`Abstract and Applied AnalysisVolume 2010, Article ID 646392, 29 pageshttp://dx.doi.org/10.1155/2010/646392`
Research Article

Ahlfors Theorems for Differential Forms

1Department of Mathematics and Statistics, University of Helsinki, 00014 Helsinki, Finland
3Department of Mathematics, University of Turku, 20014 Turku, Finland

Received 11 September 2010; Accepted 18 October 2010

Copyright © 2010 O. Martio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Some counterparts of theorems of Phragmén-Lindelöf and of Ahlfors are proved for differential forms of -classes.

1. -Forms

This paper is continuation of the earlier work [1], where the main topic was to examine the connection between quasiregular (qr) mappings and so-called -classes of differential forms. We first recall some basic notation and terminology from [1].

Let be a Riemannian manifold of class , , with or without boundary, and let be a weakly closed differential form on , that is, for each form with a compact in and such that ; we have Here , , and is the orthogonal complement of a differential form on a Riemannian manifold .

A weakly closed form of the kind (1.1) is said to be of the class on if there exists a weakly closed differential form such that almost everywhere on we have for some constant .

The differential form (1.1) is said to be of the class on if there exists a differential form (1.4) such that almost everywhere on for some constants .

Theorem 1.1.

For a proof see [1].

The following partial integration formula for differential forms is useful [1].

Lemma 1.2. Let and be differential forms, , , , , and let have a compact support . Then In particular, the form is weakly closed if and only if a.e. on .

Let and be Riemannian manifolds of dimensions , , , and with scalar products , , respectively. The Cartesian product has the natural structure of a Riemannian manifold with the scalar product We denote by and the natural projections of the manifold onto submanifolds.

If and are volume forms on and , respectively, then the differential form is a volume form on .

Let be an orthonormal system of coordinates in , . Let be a domain in , and let be an -dimensional Riemannian manifold. We consider the manifold .

2. 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 be an -dimensional noncompact Riemannian manifold without boundary. Boundary sets on are analogies to prime ends due to Carathéodory (cf. e.g., [2]).

Let , be a collection of open sets with the following properties:(i)for all ,  ,(ii).

A sequence with these properties will be called a chain on the manifold .

Let , be two chains of open sets on . We will say that the chain is contained in the chain , if for each there exists a number such that for all we have . Two chains, each of which is contained in the other one, are called equivalent. Each equivalence class of chains is called a boundary set of the manifold . To define it is enough to determine at least one representative in the equivalence class. If the boundary set is defined by the chain , then we will write .

A sequence of points converges to if for some (and, therefore, all) chain the following condition is satisfied: for every there exists an integer such that for all . A sequence lies off a boundary set , if for every there exists a number such that for all .

A boundary set is called a set of ends of the manifold if each of has a compact boundary . If in addition each of the sets is connected, then is called an end of the manifold .

2.1. Types of Boundary Set

Let be an open set on and let be closed subsets in such that . Each triple is called a condenser on .

We fix . The -capacity of the condenser is defined by where the infimum is taken over the set of all continuous functions of class such that , . It is easy to see that for a pair and with , we have

Let , be compact in . A standard approximation method shows that does not change if one restricts the class of functions in the variational problem (2.1) to Lipschitz functions equal to 0 and 1 in the sets and , respectively.

Let be an arbitrary chain on a manifold . We fix a subdomain . If is sufficiently large, the intersection and we consider the condenser . Then it is clear that for We will say that the chain on has -capacity zero, if for every subdomain we have

We will say that a boundary set is of -parabolic type if every chain is of -capacity zero. A boundary set is of -hyperbolic type if at least one of the chains is not of -parabolic type.

Let be an arbitrary exhaustion of the manifold by subdomains . The manifold is of -parabolic or -hyperbolic type depending on the -parabolicity or -hyperbolicity of the boundary set .

It is well-known, see [3], that a noncompact Riemannian manifold without boundary is of -parabolic type if and only if every solution of the inequality 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.

Example 2.1. The space is of -parabolic type for and -hyperbolic type for .

We next present a proposition that provides a convenient method of verifying the -parabolicity and -hyperbolicity of boundary sets.

Lemma 2.2 (see [4]). Let be a boundary set on . If for a chain and for a nonempty open set the condition (2.4) holds, then the boundary set is of -parabolic type.

2.2. -Solutions

Let be a Riemannian manifold and let be a mapping defined a.e. on the tangent bundle . Suppose that for a.e. the mapping is continuous on the fiber , that is, for a.e. the function is defined and continuous; the mapping is measurable for all measurable vector fields (see [5]).

Suppose that for a.e. and for all the inequalities hold with and for some constants . It is clear that we have .

We consider the equation Solutions to (2.9) are understood in the weak sense, that is, -solutions are -functions satisfying the integral identity for all with a compact support .

A function in is an -subsolution of (2.9) in if weakly in , that is, whenever , is nonnegative with a compact support in .

A basic example of such an equation is the -Laplace equation

3. Exhaustion Functions

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

3.1. Exhaustion Functions of Boundary Sets

Let , , be a locally Lipschitz function. For arbitrary we denote by the -balls and -spheres, respectively.

Let be a locally Lipschitz function such that: there exists a compact such that for a.e. . We say that the function is an exhaustion function for a boundary set of if for an arbitrary sequence of points , the function if and only if .

It is easy to see that this requirement is satisfied if and only if for an arbitrary increasing sequence the sequence of the open sets is a chain, defining a boundary set . Thus the function exhausts the boundary set in the traditional sense of the word.

The function is called the exhaustion function of the manifold if the following two conditions are satisfied(i)for all the -ball is compact;(ii)for every sequence with , the sequence of -balls generates an exhaustion of , that is,

Example 3.1. Let be a Riemannian manifold. We set where is a fixed point. Because almost everywhere on , the function defines an exhaustion function of the manifold .

3.2. Special Exhaustion Functions

Let be a noncompact Riemannian manifold with the boundary (possibly empty). Let satisfy (2.8) and let be an exhaustion function, satisfying the following conditions: (a1)there is such that is compact and is a solution of (2.9) in the open set ;(a2)for a.e. , ,

Here is the element of the -dimensional Hausdorff measure on . Exhaustion functions with these properties will be called the special exhaustion functions of with respect to . In most cases the mapping will be the -Laplace operator (2.13).

Since the unit vector is orthogonal to the -sphere , the condition (a2) means that the flux of the vector field through -spheres is constant.

Suppose that the function is continuously differentiable. If (b1) and satisfies (2.9), and (b2)at every point where has a tangent plane the condition is satisfied where is a unit vector of the inner normal to the boundary , then is a special exhaustion function of the manifold .

The proof of this statement is simple. Consider the open set with the boundary . Using the Stokes formula, we have for noncritical values (for the definition of critical values of -functions see, e.g., [6, Part II, Chapter 2, Section 10]) and (a2) follows.

Example 3.2. We fix an integer , , and set It is easy to see that everywhere in where . We will call the set a -ball and the set a -sphere in .

We will say that an unbounded domain is -admissible if for each the set has compact closure.

It is clear that every unbounded domain is -admissible. In the general case the domain is -admissible if and only if the function is an exhaustion function of . It is not difficult to see that if a domain is -admissible, then it is -admissible for all .

Fix . Let be a bounded domain in the -plane and let be a domain in .

The domain is -admissible. The -spheres are orthogonal to the boundary and therefore everywhere on the boundary. The function is a special exhaustion function of the domain . Therefore for the domain is of -parabolic type and for   -hyperbolic type.

Example 3.3. Fix . Let be a bounded domain in the plane with a piecewise smooth boundary and let be the cylinder domain with base .

The domain is -admissible. The -spheres are orthogonal to the boundary and therefore everywhere on the boundary, where is as in Example 3.2.

Let where is a -function. We have and From the equation we conclude that the function satisfies (2.13) in and thus it is a special exhaustion function of the domain .

Example 3.4. Let , where , , be the spherical coordinates in . Let , , be an arbitrary domain on the unit sphere . We fix and consider the domain As mentioned above it is easy to verify that the given domain is -admissible and the functionis a special exhaustion function of the domain for .

Example 3.5. Fix . Let be an orthonormal system of coordinates in ,. Let be an unbounded domain with piecewise smooth boundary and let be an -dimensional compact Riemannian manifold with or without boundary. We consider the manifold .

We denote by , , and the points of the corresponding manifolds. Let and be the natural projections of the manifold .

Assume now that the function is a function on the domain satisfying the conditions (b1), (b2) and (2.13). We consider the function .

We have Because is a special exhaustion function of we have

Let be an arbitrary point where the boundary has a tangent hyperplane and let be a unit normal vector to .

If , then where the vector is orthogonal to and is a vector from . Thus because is a special exhaustion function on and satisfies the property on . If , then the vector is orthogonal to and because the vector is parallel to .

The other requirements for a special exhaustion function for the manifold are easy to verify.

Therefore, the function is a special exhaustion function on the manifold .

Example 3.6. Let be a compact Riemannian manifold, , with piecewise smooth boundary or without boundary. We consider the Cartesian product , . We denote by , and the points of the corresponding spaces. It is easy to see that the function is a special exhaustion function for the manifold . Therefore, for the given manifold is of -parabolic type and for -hyperbolic type.

Example 3.7. Let , where , , be the spherical coordinates in . Let be an arbitrary domain on the unit sphere . We fix and consider the domain with the metric where are -functions on and is an element of length on .

The manifold is a warped Riemannian product. In the case , , and the manifold is isometric to a cylinder in . In the case , , and the manifold is a spherical annulus in .

The volume element in the metric (3.25) is given by the expression If , then the length of the gradient in takes the form where is the gradient in the metric of the unit sphere .

For the special exhaustion function (2.13) reduces to the following form Solutions of this equation are the functions where and are constants.

Because the function satisfies obviously the boundary condition (a2) as well as the other conditions of Section 3.2, we see that under the assumption the function is a special exhaustion function on the manifold .

Theorem 3.8. Let be a special exhaustion function of a boundary set of the manifold . Then(i)if , the set is of -parabolic type,(ii)if , the set is of -hyperbolic type.

Proof. Choose such that . We need to estimate the -capacity of the condenser . We have where is a quantity independent of . Indeed, for the variational problem (2.1) we choose the function , for , and for . Using the Kronrod-Federer formula [7, Theorem ], we get
Because the special exhaustion function satisfies (2.13) and the boundary condition (), one obtains for arbitrary , Thus we have established the inequality
By the conditions, imposed on the special exhaustion function, the function is an extremal in the variational problem (2.1). Such an extremal is unique and therefore the preceding inequality holds in fact with equality. This conclusion proves (3.32).
If , then letting in (3.32) we conclude the parabolicity of the type of . Let . Consider an exhaustion and choose such that the -ball contains the compact set .
Set . Then for we have hence and the boundary set is of -hyperbolic type.

4. 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 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.

4.1. Boundary Conditions

Let be an -dimensional Riemannian manifold with nonempty boundary . We will fix a closed differential form , , , of class and the complementary closed form , , , satisfying the condition (1.4). We assume that there exists a differential form with continuous coefficients for which .

Let be an exhaustion function of . As mentioned before we let be an -ball and an -sphere.

4.2. Dirichlet Condition with Zero Boundary Values

We will say that the form (with continuous coefficients and such that ) satisfies Dirichlet's condition with zero boundary values on if for every differential form , , and for almost every

In particular, the form satisfies the boundary condition (4.1), if its coefficients are continuous and if its support does not intersect with , that is

If is compact then (4.1) takes the form

4.3. Neumann Condition with Zero Boundary Values

We will say that a form satisfies Neumann's condition with zero boundary values, if for every differential form , , and for almost every

If is compact then (4.4) takes the form

4.4. Mixed Zero Boundary Condition

We will say that a form satisfies mixed zero boundary condition if for an arbitrary function and for almost every we have

If is compact then (4.6) takes the form

We assume that the form has the property (4.1). On the basis of Stokes' formula (the standard Stokes formula with generalized derivatives) we conclude that for almost every holds. Therefore we get This implies that the restriction of onto the boundary is the zero form, that is,

We next clarify the geometric meaning of the condition (4.11). We assume that is a point where the boundary has a tangent plane and that the form satisfies the regularity condition (4.8) in some neighborhood of the point .

Proposition 4.1. If a form is simple at a point , then the condition (4.11) is fulfilled if and only if the form is of the form where is a form, .

Proof. We give an orthonormal system of coordinates at the point such that the hyperplane is given by the equation . Let . Because the form is simple, we can represent it as follows where are some constants. The condition (4.11) can now be rewritten as follows and we easily obtain (4.12).
The proof of the converse implication is obvious.

We next clarify the geometric meaning of the Neumann condition (4.4). We fix the forms By Stokes' formula we have for almost every Because the form is closed, condition (4.4) gives Therefore at every point .

Exactly in the same way we verify that the mixed zero boundary condition (4.6) is equivalent to the condition at every point .

Consider the case of quasilinear equations (2.9). Let be a regular point and let be local coordinates in a neighborhood of this point. We have We set . In the case (4.1) we choose where is an arbitrary locally Lipschitz function. We obtain and further This condition characterizes generalized solutions of (2.9) with zero Dirichlet boundary condition on .

On the other side, choosing in the case of the Neumann condition (4.4) for an arbitrary locally Lipschitz function we get for almost every which characterizes generalized solutions of (2.9) with zero Neumann boundary conditions on .

It is easy to see that at every point of the boundary we have where is the angle between the inner normal vector to and the direction ; is the element of surface area on .

Thus, at a regular boundary point, the condition (4.18) is equivalent to the requirement

Using (4.11) we see that the condition (4.6) is equivalent to the traditional mixed boundary condition at regular boundary points.

4.5. Maximum Principle for -Forms

Let be a compact Riemannian manifold with nonempty boundary, , and let , , , be a differential form of class on . Let , , be a form complementary to the form .

Theorem 4.2. Suppose that there exists a differential form , on . If either (4.3) or (4.5) holds, then on .

Proof. We assume that (4.3) holds and set . Then (4.3) yields Because we get Using (1.5) we deduce
We assume that the boundary condition (4.5) holds. Choose . Then (4.5) gives As above, we arrive at the inequality (4.29). This inequality implies that on .

In order to illustrate Theorem 4.2 we consider the example of generalized solutions of (2.9) under the condition for all with the constants and .

Setting we get

Corollary 4.3. Suppose that the manifold is compact and the boundary is not empty. If the function satisfies the condition (4.22) or (4.23), then on .

5. Estimates for Energy Integral: Applications

This chapter is devoted to Phragmén-Lindelöf and Ahlfors theorems for differential forms.

5.1. Basic Theorem

Let be a noncompact Riemannian manifold, . We consider a class of differential forms , , such that the form satisfies the conditions (1.1) and is in the class . Let be a form satisfying the condition (1.4), complementary to .

If the boundary is nonempty then we will assume that the form satisfies on some boundary condition . In the case considered below such a boundary condition can be any of the conditions (4.1), (4.2), (4.4), and (4.6). We will denote by the set of forms , , satisfying the boundary condition on . In particular, below we will operate with the classes of the , , , and forms corresponding to the boundary conditions (4.1), (4.2), (4.4), and (4.6), respectively.

We fix a locally Lipschitz exhaustion function , . Let and let be an -ball, and its boundary sphere as before.

We introduce a characteristic setting where the infimum is taken over all , .

Some estimates of (5.1) are given in [8, 9].

Under these circumstances we have the following theorem.

Theorem 5.1. Suppose that the form satisfies one of the boundary condition (4.1), (4.4), or (4.6). Then for almost all and for an arbitrary the following relation holds where
In particular, for all we have

Proof. The Kronrod-Federer formula yields and, in particular, the function is absolutely continuous on closed intervals of . Now it is enough to prove the inequality From (5.5) we have for almost every
By (1.6) we obtain However, the form is weakly closed and satisfies one of the conditions (4.1), (4.2), or (4.4). Therefore for a.e. , Thus we get
Further from (5.1) it follows that
Combining the above inequalities we obtain This inequality together with the equality (5.7) yields We thus obtain the desired conclusion (5.6).

We will need also some other estimates of the energy integral. We now prove the first of these inequalities. Denote by the set of all differential forms such that for almost every and for an arbitrary Lipschitz function the following formula holds

Theorem 5.2. If the differential form , , satisfies the boundary condition (4.1), (4.4), or (4.6), then for all and for an arbitrary form the following relation holds

Proof. We consider the function Suppose that the form satisfies the condition (4.1). Setting in (4.1) we get or
The function is locally Lipschitz on and . Thus by (5.15) we get
Hence we arrive at the relation which by (1.6) yields
Observing that for and for we obtain Because , the inequality (5.16) follows.
Let the form satisfy the condition (4.4). We choose and observe that Then we get Further details of the proof in this case are similar to those carried out above.
We assume that the form satisfies the mixed boundary condition (4.6). Observing that we get Arguing as above we complete the proof of the theorem.

There is also an estimate for the energy integral which does not use the complementary form of . Such an estimate is given in the next theorem.

Theorem 5.3. If the form , , satisfies on one of the boundary conditions (4.1), (4.4), or (4.6), then for all and for an arbitrary form we have

Proof. We use the earlier established relation (5.22). We estimate the integral on the right hand side of (5.22). By (1.7) we get From (5.22) we get Using the facts that on and on we easily obtain (5.28).

5.2. Phragmén-Lindelöf Theorem

Let be an -dimensional noncompact Riemannian manifold with or without boundary and let be a differential form as in (1.1), , and its complementary form as in (1.4).

We assume that there exists a differential form with . If the boundary is nonempty, then we will assume that satisfies the boundary condition of Dirichlet (4.1), Neumann's condition (4.4), or the condition (4.6).

We fix a locally Lipschitz exhaustion function , . Let, as above and let where the infimum is taken over all closed forms , satisfying conditions (5.14), (5.15) on .

The following theorem exhibits a generalization of the classical Phragmén-Lindelöf principle for holomorphic functions.

Theorem 5.4. Suppose that the form , , satisfies one of the boundary conditions (4.1), (4.4) or (4.6). The following alternatives hold: either the form a.e. on the manifold , or for all we have