A counterexample to Cartan’s conjecture on holomorphic curves omitting hyperplanes

In his 1928 thesis H. Cartan proved a theorem which can be considered as an extension of Montel's normality criterion to holomorphic curves in complex projective plane p2. He also conjectured that a similar result is true for holomorphic curves in pn for any n. A counterexample to this conjecture is constructed for any n > 3. The following theorem of Borel may be considered as an extension of Picard's theorem to holomorphic mappings of the complex plane C to complex projective space. Borel's Theorem. Let fi,... , fp be a system of entire functions without zeros and (1) fi + + fp = ? Then the set of indices {1,... ,p} can be partitioned into disjoint subsets {I} sueh that III > 2, and for every I the functions fj, j E I, are proportional and their sum is zero. According to the so-called Bloch principle, to every theorem of Picard type should correspond a Montel-type theorem for families of functions in the unit disk. The following statement is known as Cartan's Conjecture ([2, 3]). Let .F be an infinite family of p-tuples of holomorphic functions f = (fi, ... , fp) without zeros in the unit disk U satisfying the Borel equation (1). Then there exists an infinite subsequence L having the following property. There exists a partition of indices P = {1, . . . , p} into disjoint sets {S} and each S contains a subset I with at least two elements, which may be equal to S itself. These satisfy the following properties for f E C: (i) For each S and j, k E I C S the sequence {fj/fk} is convergent (uniformly on compacta, to a non-zero function). (ii) If j E S\I and k E I C S then fj/fk converges to 0. (iii) Given k E I C S, jE fj/fk converges to 0. jEI Received by the editors March 29, 1995. 1991 Mathematics Subject Classification. Primary 30D45; Secondary 32H30. @)1996 American Mathematical Society 3097 This content downloaded from 157.55.39.211 on Tue, 27 Sep 2016 04:10:04 UTC All use subject to http://about.jstor.org/terms 3098 ALEXANDRE EREMENKO When p = 3 the statement is (almost) equivalent to the Montel theorem, which asserts that a family of meromorphic functions in the unit disk omitting three given values is normal. Cartan [2], see also [3, Ch. VIII], proved a partial result: Let F be as above. Then there exists a subsequence ? C Y having one of the following properties: (a) The full set P of indices satisfies (i), (ii) and (iii) (with single set S = P), or (b) There are two disjoint subsets S1 and S2 in P, each containing at least two elements, satisfying the three conditions (i), (ii) and (iii). The point is that S1 and S2 in (b) may not cover the whole set of indices P. This result implies that Cartan's conjecture is true for p = 3 and p = 4 [2]. We show that it fails for p = 5. Example. It is convenient to work in the rectangle R = {x + iy xl 12 > 4e consider the function h(z) = hn(z) = exp(nexpiz), z E R. We have log Ihn(x + iy)I = ncosxexp(-y). The set {z E R: I hn(z)I ir/2}. Choose a diffeomorphism p of the disk {w: IwI 0, 2 0} and Gn has no zeros in R n {x O and (6) logIGn(x+iy)I = (n+o(l))cosxexp(-y), x O and (9) loglHn(x+iy)-11 = (n+o(l))cosxexp(-y), x<0, as n -4 oo uniformly on R. Now we set a = 7r1/(e + 1) and define fn(z) exp n (z + a) fr(z) = exp fn(-z + a)} fn = Gn -fni fn4 =Hn fn fn5(z)--1. From this definition and (7) follows that (1) is satisfied. Furthermore we have in view of (5), (6), (8) and (9) (10) IGnl < Ifnl and IHnl < Ifnl in f for n large enough. Inequalities (10) show that all five functions fi are zero-free in R if n is large enough. Now we show that the conclusion of Cartan's conjecture is not valid for the functions of our sequence. This is because fn cannot be in the same class S with any other function fn{, 1 < j < 4. Indeed, when j is odd we have log Ifn(z)I = (n + o(l))(Rz + a), n -oX,