Generalization of a theorem of Clunie and Hayman

Clunie and Hayman proved that if the spherical derivative of an entire function has order of growth sigma then the function itself has order at most sigma+1. We extend this result to holomorphic curves in projective space of dimension n omitting n hyperplanes in general position.


Introduction
We consider holomorphic curves f : C → P n ; for the general background on the subject we refer to [7]. The Fubini-Study derivative f ′ measures the length distortion from the Euclidean metric in C to the Fubini-Study metric in P n . The explicit expression is where (f 0 , . . . , f n ) is a homogeneous representation of f (that is the f j are entire functions which never simultaneously vanish), and See [3] for a general discussion of the Fubini-Study derivative. * Supported by NSF grant DMS-055279 and by the Humboldt Foundation.
We recall that the Nevanlinna-Cartan characteristic is defined by where dm is the area element in C. So the condition lim sup Clunie and Hayman [4] found that for curves C → P 1 omitting one point in P 1 , a stronger conclusion follows from (1), namely In the most important case σ = 0, a different proof of this fact for n = 1 is due to Pommerenke [8]. Pommerenke's method gives the exact constant C(0). In this paper we prove that this phenomenon persists in all dimensions.
Theorem. For holomorphic curves f : C → P n omitting n hyperplanes in general position, condition (1) implies (3) with an explicit constant C(n, σ).
In [6], the case σ = 0 was considered. There it was proved that holomorphic curves in P n with bounded spherical derivative and omitting n hyperplanes in general position must satisfy T (r, f ) = O(r). With a stronger assumption that f omits n + 1 hyperplanes this was earlier established by Berteloot and Duval [2] and by Tsukamoto [9]. The proof in [6] has two drawbacks: it does not extend to arbitrary σ ≥ 0, and it is non-constructive; unlike Clunie-Hayman and Pommerenke's proofs mentioned above, it does not give an explicit constant in (3).
It is shown in [6] that the condition that n hyperplanes are omitted is exact: there are curves in any dimension n satisfying (1), T (r, f ) ∼ cr 2σ+2 and omitting n − 1 hyperplanes.

Preliminaries
Without loss of generality we assume that the omitted hyperplanes are given in the homogeneous coordinates by the equations {w j = 0}, 1 ≤ j ≤ n. We fix a homogeneous representation (f 0 , . . . , f n ) of our curve, where f j are entire functions, and f n = 1. Then is a positive subharmonic function, and Jensen's formula gives where n(t) = µ({z : |z| ≤ t}), and µ = µ u is the Riesz measure of u, that is the measure with the density This measure µ is also called Cartan's measure of f . Positivity of u and (2) imply that all f j are of order at most 2σ + 2, normal type. As f j (z) = 0, 1 ≤ j ≤ n we conclude that where P j are polynomials of degree at most 2σ + 2.
We need two lemmas from potential theory.

Lemma 1. [6]
Let v be a non-negative harmonic function in the closure of the disc B(a, R), and assume that v(z 1 ) = 0 for some point z 1 ∈ ∂B(a, R).

Lemma 2.
Let v be a non-negative superharmonic function in the closure of the disc B(a, R), and suppose that v(z 1 ) = 0 for some z 1 ∈ ∂B(a, R). Then Proof. Function v(a + Rz) satisfies the conditions of the lemma with R = 1. So it is enough to prove the lemma with a = 0 and R = 1. Let Minimizing |∂G/∂|z|| over |z| = 1 and |ζ| = 1/2 we obtain 1/3 which proves the lemma.

Proof of the theorem
We may assume without loss of generality that f 0 has infinitely many zeros. Indeed, we can compose f with an automorphism of P n , for example replace f 0 by f 0 + cf 1 , c ∈ C and leave all other f j unchanged. This transformation changes neither the n omitted hyperplanes nor the rate of growth of T (r, f ) and multiplies the spherical derivative by a bounded factor.
Put u j = log |f j |, and u * = max 1≤j≤n u j .
Here and in what follows max denotes the pointwise maximum of subharmonic functions.
Proposition 1. Suppose that at some point z 1 we have for some m = k and all j; m, k, j ∈ {0, . . . , n}. Then Proof.
and the conclusion of the proposition follows since |∇ log |f || = |f ′ /f |.
Proof. If u 0 (z) ≤ u * (z) for all sufficiently large |z|, then there is nothing to prove. Suppose that u 0 (a) > u * (a), and consider the largest disc B(a, R) centered at a where the inequality u 0 (z) > u * (z) persists. If z 0 is the zero of the smallest modulus of f 0 then R ≤ |a| + |z 0 | < (1 + ǫ)|a| when |a| is large enough.
Next we study the Riesz measure of the subharmonic function u * = max{u 1 , . . . , u n }.
We begin with maximum of two harmonic functions. Let u 1 and u 2 be two harmonic functions in C of the form u j = Re P j where P j = 0 are polynomials. Suppose that u 1 = u 2 . Then the set E = {z ∈ C : u 1 (z) = u 2 (z)} is a proper real-algebraic subset of C without isolated points. Apart from a finite set of ramification points, E consists of smooth curves. For every smooth point z ∈ E, we denote by J(z) the jump of the normal (to E) derivative of the function w = max{u 1 , u 2 } at the point z. This jump is always positive and the Riesz measure µ w is given by the formula which means that µ w is supported by E and has a density J(z)/2π with respect to the length element |dz| on E.
where the union is taken over all pairs 1 ≤ i, j ≤ n for which u i = u j . Then E is a proper real semi-algebraic subset of C, and ∞ is not an isolated point of E. For the elementary properties of semi-algebraic sets that we use here see, for example, [1,5]. There exists r 0 > 0 such that Γ = E ∩{r 0 < |z| < ∞} is a union of finitely many disjoint smooth simple curves, This union coincides with the support of µ u * in {z : r 0 < |z| < ∞}. Consider a point z 0 ∈ Γ. Then z 0 ∈ Γ k for some k. As Γ k is a smooth curve, there is a neighborhood D of z 0 which does not contain other curves Γ j , j = k and which is divided by Γ k into two parts, D 1 and D 2 . Then there exist i and j such that u * (z) = u i (z), z ∈ D 1 and u * (z) = u j (z), z ∈ D 2 , and u * (z) = max{u i (z), u j (z)}, z ∈ D. So the restriction of the Riesz measure µ u * on D is supported by Γ k ∩ D and has density J(z)/(2π) where |J(z)| = |∂u i /∂n − ∂u j /∂n|(z) = |∇(u i − u j )|(z), and ∂/∂n is the derivation in the direction of a normal to Γ k . Taking into account that u j = Re P j where P j are polynomials, we conclude that there exist positive numbers c k and b k such that Let b = max k b k , and among those curves Γ k for which b k = b choose one with maximal c k (which we denote by c 0 ). We denote this chosen curve by Γ 0 and fix it for the rest of the proof.
Case 2. u 0 (z) > u * (z) for all sufficiently large z ∈ Γ 0 . Let a be a point on Γ 0 , |a| > 3r 0 , and u 0 (a) > u * (a). Let B(a, R) be the largest open disc centered at a in which the inequality u 0 (z) > u * (z) holds. Then because we assume that f 0 has zeros, so u 0 (z 0 ) = −∞ for some z 0 .
In B(a, R) we consider the positive superharmonic function v = u 0 − u * . Let us check that it satisfies the conditions of Lemma 2. The existence of a point z 1 ∈ ∂B(a, R) with v(z 1 ) = 0 follows from the definition of B(a, R). The Riesz measure of µ v is estimated using (7), (8): On the other hand (1) and Proposition 1 imply that |∇v(z 1 )| ≤ K(n + 1)(|a| + R) σ Combining these two inequalities and taking (9) into account, we obtain b ≤ σ and c 0 ≤ 3 · 4 σ K(n + 1), as required.
Proof. By Jensen's formula, where ν(t) = µ u * ({z : |z| ≤ t}). The number of curves Γ k supporting the Riesz measure of u * is easily seen to be at most 2n(n − 1)(σ + 1). The density of the Riesz measure µ u * on each curve Γ k is given by (8), where c k ≤ c 0 and b k ≤ b, and the parameters c 0 and b are estimated in Proposition 3. Combining all these data we obtain the result.
It remains to combine Propositions 2 and 4 to obtain the final result.
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