity of holomorphic curves,Contemp. Math. 25 (1983), 149-154.[10] W. Stoll, On the propagation of dependences, Pacific J. Math. 139(1989), 311-337.[11] D. D. Thai and S. D. Quang, Uniqueness problem with truncated mul-tiplicities of meromorphic mappings in several complex variables, Inter.J. Math., 17 (2006), 1223-1257.Gerd Dethloff1−21 Universite´ Europe´enne de Bretagne, France122 Universite´ de BrestLaboratoire de mathe´matiquesUMR CNRS 62056, avenue Le Gorgeu, BP 45229275 Brest Cedex, Francee-mail: gerd.dethloff@univ-brest.frSi Duc Quang and Tran Van TanDepartment of MathematicsHanoi National University of Education136-Xuan Thuy street, Cau Giay, Hanoi, Vietname-mails: ducquang.s@gmail.com and tranvantanhn@yahoo.com13

A Uniqueness Theorem for Meromorphic Mappings with Two Families of Hyperplanes

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