Ahlfors theorems for differential forms

Some counterparts of theorems of Phragm\'en-Lindel\"of and of Ahlfors are proved for differential forms of ${\cal WT}$--classes.}

with a compact supp ϕ in M and such that supp ϕ ∩ ∂M = ∅, we have M w, δϕ dv = 0 .
Here δϕ = (−1) k ⋆ −1 d ⋆ ϕ, k = deg ϕ and ⋆α is the orthogonal complement of a differential form α on a Riemannian manifold M. A weakly closed form w of the kind (1.1) is said to be of the class WT 1 on M if there exists a weakly closed differential form θ ∈ L q loc (M), deg θ = n − k, such that almost everywhere on M we have ν 0 |θ| q ≤ w, * θ (1. 3) for some constant ν 0 .
For a proof see [4]. The following partial integration formula for differential forms is useful [4].
Let A and B be Riemannian manifolds of dimensions dim A = k, 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 , = , A + , B .
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 .
Let y 1 , . . . , y k be an orthonormal system of coordinates in R k , 1 ≤ k ≤ n. Let A be a domain in R k and let B be an (n − k)-dimensional Riemannian manifold. We consider the manifold N = A × B.

Boundary sets
Below we introduce the notions of parabolic and hyperbolic type of boundary sets on noncompact Riemannian manifolds and study exhaustion functions of such sets. We also present some illuminating examples.
Let M be an n-dimensional noncompact Riemannian manifold without boundary. Boundary sets on M are analogies to prime ends due to Caratheodory (cf. e.g. [16]).
Let {U k }, k = 1, 2, . . . be a collection of open sets U k ⊂ M with the following properties: (i) for all k = 1, 2, . . . U k+1 ⊂ U k , A sequence with these properties will be called a chain on the manifold M.
Let {U ′ k }, {U ′′ k } be two chains of open sets on M. We shall say that the chain U ′ k is contained in the chain {U ′′ k }, if for each m ≥ 1 there exists a number k(m) such that for all k > k(m) we have U ′ k ⊂ U ′′ m . Two chains, each of which is contained in the other one, are called equivalent. Each equivalence class ξ of chains is called a boundary set of the manifold M. To define ξ it is enough to determine at least one representative in the equivalence class. If the boundary set ξ is defined by the chain {U k }, then we shall write ξ ≍ {U k }.
A sequence of points m k ∈ M converges to ξ if for some (and, therefore, all) chain {U k } ∈ ξ the following condition is satisfied: for every k = 1, 2, . . . there exists an integer n(k) such that m n ∈ U k for all n > n(k). A sequence (m n ) lies off a boundary set ξ ≍ {U k }, if for every k = 1, 2, . . . there exists a number n(k) such that for all n > n(k) m n / ∈ U k .
A boundary set ξ ≍ {U k } is called a set of ends of the manifold M if each of {U k } has a compact boundary ∂U k . If in addition each of the sets U k is connected, then ξ ≍ {U k } is called an end of the manifold M. We fix p ≥ 1. The p-capacity of the condenser (A, B; D) is defined by where the infimum is taken over the set of all continuous functions of class It is easy to see that for a pair (A, B; D) and ( Let A, B be compact in D. A standard approximation method shows that cap p (A, B; D) does not change if one restricts the class of functions in the variational problem (2.2) to Lipschitz functions equal to 0 and 1 in the sets A and B, respectively.
Let {U k } be an arbitrary chain on a manifold M. We fix a subdomain H ⊂⊂ M. If k is sufficiently large, the intersection H ∩ U k = ∅ and we consider the condenser (H, U k ; M). Then it is clear that for k = 1, 2, . . .
We shall say that the chain {U k } on M has p-capacity zero, if for every subdomain H ⊂⊂ M we have We shall say that a boundary set ξ is of p-parabolic type if every chain {U k } ≍ ξ is of p-capacity zero. A boundary set ξ is of α-hyperbolic type if at least one of the chains {U k } ∈ ξ is not of p-parabolic type. Let be an arbitrary exhaustion of the manifold M by subdomains {U k }. The manifold M is of p-parabolic or p-hyperbolic type depending on the pparabolicity or p-hyperbolicity of the boundary set {M \ U k }. It is well-known, see [7], that a noncompact Riemannian manifold M without boundary is of p-parabolic type if and only if every solution of the inequality div M (|∇u| p−2 ∇u) ≥ 0 which is bounded from above is a constant. The classical parabolicity and hyperbolicity coincides with 2-parabolicity and 2-hyperbolicity, respectively. Therefore whenever we refer to parabolic or hyperbolic type (of a manifold or a boundary set) we mean 2-parabolicity or 2-hyperbolicity.
2.4. Example. The space R n is of p-parabolic type for p ≥ n and p-hyperbolic type for p < n.
We next present a proposition that provides a convenient method of verifying the p-parabolicity and p-hyperbolicity of boundary sets. [14]. Let ξ be a boundary set on M. If for a chain {U k ≍ ξ} and for a nonempty open set H 0 ⊂⊂ M the condition (2.3) holds, then the boundary set ξ is of p-parabolic type.

A-solutions .
Let M be a Riemannian manifold and let be a mapping defined a.e. on the tangent bundle T (M). Suppose that for a.e. m ∈ M the mapping A is continuous on the fiber T m , i.e. for a.e. m ∈ M the function A(m, ·) : ξ ∈ T m → T m is defined and continuous; the mapping m → A m (X) is measurable for all measurable vector fields X (see [6]). Suppose that for a.e. m ∈ M and for all ξ ∈ T m the inequalities and |A(m, ξ)| ≤ ν 2 |ξ| p−1 (2.8) hold with p > 1 and for some constants ν 1 , ν 2 > 0. It is clear that we have We consider the equation div A(m, ∇f ) = 0. (2.9) Solutions to (2.9) are understood in the weak sense, that is, A-solutions are W 1 p,loc -functions satisfying the integral identity

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

Special exhaustion functions .
Let M be a noncompact Riemannian manifold with the boundary ∂M (possibly empty). Let A satisfy (2.7) and (2.8) and let h : M → (0, h 0 ) be an exhaustion function, satisfying the following 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 (2.13).
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 with the boundary ∂M(t 1 , t 2 ). Using the Stokes formula, we have for noncritical values t 1 < t 2 (for the definition of critical values of C k -functions see, for example, [2, Part II, Chapter 2, §10]) and a 2 ) follows.

Example.
We fix an integer k, 1 ≤ k ≤ n, and set 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. The function is a special exhaustion function of the domain D. Therefore for p ≥ k the domain D is of p-parabolic type and for p < k p-hyperbolic type.

Example.
Fix 1 ≤ k < n. Let ∆ be a bounded domain in the plane x 1 = . . . = x k = 0 with a piecewise smooth boundary and let be the cylinder domain with base ∆.
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, where d k is as in Example 3.4.
From the equation satisfies the equation (2.13) in D \ K and thus it is a special exhaustion function of the domain D.
, be the spherical coordinates in R n . Let U ⊂ S n−1 (1), ∂U = ∅, be an arbitrary domain on the unit sphere S n−1 (1). We fix 0 ≤ r 1 < ∞ and consider the domain As above it is easy to verify that the given domain is n-admissible and the function is a special exhaustion function of the domain D for p = n.
3.11. Example. Fix 1 ≤ n ≤ p. Let x 1 , x 2 , . . . , 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 an (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 the equation (2.13). We consider the function We have Because h is a special exhaustion function of D we have div (|∇h * | p−2 ∇h * ) = 0.
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 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.
3.13. Example. Let A be a compact Riemannian manifold, dim A = k, with piecewise smooth boundary or without boundary. We consider the Cartesian product M = A × R n , n ≥ 1. We denote by a ∈ A, x ∈ R n and (a, x) ∈ M the points of the corresponding spaces. It is easy to see that the function h(a, x) = log |x|, p = n, |x| p−n p−1 , p = n, is a special exhaustion function for the manifold M. Therefore, for p ≥ n the given manifold is of p-parabolic type and for p < n p-hyperbolic type.
The manifold M = (D, ds 2 M ) is a warped Riemannian product. In the case 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 case 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 (3.15) is given by the expression 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) the equation (2.13) reduces to the following form d dr Solutions of this equation are the functions where C 1 and C 2 are constants. Because the function h satisfies obviously the boundary condition a) 2 as well as the other conditions of (3.3), we see that under the assumption is a special exhaustion function on the manifold M. Proof. Choose 0 < t 1 < t 2 < h 0 such that K ⊂ B h (t 1 ). We need to estimate the p-capacity of the condenser (B h (t 1 ), M \ B h (t 2 ); M). We have is a quantity independent of t > h(K) = sup{h(m) : m ∈ K}. Indeed, for the variational problem (2.2) we choose the function ϕ 0 , ϕ 0 (m) = 0 for m ∈ B h (t 1 ), and ϕ 0 (m) = 1 for m ∈ M \ B h (t 2 ). Using the Kronrod-Federer formula [3, Theorem 3.2.22], we get Because the special exhaustion function satisfies the equation (2.13) and the boundary condition a) 2 , one obtains for arbitrary 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 (2.2). Such an extremal is unique and therefore the preceding inequality holds in fact with equality. This conclusion proves the equation (3.19).
If h 0 = ∞, then letting t 2 → ∞ in (3.19) 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 the boundary set ξ is of p-hyperbolic type. 2

Energy integral
The fundamental result of this section is an estimate for the rate of growth of the energy integral of forms of the class WT 2 on noncompact manifolds under various boundary conditions for the forms. As an application we get Phragmén-Lindelöf type theorems for the forms of this class and we prove some generalizations of the classical theorem of Ahlfors concerning the number of distinct asymptotic tracts of an entire function of finite order.

Boundary conditions .
Let M be an n-dimensional Riemannian manifold with nonempty boundary ∂M. We will fix a closed differential form w, deg w = k, 1 ≤ k ≤ n, w ∈ L p loc (M) of class WT 1 and the complementary closed form θ, deg θ = n − k, θ ∈ L q loc (M), satisfying the condition (1.2). We assume that there exists a differential form Z ∈ W 1 p,loc with continuous coefficients for which dZ = w.
Let h : M → (0, h 0 ) be an exhaustion function of M. As before we let B h (τ ) be an h-ball and Σ h (τ ) an h-sphere.

Dirichlet condition with zero boundary values .
We shall say that the form Z ∈ W 1 p,loc 1 satisfies Dirichlet's condition with zero boundary values on ∂M if for every differential form v ∈ L q loc (M), deg v = n − k, and for almost every τ ∈ (0, h 0 ) 4.6. Neumann condition with zero boundary values . We shall say that a form Z satisfies Neumann's condition with zero boundary values, if for every differential form v ∈ W 1 p,loc (M), deg v = k − 1, and for almost every τ ∈ (0, h 0 ) If M is compact then (4.7) takes the form M dv ∧ θ = 0. (4.8)

Mixed zero boundary condition .
We shall say that a form Z satisfies mixed zero boundary condition if for an arbitrary function φ ∈ C 1 (M) and for almost every τ ∈ (0, h 0 ) we have We assume that the form has the property (4.3). On the basis of Stokes' formula (the standard Stokes formula with generalized derivatives) we conclude that for almost every τ ∈ (0, h 0 ) This implies that the restriction of Z onto the boundary ∂M is the zero form, i.e. Z | ∂M (m) = 0 at every point m ∈ ∂M. (4.13) We next clarify the geometric meaning of the condition (4.13). We assume that m ∈ ∂M is a point where the boundary ∂M has a tangent plane T m (∂M) and that the form Z satisfies the regularity condition (4.12) in some neighborhood of the point m. 15) where ω is a form, deg ω = deg Z − 1.

Proposition. If a form Z is simple at a point m ∈ M, then the condition (4.13) is fulfilled if and only if the form Z is of the form
Proof. We give an orthonormal system of coordinates x 1 , . . . , x n at the point m such that the hyperplane T m (∂M) is given by the equation x n = 0. Let deg Z = l. Because the form Z is simple, we can represent it as follows where a i,j = a i,j (m) are some constants. The condition (4.13) can now be rewritten as follows and we easily obtain (4.15).
The proof of the converse implication is obvious. 2 We next clarify the geometric meaning of the Neumann condition (4.7). We fix the forms Z, v ∈ C 2 (intM) ∩ C 1 (∂M).
By Stokes' formula we have for almost every τ ∈ (0, h 0 ) Because the form θ is closed, condition (4.7) gives Consider the case of quasilinear equations (2.9). Let m ∈ ∂M be a regular point and let x 1 , . . . , x n be local coordinates in a neighborhood of this point. We have We set Z = f . In the case (4.3) we choose v = φθ where θ is an arbitrary locally Lipschitz function. We obtain φf θ, for all φ. On the other side, choosing in the case of the Neumann condition (4.7) for v an arbitrary locally Lipschitz function φ we get for almost every τ ∈ (0, h) which characterizes generalized solutions of the equation (2.9) with zero Neumann boundary conditions on ∂M.
It is easy to see that at every point of the boundary we have where (ν, x i ) is the angle between the inner normal vector ν to ∂M and the direction 0x i ; dH n−1 M is the element of surface area on M.. Thus, at a regular boundary point, the condition (4.17) is equivalent to the requirement A(m, ∇f (m)), ν = 0.
Using (4.13) we see that the condition (4.10) is equivalent to the traditional mixed boundary condition at regular boundary points. Proof. We assume that (4.5) holds and set v = θ. Then (4.5) yields We assume that the boundary condition (4.8) holds. Choose v = Z. Then (4.8) gives M w ∧ θ = 0.
As above, we arrive at the inequality (4.24). This inequality implies that θ ≡ 0 on M. Setting Z = f we get 4.26. Corollary. Suppose that the manifold M is compact and the boundary ∂M is not empty. If the function f satisfies the condition (4.20) or (4.21), then f ≡ const on M.

Estimates for energy integral. Applications
This Chapter is devoted to Phragmén-Lindelöf and Ahlfors theorems for differential forms.

Basic theorem .
Let M be a noncompact Riemannian manifold, dim M = n. We consider a class F of differential forms Z ∈ W 1 p,loc (M), deg Z = k − 1, such that the form dZ = w satisfies the conditions (1.1) and is in the class WT 2 . Let θ ∈ L q loc be a form satisfying the condition (1.2), complementary to w.
If the boundary ∂M is nonempty then we shall assume that the form Z satisfies on ∂M some boundary condition B. In the case considered below such a boundary condition can be any of the conditions (4.3), (4.4), (4.7), (4.10). We shall denote by F B (M) the set of forms Z, dZ ∈ WT 2 , satisfying the boundary condition B on M. In particular, below we shall operate with the classes F D , F 0 , F N , and F DN forms corresponding to the boundary conditions (4.3), (4.4), (4.7), (4.10), respectively.
We introduce a characteristic ε(τ ) setting where the infimum is taken over all Z ∈ F B (M), Z = 0. Some estimates of (5.2) are given in [8] and [12]. Under these circumstances we have 5.3. Theorem. Suppose that the form Z ∈ F B (M) satisfies one of the boundary condition (4.3), (4.7), or (4.10). Then for almost all τ ∈ (0, h 0 ) and for an arbitrary τ 0 the following relation holds |w| p * 1 1.
In particular, for all τ 1 < τ 2 we have Proof. The Kronrod-Federer formula yields and, in particular, the function I(τ ) is absolutely continuous on closed intervals of (0, h 0 ). Now it is enough to prove the inequality From (5.6) we have for almost every τ ∈ (0, h 0 )

By (1.4) we obtain
However, the form Z is weakly closed and satisfies one of the conditions (4.3), (4.4), or (4.7). Therefore for a.e. τ ∈ (0, h 0 ), Thus we get Further from (5.2) it follows that Combining the above inequalities we obtain This inequality together with the equality (5.8) yields We thus obtain the desired conclusion (5.7). 2 We shall need also some other estimates of the energy integral. We now prove the first of these inequalities. Denote by F (B h (τ )) the set of all differential forms (5.9) such that for almost every τ ∈ (0, h 0 ) and for an arbitrary Lipschitz function φ the following formula holds 5.11. Theorem. If the differential form Z ∈ F B (M), dZ ∈ WT 2 , satisfies the boundary condition (4.3), (4.7), or (4.10), then for all τ 1 < τ 2 < h 0 and for an arbitrary form Z 0 ∈ F (B h (τ 2 )) the following relation holds Proof. We consider the function Suppose that the form Z satisfies the condition (4.3). Setting in (4. The function (φ) p is locally Lipschitz on B h (τ 2 ) and φ| Σ h (τ 2 ) = 0. Thus by (5.10) we get Hence we arrive at the relation Observing that φ(m) = 1 for m ∈ B h (τ 1 ) and φ(m) = 0 for m ∈ M \ B h (τ 2 ) we obtain Because |φ| ≤ 1, the inequality (5.12) follows. Let the form Z satisfy the condition (4.7). We choose v = (φ) p Z and observe that v| Σ h (τ 2 ) = 0.
Then we get Further details of the proof in this case are similar to those carried out above. We assume that the form Z satisfies the mixed boundary condition (4.10). Observing that Arguing as above we complete the proof of the theorem. 2 There is also an estimate for the energy integral which does not use the complementary form θ of dZ = w. Such an estimate is given in the next theorem.
The following theorem exhibits a generalization of the classical Phragmén-Lindelöf principle for holomorphic functions.
5.17. Theorem. Suppose that the form Z, dZ ∈ WT 2 (M), satisfies one of the boundary conditions (4.3), (4.7) or (4.10). The following alternatives hold: either the form dZ = 0 a.e. on the manifold M, or for all Proof. The property (5.18) follows readily from (5.5) and is presented here only for the sake of completeness. By (5.5) and (5.12), for a.e. τ ∈ (τ 0 , h 0 ) we have Therefore we get Analogously, using (5.15) we get If we now assume that the form w ≡ 0, then I(τ 0 ) > 0 for some τ 0 ∈ (0, h 0 ). From this there easily follow (5.19) and (5.20). 2 5.21. Integral of energy and allocation of finite forms . There is another application of the above estimates of energy integrals connected with a generalization of the classical Denjoy-Carleman-Ahlfors theorem about the number of different asymptotic tracts of an entire function of a given order. In the present case this theorem can be interpreted as a statement concerning the connection between the number of finite forms in the class WT 2 defined on the manifold M and the rate of growth of their energy integrals.
Let M be an n-dimensional noncompact Riemannian manifold with or without boundary. We fix a locally Lipschitz exhaustion function h : M → (0, h 0 ), 0 < h 0 ≤ ∞, of the manifold M.
We assume that there are L ≥ 1 mutually disjoint domains O 1 , O 2 , . . . , O L on M such that O i ∩ ∂M = ∅ if the boundary ∂M is nonempty. We also assume that on each domain O i is given a differential form Z i with continuous coefficients and the properties: deg Z i = k − 1, dZ ≡ 0, dZ i = w ∈ WT 2 with structure constants p, ν 1 , ν 2 , independent of i = 1, 2, . . . , L, Z i satisfies on ∂O i the zero boundary condition (4.4).
We define a form Z on M by setting Z| O i = Z i and Z = 0 on M\∪ L i=1 O i . According to Theorem 4.23 each of the domains O i has a noncompact closure. Then by Theorem 5.17 the "narrower" the intersection of the domains O i with h-spheres Σ h (t) for t → ∞, the higher is the rate of growth of the form Z. Below we shall consider the Denjoy-Carleman-Ahlfors theorem as a statement on the connection between the number L of mutually disjoint domains O i on M 0 and the rate of growth of the energy of the form Z (or of the form Z itself) with respect to an exhaustion function h(m) of the manifold M. We shall prove that such a formulation of the problem contains, in particular, the classical Denjoy-Carleman-Ahlfors problem for holomorphic functions of the complex plane. In the case of harmonic functions of R n see [5] for the history of the problem.
We next introduce some necessary notation. We consider an open subset D ⊂ M with a noncompact closure and we assume that the restriction of the form Z to D satisfies condition (4.4).
The function h| D : D → (0, ∞) is an exhaustion function of D. We fix an h-ball B h (τ ). Considering the variational problem (5.2) for the class of forms Z, satisfying the boundary condition (4.4) on D 1 we define the characteristic where F 0 is defined in 5.1 and in (4.3).
Following [13] we introduce the N-mean E(t; N) = inf 1 N Proof. It is enough to observe that each pair of forms Z, Z 0 admissible for the variational problem (5.2) for the set D 1 is also admissible for this problem for the set D 2 .
From (5.24) we get (5.25). 2 We next derive a more general assertion about the monotonicity of Nmeans.  The next theorem provides a solution to the aforementioned problem concerning the connection between the number L of finite forms on M, and the rate of growth of the total energy of these forms or the sum of their L p -norms on an h-ball B h (τ ).  Fix τ 0 > d. Using the inequality (5.5) from Theorem 5.3 for an arbitrary k = 1, 2, . . . , N and a.e. 0 < τ 0 < τ ′ < h 0 we have Adding these inequalities, we get |dZ| p * 1 1.
Applying the arithmetic-geometric mean inequality The domains O 1 , O 2 , . . . , O N are nonintersecting. Therefore for all τ 0 < t < h 0 we have 1 N