Wiman and Arima theorems for quasiregular mappings

Generalizations of the theorems of Wiman and of Arima on entire functions are proved for spatial quasiregular mappings.


Main Results
It follows from the Ahlfors theorem that an entire holomorphic function f of order ρ has no more than 2ρ distinct asymptotic curves where r stands for the largest integer ≤ r. This theorem does not give any information if ρ < 1/2, This case is covered by two theorems: if an entire holomorphic function f has order ρ < 1/2 then lim sup r → ∞ min |z| r |f z | ∞ Wiman 1 and if f is an entire holomorphic function of order ρ > 0 and l is a number satisfying the conditions 0 < l ≤ 2π, l < π/ρ, then there exists a sequence of circular arcs {|z| r k , θ k ≤ arg z ≤ θ k l}, r k → ∞, 0 ≤ θ k < 2π, along which |f z | tends to ∞ uniformly with respect to arg z Arima 2 .
Below we prove generalizations of these theorems for quasiregular mappings for n ≥ 2. The next two theorems are generalizations of the theorems of Wiman and of Arima for quasiregular mappings on manifolds.
The proofs of these results are based upon Phragmén-Lindelöf's and Ahlfors' theorems for differential forms of WT-classes obtained in 3 .
For n-harmonic functions on abstract cones, similar theorems were obtained in 4 . Our notation is as in 3, 5 . We assume that the results of 3 are known to the reader and we only recall some results on qr-mappings. The quantity is called the maximal dilatation of F and if K F ≤ K, then the mapping F is called Kquasiregular.
If F : M → N is a quasiregular homeomorphism, then the mapping F is called quasiconformal. In this case, the inverse mapping F −1 is also quasiconformal in the domain F M ⊂ N and K F −1 K F . Let A and B be Riemannian manifolds of dimensions dim A k and dim B n − k, 1 ≤ k < n, and with scalar products , A , , B , respectively. The Cartesian product N A×B has the natural structure of a Riemannian manifold with the scalar product We denote by π : A × B → A and η : A × B → B the natural projections of the manifold N onto submanifolds.
If w A and w B are volume forms on A and B, respectively, then the differential form w N π * w A ∧ η * w B is a volume form on N.
Theorem 2.1 see 5 . Let F : M → N be a quasiregular mapping and let f π • F : M → A.
Then the differential form f * w A is of the class WT 2 on M with the structure constants p n/k, ν 1 ν 1 n, k, K O , and ν 2 ν 2 n, k, K O .
Let D be an unbounded domain in R n and let f f 1 , f 2 , . . . , f n : D → R n be a quasiregular mapping. We assume that f ∈ C 0 D . It is natural to consider the Phragmén-Lindelöf alternative under the following assumptions: Several formulations of the Phragmén-Lindelöf theorem under various assumptions can be found in 7-11 . However, these results are mainly of qualitative character. Here we give a new approach to Phragmén-Lindelöf type theorems for quasiregular mappings, based on isoperimetry, that leads to almost sharp results. Our approach can be used to prove Phragmén-Lindelöf type results for quasiregular mappings of Riemannian manifolds.
Let N be an n-dimensional noncompact Riemannian C 2 -manifold with piecewise smooth boundary ∂N possibly empty . A function u ∈ C 0 N ∩ W 1 n,loc N is called a growth function with N as a domain of growth if i u ≥ 1, ii u | ∂N 1 if ∂N / ∅, and sup y∈N u y ∞. Inequalities and Applications   5 We consider a quasiregular mapping f : M → N, f ∈ C 0 M ∪ ∂M , where M is a noncompact Riemannian C 2 -manifold, dim M n, and ∂M / ∅. We assume that f ∂M ⊂ ∂N.

Journal of
In what follows, we mean by the Phragmén-Lindelöf principle an alternative of the form: either the function u f m has a certain rate of growth in M or f m ≡ const.
By choosing the domain of growth N and the growth function u in a special way, we can obtain several formulations of Phragmén-Lindelöf theorems for quasiregular mappings. In view of the examples in 7 , the best results are obtained if an n-harmonic function is chosen as a growth function. In the case a , the domain of growth is N {y y 1 , . . . , y n ∈ R n : y 1 ≥ 0} and as the function of growth, it is natural to choose u y y 1 1; in the case b , the domain N is the set {y y 1 , . . . , y n ∈ R n :

Exhaustion Functions
Below we introduce exhaustion and special exhaustion functions on Riemannian manifolds and give illustrating examples.

Exhaustion Functions of Boundary Sets
Let h : Let h : M → R be a locally Lipschitz function such that there exists a compact K ⊂ M with |∇h x | > 0 for a.e. m ∈ M \ K. We say that the function h is an exhaustion function for a boundary set Ξ of M if for an arbitrary sequence of points m k ∈ M, k 1, 2, . . . , the function h m k → h 0 if and only if m k → ξ.
It is easy to see that this requirement is satisfied if and only if for an arbitrary increasing sequence t 1 < t 2 < · · · < h 0 , the sequence of the open sets V k {m ∈ M : h m > t k } is a chain, defining a boundary set ξ. Thus the function h exhausts the boundary set ξ in the traditional sense of the word.
The function h : M → 0, h 0 is called the exhaustion function of the manifold M if the following two conditions are satisfied: {B h t k } generates an exhaustion of M, that is,

Special Exhaustion Functions
Let M be a noncompact Riemannian manifold with the boundary ∂M possibly empty . Let A satisfy 3.2 and 3.3 and let h : M → 0, h 0 be an exhaustion function, satisfying the following additional conditions: Here dH n−1 is the element of the n − 1 -dimensional Hausdorff measure on Σ h . Exhaustion functions with these properties will be called the special exhaustion functions of M with respect to A. In most cases, the mapping A will be the p-Laplace operator 3.8 and, unless otherwise stated, A is the p-Laplace operator.
Since the unit vector ν ∇h/|∇h| is orthogonal to the h-sphere Σ h , the condition a 2 means that the flux of the vector field A m, ∇h through h-spheres Σ h t is constant.
In the following, we consider domains D in R n as manifolds M. However, the boundaries ∂D of D are allowed to be rather irregular. To handle this situation, we introduce A, h -transversality property for M.
Let h : M → 0, h 0 be a C 2 -exhaustion function. We say that M satisfies the A, htransversality property if for a.e. t 1 , t 2 , h < t 1 < t 2 < h 0 , and for every ε > 0, there exists an open set with piecewise regular boundary such that where v is the unit inner normal to ∂G.
be a cylinder with base D. The function h : 0, ∞ → R, h x x 3 , is an exhaustion function for M. Since every domain D in R 2 can be approximated by smooth domains D from inside, it is easy to see that for 0 < t 1 < t 2 < ∞ the domain G D × t 1 , t 2 can be used as an approximating domain G ε t 1 , t 2 . Note that the transversality condition 4.8 is automatically satisfied for the p-Laplace operator A m, ξ |ξ| p−2 ξ.
. . , x n be an orthonormal system of coordinates in R n , 1 ≤ n < p. Let D ⊂ R n be an unbounded domain with piecewise smooth boundary and let B be a p − n -dimensional compact Riemannian manifold with or without boundary. We consider the manifold M D × B. We denote by x ∈ D, b ∈ B, and x, b ∈ M the points of the corresponding manifolds. Let π : D × B → D and η : D × B → B be the natural projections of the manifold M.
Assume now that the function h is a function on the domain D satisfying the conditions b 1 , b 2 , and 3.8 . We consider the function h * h • π : M → 0, ∞ .
We have

4.12
Journal of Inequalities and Applications 9 Because h is a special exhaustion function of D, we have div |∇h * | p−2 ∇h * 0.

4.13
Let x, b ∈ ∂M be an arbitrary point where the boundary ∂M has a tangent hyperplane and let ν be a unit normal vector to ∂M.
If x ∈ ∂D, then ν ν 1 ν 2 where the vector ν 1 ∈ R k is orthogonal to ∂D and ν 2 is a vector from T b B . Thus 14 because h is a special exhaustion function on D and satisfies the property b 2 on ∂D. If b ∈ ∂B, then the vector ν is orthogonal to ∂B × R n and The other requirements for a special exhaustion function for the manifold M are easy to verify. Therefore, the function is a special exhaustion function on the manifold M D × B.
Example 4.5. We fix an integer k, 1 ≤ k ≤ n, and set

4.17
It is easy to see that |∇d k x | 1 everywhere in R n \ Σ 0 , where Σ 0 {x ∈ R n : d k x 0}. We shall call the set a k-ball and the set We shall say that an unbounded domain D ⊂ R n is k-admissible if for each t > inf x∈D d k x , the set D ∩ B k t has compact closure.
It is clear that every unbounded domain D ⊂ R n is n-admissible. In the general case, the domain D is k-admissible if and only if the function d k x is an exhaustion function of D.
It is not difficult to see that if a domain D ⊂ R n is k-admissible, then it is l-admissible for all k < l < n.
Fix 1 ≤ k < n. Let Δ be a bounded domain in the n − k -plane x 1 · · · x k 0 and let The domain D is k-admissible. The k-spheres Σ k t are orthogonal to the boundary ∂D and therefore ∇d k , ν 0 everywhere on the boundary.  Let h φ d k where φ is a C 2 -function with φ ≥ 0. We have ∇h φ ∇d k and since |∇d k | 1, we obtain

4.23
From the equation is a special exhaustion function for the manifold M. Therefore, for p ≥ n, the given manifold has p-parabolic type and for p < n, p-hyperbolic type.
Example 4.9. Let r, θ , where r ≥ 0, θ ∈ S n−1 1 , be the spherical coordinates in R n . Let U ⊂ S n−1 1 be an arbitrary domain on the unit sphere S n−1 1 . We fix 0 ≤ r 1 < r 2 < ∞ and consider the domain with the metric where α r , β r > 0 are C 0 -functions on r 1 , r 2 and dl θ is an element of length on S n−1 1 . The manifold M D, ds 2 M is a warped Riemannian product. In the cases, α r ≡ 1, β r 1, and U S n−1 the manifold M is isometric to a cylinder in R n 1 . In the cases, α r ≡ 1, β r r, U S n−1 the manifold M is a spherical annulus in R n . The volume element in the metric 4.30 is given by the expression dσ M α r β n−1 r dr dS n−1 1 .

4.31
If φ r, θ ∈ C 1 D , then the length of the gradient ∇φ in M takes the form where ∇ θ φ is the gradient in the metric of the unit sphere S n−1 1 . For the special exhaustion function h r, θ ≡ h r , 3.8 reduces to the following form:  ii if h 0 < ∞, the set ξ has p-hyperbolic type.

4.38
Journal of Inequalities and Applications 13 is a quantity independent of t > h K sup{h m : m ∈ K}. Indeed, for the variational problem 3, 2.9 , we choose the function ϕ 0 , ϕ 0 m 0 for m ∈ B h t 1 ,

4.40
Because the special exhaustion function satisfies 3.8 and the boundary condition a 2 , one obtains for arbitrary τ 1 , τ 2

4.41
Thus we have established the inequality By the conditions, imposed on the special exhaustion function, the function ϕ 0 is an extremal in the variational problem 3, 2.9 . Such an extremal is unique and therefore the preceding inequality holds as an equality. This conclusion proves 4.37 .
If h 0 ∞, then letting t 2 → ∞ in 4.37 we conclude the parabolicity of the type of ξ. Let h 0 < ∞. Consider an exhaustion {U k } and choose t 0 > 0 such that the h-ball B h t 0 contains the compact set K.
Set t k sup m∈∂U k h m . Then for t k > t 0 , we have and hence lim inf 4.44 and the boundary set ξ has p-hyperbolic type.

Wiman Theorem
Now we will prove Theorem 1.1.

Fundamental Frequency
Let U ⊂ Σ h τ be an open set. We need further the following quantity: where the infimum is taken over all functions ϕ ∈ W 1 p U with supp ϕ ⊂ U By the definition, ϕ is a W 1 p -function on an open set U, if ϕ belongs to this class on every component of U. . Here ∇ 2 ϕ is the gradient of ϕ on the surface Σ h τ .
In the case |∇h| ≡ 1, this quantity is well-known and can be interpreted, in particular, as the best constant in the Poincaré inequality. Following 14 , we shall call this quantity the fundamental frequency of the rigidly supported membrane U.
Observe a useful property of the fundamental frequency.
We also need the following statement.

Lemma 5.2.
Under the above assumptions for a.e. τ ∈ 0, h 0 , we have where λ p is the fundamental frequency of the membrane Σ h τ defined by formula 5.1 and

5.11
For the proof, see Lemma 4.3 in 10 .
We now use these estimates for proving Phragmén-Lindelöf type theorems for the solutions of quasilinear equations on manifolds.

5.13
Then either f m ≤ 0 everywhere on M or In particular, if h is a special exhaustion function on M, then Here where O τ {m ∈ O : τ < h m < τ 1}. By Lemma 5.2, the following inequality holds

5.21
Thus using the requirement 3.3 for 3.4 , we arrive at the estimate Further we observe that from the condition f m > f m 1 |∇h| f m ∇f m p−1 * 1.

5.23
From this relation, we arrive at 5.14 .
The proof of 5.15 is carried out exactly in the same way by means of the inequality 3, 5.75 .
In order to convince ourselves of the validity of 5.16 , we observe that by the maximum principle we have |∇h m | p * 1.

5.24
But h is a special exhaustion function and therefore by 4.37 we can write where J is a number independent of τ. The relation 5.15 implies then that 5.16 holds.
Example 5.4. Let A be a compact Riemannian manifold with nonempty piecewise smooth boundary, dim A k ≥ 1, and let M A × R n , n ≥ 1. Choosing as a special exhaustion function of M the function h a, x , defined in Example 4.8, we have

5.26
Then using the fact that h a, x | Σ h t t, we find for p n p − n p − 1 t 1−n / p−n for p / n.

Journal of Inequalities and Applications
Therefore, on the basis of 5.1 we get

5.29
where dσ A is an element of k-dimensional area on A. Therefore, and we obtain where the infimum is taken over all functions ψ ψ a, x with ψ a, x ∈ W 1 p A × S n−1 1 , ψ a, x | a∈∂A 0, ∀x ∈ S n−1 1 .

5.32
In the particular case n 1, Theorem 5.3 has a particularly simple content.
Here h x is a function of one variable, and Σ h t A × S 0 t is isometric to Σ h 1 . Therefore, h t ≡ 1 and by 5.31 we have

5.33
In the same way, 5.16 can be written in the form Let n ≥ 2. We do not know of examples where the quantity 5.31 had been exactly computed. Some idea about the rate of growth of the quantity M τ in the Phragmén-Lindelöf alternative can be obtained from the following arguments. Simplifying the numerator of 5.31 by ignoring the second summand, we get the estimate

5.35
For each fixed x ∈ S n−1 1 , the function ψ a, x is finite on A, because from the definition of the fundamental frequency it follows that

Proof of Theorem 1.1
We assume that

22
Journal of Inequalities and Applications Example 5.6. As the first corollary, we shall now prove a generalization of Wiman's theorem for the case of quasiregular mappings f : M → R n where M is a warped Riemannian product.
For 0 ≤ r 1 < r 2 ≤ ∞, let D m r, θ ∈ R n : r 1 < r < r 2 , θ ∈ S n−1 1 5.53 be a ring domain in R n and let M r 1 , r 2 × S n−1 1 be an n-dimensional Riemannian manifold on D with the metric is a special exhaustion function on M.
Let f : M → R n be a quasiregular mapping. We set u y log |y|. This function is a subsolution of 3.4 with p n and also satisfies all the other requirements imposed on a growth function.

5.57
Therefore, the requirement 1.1 on the manifold will be fulfilled if In this way, we get the following corollary.

5.63
We assume that in Example 5.8 the quantities r 1 0, r 2 ∞, and the functions α r ≡ 1, β r r, that is, the manifold is R n . As the special exhaustion function, we choose h log |x|. This function satisfies 3.6 with p n and ν 1 ν 2 1. The condition 5.58 for the manifold is obviously fulfilled.
The condition 5.62 attains the form

5.65
We have the following corollary.

Asymptotic Tracts and Their Sizes
Wiman's theorem for the quasiregular mappings f : R n → R n asserts the existence of a sequence of spheres S n−1 r k , r k → ∞, along which the mapping f x tends to ∞. It is possible to further strengthen the theorem and to specify the sizes of the sets along which such a convergence takes place. For the formulation of this result it is convenient to use the language of asymptotic tracts discussed by MacLane 15 .

Tracts
Let Let f : M → R n be a quasiregular mapping having a point a ∈ R n as a Picard exceptional value, that is, f m / a and f m attains on M all values of B a, r \ {a} for some r > 0.
The set {∞} ∪ {a} has n-capacity zero in R n and there is a solution g y in R n \ {a} of 3.4 such that g y → ∞ as y → a or y → ∞ cf. 12, Chapter 10, polar sets . As the growth function on R n \ {a}, we choose the function u y max 0, g y . It is clear that this function is a subsolution of 3.4 in R n \ {a}.
The there exists at least one M s having a nonempty intersection with f −1 B a, r . Then by the maximum principle for subsolutions, such a component cannot be relatively compact. Letting s → ∞, we find an asymptotic tract {M s }, along which a quasiregular mapping tends to a Picard exceptional value a ∈ R n .
Because one can find in every asymptotic tract a curve Γ along which u f m → ∞, we obtain the following generalization of Iversen's theorem 16 .
Theorem 6.1. Every Picard exceptional value of a quasiregular mapping f : M → R n is an asymptotic value.
The classical form of Iversen's theorem asserts that if f is an entire holomorphic function of the plane, then there exists a curve Γ tending to infinity such that f z −→ ∞ as z −→ ∞ on Γ. 6.4 We prove a generalization of this theorem for quasiregular mappings f : M → N of Riemannian manifolds.
The following result holds.

Proof of Theorem 1.2
We fix a growth function u and a special exhaustion function h as in Section 4. Let f : M → N be a nonconstant quasiregular mapping. We set

M τ max
h m τ u f m . 6.7