# Ahlfors Theorems for Differential Forms

Authors: O. Martio; V. M. Miklyukov; M. Vuorinen
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

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


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## Body

## 1.𝒲𝒯-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 𝒲𝒯-classes of differential forms. We first recall some basic notation and terminology from [1].Letℳ be a Riemannian manifold of class C3, dimℳ=n, with or without boundary, and let
(1.1)w∈Lloc⁡p(M),deg⁡w=k,0≤k≤n,p>1,
be a weakly closed differential form on ℳ, that is, for each form (1.2)φ∈Wq,loc⁡1(M),deg⁡φ=k+1,1p+1q=1,
with a compact suppφ in ℳ and such that suppφ∩∂ℳ=∅; we have (1.3)∫M〈w,δφ〉dv=0.
Here δφ=(-1)k⋆-1d⋆φ, k=degφ, and ⋆α is the orthogonal complement of a differential form α on a Riemannian manifold ℳ.A weakly closed formw of the kind (1.1) is said to be of the class 𝒲𝒯1 on ℳ if there exists a weakly closed differential form
(1.4)θ∈Lloc⁡q(M),deg⁡θ=n-k,1p+1q=1,
such that almost everywhere on ℳ we have
(1.5)ν0|θ|q≤〈w,*θ〉
for some constant ν0>0.The differential form (1.1) is said to be of the class 𝒲𝒯2 on ℳ if there exists a differential form (1.4) such that almost everywhere on ℳ(1.6)ν1|w|p≤〈w,*θ〉,(1.7)|θ|≤ν2|w|p-1,
for some constants ν1,ν2>0.Theorem 1.1.
𝒲𝒯2⊂𝒲𝒯1For a proof see [1].The following partial integration formula for differential forms is useful [1].Lemma 1.2.
Letα∈Wp,loc1(ℳ) and β∈Wq1(ℳ) be differential forms, degα+degβ=n-1, 1/p+1/q=1, 1≤p, q≤∞, and let β have a compact support suppβ⊂ℳ. Then
(1.8)∫Mdα∧β=(-1)deg⁡α+1∫Mα∧dβ.
In particular, the form α is weakly closed if and only if dα=0 a.e. on ℳ.Let𝒜 and ℬ be Riemannian manifolds of dimensions dim𝒜=k, dimℬ=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
(1.9)〈⋅,⋅〉=〈⋅,⋅〉A+〈⋅,⋅〉B.
We denote by π:𝒜×ℬ→𝒜 and η:𝒜×ℬ→ℬ the natural projections of the manifold 𝒩 onto submanifolds.Ifw𝒜 and wℬ are volume forms on 𝒜 and ℬ, respectively, then the differential form w𝒩=π*w𝒜∧η*wℬ is a volume form on 𝒩.Lety1,…,yk be an orthonormal system of coordinates in Rk, 1≤k≤n. Let 𝒜 be a domain in Rk, and let ℬ be an (n-k)-dimensional Riemannian manifold. We consider the manifold 𝒩=𝒜×ℬ.


## 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.Letℳ be an n-dimensional noncompact Riemannian manifold without boundary. Boundary sets on ℳ are analogies to prime ends due to Carathéodory (cf. e.g., [2]).Let{𝒰k}, k=1,2,… be a collection of open sets 𝒰k⊂ℳ with the following properties:(i)
for allk=1,2,…,  𝒰¯k+1⊂𝒰k,(ii)
⋂k=1∞𝒰¯k=∅.A sequence with these properties will be called a chain on the manifoldℳ.Let{𝒰k′}, {𝒰k′′} be two chains of open sets on M. We will say that the chain 𝒰k′ is contained in the chain {𝒰k′′}, if for each m≥1 there exists a number k(m) such that for all k>k(m) we have 𝒰k′⊂𝒰m′′. 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 {𝒰k}, then we will write ξ≍{𝒰k}.A sequence of pointsmk∈ℳ converges to ξ if for some (and, therefore, all) chain {𝒰k}∈ξ the following condition is satisfied: for every k=1,2,… there exists an integer n(k) such that mn∈𝒰k for all n>n(k). A sequence (mn) lies off a boundary set ξ≍{𝒰k}, if for every k=1,2,… there exists a number n(k) such that for all n>n(k)mn∉𝒰k.A boundary setξ≍{𝒰k} is called a set of ends of the manifold ℳ if each of {𝒰k} has a compact boundary ∂𝒰k. If in addition each of the sets Uk is connected, then ξ≍{Uk} is called an end of the manifoldℳ.


### 2.1. Types of Boundary Set

LetD be an open set on ℳ and let A,B⊂D be closed subsets in D such that A¯∩B¯=∅. Each triple (A,B;D) is called a condenser on ℳ.We fixp≥1. The p-capacity of the condenser (A,B;D) is defined by (2.1)capp(A,B;D)=inf⁡∫D|∇φ|p*1M,
where the infimum is taken over the set of all continuous functions of class Wp,loc1(D) such that φ(m)|A=0, φ(m)|B=1. It is easy to see that for a pair (A,B;D) and (A1,B1;D) with A1⊂A, B1⊂B we have (2.2)capp(A1,B1;D)≤capp(A,B;D).LetA¯, B¯ be compact in D. A standard approximation method shows that capp(A,B;D) 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 A and B, respectively.Let{𝒰k} 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¯,𝒰k¯;ℳ). Then it is clear that for k=1,2,…(2.3)capp(H¯,Uk¯;M)≥capp(H¯,Uk+1¯;M).
We will say that the chain {𝒰k} on ℳ has p-capacity zero, if for every subdomain H⊂⊂ℳ we have
(2.4)lim⁡k→∞capp(H¯,Uk¯;M)=0.We will say that a boundary setξ is of p-parabolic type if every chain {𝒰k}≍ξ is of p-capacity zero. A boundary set ξ is of α-hyperbolic type if at least one of the chains {𝒰k}∈ξ is not of p-parabolic type.Let(2.5){Uk}k=1∞,U¯k⊂Uk+1,⋃k=1∞Uk=M
be an arbitrary exhaustion of the manifold ℳ by subdomains {𝒰k}. The manifold ℳ is of p-parabolic or p-hyperbolic type depending on the p-parabolicity or p-hyperbolicity of the boundary set {ℳ∖𝒰¯k}.It is well-known, see [3], that a noncompact Riemannian manifold ℳ without boundary is of p-parabolic type if and only if every solution of the inequality
(2.6)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.Example 2.1.
The spaceRn 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 thep-parabolicity and p-hyperbolicity of boundary sets.Lemma 2.2 (see [4]).
Letξ be a boundary set on ℳ. If for a chain {𝒰k≍ξ} and for a nonempty open set H0⊂⊂ℳ the condition (2.4) holds, then the boundary set ξ is of p-parabolic type.


### 2.2.A-Solutions

Letℳ be a Riemannian manifold and let
(2.7)A:T(M)⟶T(M)
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 Tm, that is, for a.e. m∈ℳ the function A(m,·):ξ∈Tm→Tm is defined and continuous; the mapping m→Am(X) is measurable for all measurable vector fields X (see [5]).Suppose that for a.e.m∈ℳ and for all ξ∈Tm the inequalities
(2.8)ν1|ξ|p≤〈ξ,A(m,ξ)〉,|A(m,ξ)|≤ν2|ξ|p-1
hold with p>1 and for some constants ν1,ν2>0. It is clear that we have ν1≤ν2.We consider the equation(2.9)div⁡A(m,∇f)=0.
Solutions to (2.9) are understood in the weak sense, that is, A-solutions are Wp,loc1-functions satisfying the integral identity
(2.10)∫M〈∇θ,A(m,∇f)〉*1M=0
for all θ∈Wp1(ℳ) with a compact support suppθ⊂ℳ.A functionf in Wp,loc1(ℳ) is an A-subsolution of (2.9) in ℳ if (2.11)div⁡A(m,∇f)≥0
weakly in ℳ, that is, (2.12)∫M〈∇θ,A(m,∇f)〉*1M≤0
whenever θ∈Wp1(ℳ), is nonnegative with a compact support in ℳ.A basic example of such an equation is thep-Laplace equation (2.13)div⁡(|∇f|p-2∇f)=0.


## 2.1. Types of Boundary Set

LetD be an open set on ℳ and let A,B⊂D be closed subsets in D such that A¯∩B¯=∅. Each triple (A,B;D) is called a condenser on ℳ.We fixp≥1. The p-capacity of the condenser (A,B;D) is defined by (2.1)capp(A,B;D)=inf⁡∫D|∇φ|p*1M,
where the infimum is taken over the set of all continuous functions of class Wp,loc1(D) such that φ(m)|A=0, φ(m)|B=1. It is easy to see that for a pair (A,B;D) and (A1,B1;D) with A1⊂A, B1⊂B we have (2.2)capp(A1,B1;D)≤capp(A,B;D).LetA¯, B¯ be compact in D. A standard approximation method shows that capp(A,B;D) 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 A and B, respectively.Let{𝒰k} 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¯,𝒰k¯;ℳ). Then it is clear that for k=1,2,…(2.3)capp(H¯,Uk¯;M)≥capp(H¯,Uk+1¯;M).
We will say that the chain {𝒰k} on ℳ has p-capacity zero, if for every subdomain H⊂⊂ℳ we have
(2.4)lim⁡k→∞capp(H¯,Uk¯;M)=0.We will say that a boundary setξ is of p-parabolic type if every chain {𝒰k}≍ξ is of p-capacity zero. A boundary set ξ is of α-hyperbolic type if at least one of the chains {𝒰k}∈ξ is not of p-parabolic type.Let(2.5){Uk}k=1∞,U¯k⊂Uk+1,⋃k=1∞Uk=M
be an arbitrary exhaustion of the manifold ℳ by subdomains {𝒰k}. The manifold ℳ is of p-parabolic or p-hyperbolic type depending on the p-parabolicity or p-hyperbolicity of the boundary set {ℳ∖𝒰¯k}.It is well-known, see [3], that a noncompact Riemannian manifold ℳ without boundary is of p-parabolic type if and only if every solution of the inequality
(2.6)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.Example 2.1.
The spaceRn 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 thep-parabolicity and p-hyperbolicity of boundary sets.Lemma 2.2 (see [4]).
Letξ be a boundary set on ℳ. If for a chain {𝒰k≍ξ} and for a nonempty open set H0⊂⊂ℳ the condition (2.4) holds, then the boundary set ξ is of p-parabolic type.


## 2.2.A-Solutions

Letℳ be a Riemannian manifold and let
(2.7)A:T(M)⟶T(M)
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 Tm, that is, for a.e. m∈ℳ the function A(m,·):ξ∈Tm→Tm is defined and continuous; the mapping m→Am(X) is measurable for all measurable vector fields X (see [5]).Suppose that for a.e.m∈ℳ and for all ξ∈Tm the inequalities
(2.8)ν1|ξ|p≤〈ξ,A(m,ξ)〉,|A(m,ξ)|≤ν2|ξ|p-1
hold with p>1 and for some constants ν1,ν2>0. It is clear that we have ν1≤ν2.We consider the equation(2.9)div⁡A(m,∇f)=0.
Solutions to (2.9) are understood in the weak sense, that is, A-solutions are Wp,loc1-functions satisfying the integral identity
(2.10)∫M〈∇θ,A(m,∇f)〉*1M=0
for all θ∈Wp1(ℳ) with a compact support suppθ⊂ℳ.A functionf in Wp,loc1(ℳ) is an A-subsolution of (2.9) in ℳ if (2.11)div⁡A(m,∇f)≥0
weakly in ℳ, that is, (2.12)∫M〈∇θ,A(m,∇f)〉*1M≤0
whenever θ∈Wp1(ℳ), is nonnegative with a compact support in ℳ.A basic example of such an equation is thep-Laplace equation (2.13)div⁡(|∇f|p-2∇f)=0.


## 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

Leth:ℳ→(0,h0), 0<h0≤∞, be a locally Lipschitz function. For arbitrary t∈(0,h0) we denote by (3.1)Bh(t)={m∈M:h(m)<t},Σh(t)={m∈M:h(m)=t}
the h-balls and h-spheres, respectively.Leth:ℳ→R be a locally Lipschitz function such that: there exists a compact K⊂ℳ such that |∇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 mk∈ℳ, k=1,2,… the function h(mk)→h0 if and only if mk→ξ.It is easy to see that this requirement is satisfied if and only if for an arbitrary increasing sequencet1<t2<⋯<h0 the sequence of the open sets Vk={m∈ℳ:h(m)>tk} 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 functionh:ℳ→(0,h0) is called the exhaustion function of the manifold ℳ if the following two conditions are satisfied(i)
for allt∈(0,h0) the h-ball Bh(t)¯ is compact;(ii)
for every sequencet1<t2<⋯<h0 with limk→∞tk=h0, the sequence of h-balls {Bh(tk)} generates an exhaustion of ℳ, that is,
(3.2)Bh(t1)⊂Bh(t2)⊂⋯⊂Bh(tk)⊂⋯,⋃kBh(tk)=M.Example 3.1.
Letℳ be a Riemannian manifold. We set h(m)=dist(m,m0) where m0∈ℳ is a fixed point. Because |∇h(m)|=1 almost everywhere on ℳ, the function h defines an exhaustion function of the manifold ℳ.


### 3.2. Special Exhaustion Functions

Letℳ be a noncompact Riemannian manifold with the boundary ∂ℳ (possibly empty). Let A satisfy (2.8) and let h:ℳ→(0,h0) be an exhaustion function, satisfying the following conditions: (a1)
there ish′>0 such that h-1([0,h′]) is compact and h is a solution of (2.9) in the open set h-1((h′,h0));(a2)
for a.e.t1,t2∈(h′,h0),  t1<t2,
(3.3)∫Σh(t2)〈∇h|∇h|,A(x,∇h)〉dHn-1=∫Σh(t1)〈∇h|∇h|,A(x,∇h)〉dHn-1.Heredℋn-1 is the element of the (n-1)-dimensional Hausdorff measure on Σh. Exhaustion functions with these properties will be called the special exhaustion functions ofℳwith respect toA. In most cases the mapping A will be the p-Laplace operator (2.13).Since the unit vectorν=∇h/|∇h| is orthogonal to the h-sphere Σh, the condition (a2) means that the flux of the vector field A(m,∇h) through h-spheres Σh(t) is constant.Suppose that the functionA(m,ξ)is continuously differentiable. If(b1)
h∈C2(ℳ∖K)and satisfies (2.9), and(b2)
at every pointm∈ℳwhere∂ℳhas a tangent planeTm(∂ℳ)the condition(3.4)〈A(m,∇h(m)),ν〉=0is satisfied whereνis a unit vector of the inner normal to the boundary∂ℳ, thenhis a special exhaustion function of the manifoldℳ.The proof of this statement is simple. Consider the open set(3.5)M(t1,t2)={m∈M:t1<h(m)<t2},0<t1<t2<∞,
with the boundary ∂ℳ(t1,t2). Using the Stokes formula, we have for noncritical values t1<t2 (for the definition of critical values of Ck-functions see, e.g., [6, Part II, Chapter 2, Section 10]) (3.6)∫Σh(t2)〈∇h|∇h|,A(m,∇h)〉dHn-1-∫Σh(t1)〈∇h|∇h|,A(m,∇h)〉dHn-1=∫∂M(t1,t2)∪∪i=1,2Σh(ti)〈ν,A(m,∇h)〉dHn-1=∫∂M(t1,t2)〈ν,A(m,∇h)〉dHn-1=∫M(t1,t2)div⁡A(m,∇h)*1=0,
and (a2) follows.Example 3.2.
We fix an integerk, 1≤k≤n, and set
(3.7)dk(x)=(∑i=1kxi2)1/2.
It is easy to see that |∇dk(x)|=1 everywhere in Rn∖Σ0 where Σ0={x∈Rn:dk(x)=0}. We will call the set
(3.8)Bk(t)={x∈Rn:dk(x)<t}
a k-ball and the set
(3.9)Σk(t)={x∈Rn:dk(x)=t}
a k-sphere in Rn.We will say that an unbounded domainD⊂Rn is k-admissible if for each t>infx∈Ddk(x) the set D∩Bk(t) has compact closure.It is clear that every unbounded domainD⊂Rn is n-admissible. In the general case the domain D is k-admissible if and only if the function dk(x) is an exhaustion function of D. It is not difficult to see that if a domain D⊂Rn is k-admissible, then it is l-admissible for all k<l<n.Fix1≤k<n. Let Δ be a bounded domain in the (n-k)-plane x1=⋯=xk=0 and let (3.10)D={x=(x1,…,xk,xk+1,…,xn)∈Rn:(xk+1,…,xn)∈Δ}
be a domain in Rn.The domainD is k-admissible. The k-spheres Σk(t) are orthogonal to the boundary ∂D and therefore 〈∇dk,ν〉=0 everywhere on the boundary. The function (3.11)h(x)={log⁡dk(x),p=k,dk(p-n)/(p-1)(x),p≠k,
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 3.3.
Fix1≤k<n. Let Δ be a bounded domain in the plane x1=⋯=xk=0 with a piecewise smooth boundary and let
(3.12)D={x=(x1,…,xn)∈Rn:(xk+1,…,xn)∈Δ}=Rn-k×Δ
be the cylinder domain with base Δ.The domainD is k-admissible. The k-spheres Σk(t) are orthogonal to the boundary ∂D and therefore 〈∇dk,ν〉=0 everywhere on the boundary, where dk is as in Example 3.2.Leth=ϕ(dk) where ϕ is a C2-function. We have ∇h=ϕ′∇dk and (3.13)∑i=1n∂∂xi(|∇h|n-2∂h∂xi)=∑i=1k∂∂xi((ϕ′)n-1∂dk∂xi)=(n-1)(ϕ′)n-2ϕ′′+k-1dk(ϕ′)n-1.
From the equation
(3.14)(n-1)ϕ′′+k-1dkϕ′=0
we conclude that the function(3.15)h(x)=(dk(x))(n-k)/(n-1)satisfies(2.13) inD∖Kand thus it is a special exhaustion function of the domainD.Example 3.4.
Let(r,θ), where r≥0, θ∈Sn-1(1), be the spherical coordinates in Rn. Let U⊂Sn-1(1), ∂U≠∅, be an arbitrary domain on the unit sphere Sn-1(1). We fix 0≤r1<∞ and consider the domain
(3.16)D={(r,θ)∈Rn:r1<r<∞,θ∈U}.
As mentioned above it is easy to verify that the given domain is n-admissible and the function(3.17)h(|x|)=log⁡|x|r1is a special exhaustion function of the domainD for p=n.Example 3.5.
Fix1≤n≤p. Let x1,x2,…,xn be an orthonormal system of coordinates in Rn,1≤n<p. Let D⊂Rn 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 byx∈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 functionh is a function on the domain D satisfying the conditions (b1), (b2) and (2.13). We consider the function h*=h∘π:ℳ→(0,∞).We have(3.18)∇h*=∇(h∘π)=(∇xh)∘π,div⁡(|∇h*|p-2∇h*)=div⁡(|∇(h∘π)|p-2∇(h∘π))=div⁡(|∇xh|p-2∘π(∇xh)∘π)=(∑i=1n∂∂xi(|∇xh|p-2∂h∂xi))∘π.
Because h is a special exhaustion function of D we have (3.19)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.Ifx∈∂D, then ν=ν1+ν2 where the vector ν1∈Rk is orthogonal to ∂D and ν2 is a vector from Tb(ℬ). Thus
(3.20)〈∇h*,ν〉=〈(∇xh)∘π,ν1〉=0,
because h is a special exhaustion function on D and satisfies the property (b2) on ∂D. If b∈∂ℬ, then the vector ν is orthogonal to ∂ℬ×Rn and (3.21)〈∇h*,ν〉=〈(∇xh)∘π,ν〉=0,
because the vector (∇xh)∘π is parallel to Rn.The other requirements for a special exhaustion function for the manifoldℳ are easy to verify.Therefore,the function(3.22)h*=h*(x,b)=h∘π:M⟶(0,∞)is a special exhaustion function on the manifoldℳ=D×ℬ.Example 3.6.
Let𝒜 be a compact Riemannian manifold, dim𝒜=k, with piecewise smooth boundary or without boundary. We consider the Cartesian product ℳ=𝒜×Rn, n≥1. We denote by a∈𝒜, x∈Rn and (a,x)∈ℳ the points of the corresponding spaces. It is easy to see that the function
(3.23)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<np-hyperbolic type.Example 3.7.
Let(r,θ), where r≥0, θ∈Sn-1(1), be the spherical coordinates in Rn. Let U⊂Sn-1(1) be an arbitrary domain on the unit sphere Sn-1(1). We fix 0≤r1<r2<∞ and consider the domain
(3.24)D={(r,θ)∈Rn:r1<r<r2,θ∈U}
with the metric
(3.25)dsM2=α2(r)dr2+β2(r)dlθ2,
where α(r),β(r)>0 are C0-functions on [r1,r2) and dlθ is an element of length on Sn-1(1).The manifoldℳ=(D,dsℳ2) is a warped Riemannian product. In the case α(r)≡1, β(r)=1, and U=Sn-1 the manifold ℳ is isometric to a cylinder in Rn+1. In the case α(r)≡1, β(r)=r, and U=Sn-1 the manifold ℳ is a spherical annulus in Rn.The volume element in the metric (3.25) is given by the expression (3.26)dσM=α(r)βn-1(r)drdSn-1(1).
If ϕ(r,θ)∈C1(D), then the length of the gradient ∇ϕ in ℳ takes the form (3.27)|∇ϕ|2=1α2(ϕr′)2+1β2|∇θϕ|2,
where ∇θϕ is the gradient in the metric of the unit sphere Sn-1(1).For the special exhaustion functionh(r,θ)≡h(r) (2.13) reduces to the following form (3.28)ddr((1α(r))p-1(hr′(r))p-1βn-1(r))=0.
Solutions of this equation are the functions (3.29)h(r)=C1∫r1rα(t)β(n-1)/(p-1)(t)dt+C2
where C1 and C2 are constants.Because the functionh satisfies obviously the boundary condition (a2) as well as the other conditions of Section 3.2, we see that under the assumption (3.30)∫r2α(t)β(n-1)/(p-1)(t)dt=∞
the function (3.31)h(r)=∫r1rα(t)β(n-1)/(p-1)(t)dt
is a special exhaustion function on the manifold ℳ.Theorem 3.8.
Leth:ℳ→(0,h0) be a special exhaustion function of a boundary set ξ of the manifold ℳ. Then(i)
ifh0=∞, the set ξ is of p-parabolic type,(ii)
ifh0<∞, the set ξ is of p-hyperbolic type.Proof.
Choose0<t1<t2<h0 such that K⊂Bh(t1). We need to estimate the p-capacity of the condenser (Bh(t1),ℳ∖Bh(t2);ℳ). We have
(3.32)capp(B¯h(t1),M∖Bh(t2);M)=J(t2-t1)p-1,
where
(3.33)J=∫Σh(t)|∇h|p-1dHMn-1
is a quantity independent of t>h(K)=sup{h(m):m∈K}. Indeed, for the variational problem (2.1) we choose the function φ0, φ0(m)=0 for m∈Bh(t1),
(3.34)φ0(m)=h(m)-t1t2-t1,m∈Bh(t2)∖Bh(t1)
and φ0(m)=1 for m∈ℳ∖Bh(t2). Using the Kronrod-Federer formula [7, Theorem 3.2.22], we get
(3.35)capp(Bh(t1),M∖Bh(t2);M)≤∫M|∇φ0|p*1M≤1(t2-t1)p∫t1<h(m)<t2|∇h(m)|p*1M=∫t1t2dt∫Σh(t)|∇h(m)|p-1dHMn-1.
Because the special exhaustion function satisfies (2.13) and the boundary condition (a2), one obtains for arbitrary τ1,τ2, h(K)<τ1<τ2<h0(3.36)∫Σh(t2)|∇h|p-1dHMn-1-∫Σh(t1)|∇h|p-1dHMn-1=∫Σh(t2)|∇h|p-2〈∇h,ν〉dHMn-1-∫Σh(t1)|∇h|p-2〈∇h,ν〉dHMn-1=∫t1<h(m)<t2div⁡M(|∇h|p-2∇h)*1M=0.
Thus we have established the inequality
(3.37)capp(Bh(t1),M∖Bh(t2);M)≤J(t2-t1)p-1.
By the conditions, imposed on the special exhaustion function, the functionφ0 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).
Ifh0=∞, then letting t2→∞ in (3.32) we conclude the parabolicity of the type of ξ. Let h0<∞. Consider an exhaustion {𝒰k} and choose t0>0 such that the h-ball Bh(t0) contains the compact set K.
Settk=supm∈∂𝒰kh(m). Then for tk>t0 we have (3.38)capp(U¯k0,Uk;M)≥capp(Bh(t0),Bh(tk);M)=J(tk-t0)p-1,
hence
(3.39)lim⁡inf⁡k→∞capp(U¯k0,Uk;M)≥J(h0-t0)p-1>0,
and the boundary set ξ is of p-hyperbolic type.


## 3.1. Exhaustion Functions of Boundary Sets

Leth:ℳ→(0,h0), 0<h0≤∞, be a locally Lipschitz function. For arbitrary t∈(0,h0) we denote by (3.1)Bh(t)={m∈M:h(m)<t},Σh(t)={m∈M:h(m)=t}
the h-balls and h-spheres, respectively.Leth:ℳ→R be a locally Lipschitz function such that: there exists a compact K⊂ℳ such that |∇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 mk∈ℳ, k=1,2,… the function h(mk)→h0 if and only if mk→ξ.It is easy to see that this requirement is satisfied if and only if for an arbitrary increasing sequencet1<t2<⋯<h0 the sequence of the open sets Vk={m∈ℳ:h(m)>tk} 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 functionh:ℳ→(0,h0) is called the exhaustion function of the manifold ℳ if the following two conditions are satisfied(i)
for allt∈(0,h0) the h-ball Bh(t)¯ is compact;(ii)
for every sequencet1<t2<⋯<h0 with limk→∞tk=h0, the sequence of h-balls {Bh(tk)} generates an exhaustion of ℳ, that is,
(3.2)Bh(t1)⊂Bh(t2)⊂⋯⊂Bh(tk)⊂⋯,⋃kBh(tk)=M.Example 3.1.
Letℳ be a Riemannian manifold. We set h(m)=dist(m,m0) where m0∈ℳ is a fixed point. Because |∇h(m)|=1 almost everywhere on ℳ, the function h defines an exhaustion function of the manifold ℳ.


## 3.2. Special Exhaustion Functions

Letℳ be a noncompact Riemannian manifold with the boundary ∂ℳ (possibly empty). Let A satisfy (2.8) and let h:ℳ→(0,h0) be an exhaustion function, satisfying the following conditions: (a1)
there ish′>0 such that h-1([0,h′]) is compact and h is a solution of (2.9) in the open set h-1((h′,h0));(a2)
for a.e.t1,t2∈(h′,h0),  t1<t2,
(3.3)∫Σh(t2)〈∇h|∇h|,A(x,∇h)〉dHn-1=∫Σh(t1)〈∇h|∇h|,A(x,∇h)〉dHn-1.Heredℋn-1 is the element of the (n-1)-dimensional Hausdorff measure on Σh. Exhaustion functions with these properties will be called the special exhaustion functions ofℳwith respect toA. In most cases the mapping A will be the p-Laplace operator (2.13).Since the unit vectorν=∇h/|∇h| is orthogonal to the h-sphere Σh, the condition (a2) means that the flux of the vector field A(m,∇h) through h-spheres Σh(t) is constant.Suppose that the functionA(m,ξ)is continuously differentiable. If(b1)
h∈C2(ℳ∖K)and satisfies (2.9), and(b2)
at every pointm∈ℳwhere∂ℳhas a tangent planeTm(∂ℳ)the condition(3.4)〈A(m,∇h(m)),ν〉=0is satisfied whereνis a unit vector of the inner normal to the boundary∂ℳ, thenhis a special exhaustion function of the manifoldℳ.The proof of this statement is simple. Consider the open set(3.5)M(t1,t2)={m∈M:t1<h(m)<t2},0<t1<t2<∞,
with the boundary ∂ℳ(t1,t2). Using the Stokes formula, we have for noncritical values t1<t2 (for the definition of critical values of Ck-functions see, e.g., [6, Part II, Chapter 2, Section 10]) (3.6)∫Σh(t2)〈∇h|∇h|,A(m,∇h)〉dHn-1-∫Σh(t1)〈∇h|∇h|,A(m,∇h)〉dHn-1=∫∂M(t1,t2)∪∪i=1,2Σh(ti)〈ν,A(m,∇h)〉dHn-1=∫∂M(t1,t2)〈ν,A(m,∇h)〉dHn-1=∫M(t1,t2)div⁡A(m,∇h)*1=0,
and (a2) follows.Example 3.2.
We fix an integerk, 1≤k≤n, and set
(3.7)dk(x)=(∑i=1kxi2)1/2.
It is easy to see that |∇dk(x)|=1 everywhere in Rn∖Σ0 where Σ0={x∈Rn:dk(x)=0}. We will call the set
(3.8)Bk(t)={x∈Rn:dk(x)<t}
a k-ball and the set
(3.9)Σk(t)={x∈Rn:dk(x)=t}
a k-sphere in Rn.We will say that an unbounded domainD⊂Rn is k-admissible if for each t>infx∈Ddk(x) the set D∩Bk(t) has compact closure.It is clear that every unbounded domainD⊂Rn is n-admissible. In the general case the domain D is k-admissible if and only if the function dk(x) is an exhaustion function of D. It is not difficult to see that if a domain D⊂Rn is k-admissible, then it is l-admissible for all k<l<n.Fix1≤k<n. Let Δ be a bounded domain in the (n-k)-plane x1=⋯=xk=0 and let (3.10)D={x=(x1,…,xk,xk+1,…,xn)∈Rn:(xk+1,…,xn)∈Δ}
be a domain in Rn.The domainD is k-admissible. The k-spheres Σk(t) are orthogonal to the boundary ∂D and therefore 〈∇dk,ν〉=0 everywhere on the boundary. The function (3.11)h(x)={log⁡dk(x),p=k,dk(p-n)/(p-1)(x),p≠k,
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 3.3.
Fix1≤k<n. Let Δ be a bounded domain in the plane x1=⋯=xk=0 with a piecewise smooth boundary and let
(3.12)D={x=(x1,…,xn)∈Rn:(xk+1,…,xn)∈Δ}=Rn-k×Δ
be the cylinder domain with base Δ.The domainD is k-admissible. The k-spheres Σk(t) are orthogonal to the boundary ∂D and therefore 〈∇dk,ν〉=0 everywhere on the boundary, where dk is as in Example 3.2.Leth=ϕ(dk) where ϕ is a C2-function. We have ∇h=ϕ′∇dk and (3.13)∑i=1n∂∂xi(|∇h|n-2∂h∂xi)=∑i=1k∂∂xi((ϕ′)n-1∂dk∂xi)=(n-1)(ϕ′)n-2ϕ′′+k-1dk(ϕ′)n-1.
From the equation
(3.14)(n-1)ϕ′′+k-1dkϕ′=0
we conclude that the function(3.15)h(x)=(dk(x))(n-k)/(n-1)satisfies(2.13) inD∖Kand thus it is a special exhaustion function of the domainD.Example 3.4.
Let(r,θ), where r≥0, θ∈Sn-1(1), be the spherical coordinates in Rn. Let U⊂Sn-1(1), ∂U≠∅, be an arbitrary domain on the unit sphere Sn-1(1). We fix 0≤r1<∞ and consider the domain
(3.16)D={(r,θ)∈Rn:r1<r<∞,θ∈U}.
As mentioned above it is easy to verify that the given domain is n-admissible and the function(3.17)h(|x|)=log⁡|x|r1is a special exhaustion function of the domainD for p=n.Example 3.5.
Fix1≤n≤p. Let x1,x2,…,xn be an orthonormal system of coordinates in Rn,1≤n<p. Let D⊂Rn 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 byx∈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 functionh is a function on the domain D satisfying the conditions (b1), (b2) and (2.13). We consider the function h*=h∘π:ℳ→(0,∞).We have(3.18)∇h*=∇(h∘π)=(∇xh)∘π,div⁡(|∇h*|p-2∇h*)=div⁡(|∇(h∘π)|p-2∇(h∘π))=div⁡(|∇xh|p-2∘π(∇xh)∘π)=(∑i=1n∂∂xi(|∇xh|p-2∂h∂xi))∘π.
Because h is a special exhaustion function of D we have (3.19)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.Ifx∈∂D, then ν=ν1+ν2 where the vector ν1∈Rk is orthogonal to ∂D and ν2 is a vector from Tb(ℬ). Thus
(3.20)〈∇h*,ν〉=〈(∇xh)∘π,ν1〉=0,
because h is a special exhaustion function on D and satisfies the property (b2) on ∂D. If b∈∂ℬ, then the vector ν is orthogonal to ∂ℬ×Rn and (3.21)〈∇h*,ν〉=〈(∇xh)∘π,ν〉=0,
because the vector (∇xh)∘π is parallel to Rn.The other requirements for a special exhaustion function for the manifoldℳ are easy to verify.Therefore,the function(3.22)h*=h*(x,b)=h∘π:M⟶(0,∞)is a special exhaustion function on the manifoldℳ=D×ℬ.Example 3.6.
Let𝒜 be a compact Riemannian manifold, dim𝒜=k, with piecewise smooth boundary or without boundary. We consider the Cartesian product ℳ=𝒜×Rn, n≥1. We denote by a∈𝒜, x∈Rn and (a,x)∈ℳ the points of the corresponding spaces. It is easy to see that the function
(3.23)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<np-hyperbolic type.Example 3.7.
Let(r,θ), where r≥0, θ∈Sn-1(1), be the spherical coordinates in Rn. Let U⊂Sn-1(1) be an arbitrary domain on the unit sphere Sn-1(1). We fix 0≤r1<r2<∞ and consider the domain
(3.24)D={(r,θ)∈Rn:r1<r<r2,θ∈U}
with the metric
(3.25)dsM2=α2(r)dr2+β2(r)dlθ2,
where α(r),β(r)>0 are C0-functions on [r1,r2) and dlθ is an element of length on Sn-1(1).The manifoldℳ=(D,dsℳ2) is a warped Riemannian product. In the case α(r)≡1, β(r)=1, and U=Sn-1 the manifold ℳ is isometric to a cylinder in Rn+1. In the case α(r)≡1, β(r)=r, and U=Sn-1 the manifold ℳ is a spherical annulus in Rn.The volume element in the metric (3.25) is given by the expression (3.26)dσM=α(r)βn-1(r)drdSn-1(1).
If ϕ(r,θ)∈C1(D), then the length of the gradient ∇ϕ in ℳ takes the form (3.27)|∇ϕ|2=1α2(ϕr′)2+1β2|∇θϕ|2,
where ∇θϕ is the gradient in the metric of the unit sphere Sn-1(1).For the special exhaustion functionh(r,θ)≡h(r) (2.13) reduces to the following form (3.28)ddr((1α(r))p-1(hr′(r))p-1βn-1(r))=0.
Solutions of this equation are the functions (3.29)h(r)=C1∫r1rα(t)β(n-1)/(p-1)(t)dt+C2
where C1 and C2 are constants.Because the functionh satisfies obviously the boundary condition (a2) as well as the other conditions of Section 3.2, we see that under the assumption (3.30)∫r2α(t)β(n-1)/(p-1)(t)dt=∞
the function (3.31)h(r)=∫r1rα(t)β(n-1)/(p-1)(t)dt
is a special exhaustion function on the manifold ℳ.Theorem 3.8.
Leth:ℳ→(0,h0) be a special exhaustion function of a boundary set ξ of the manifold ℳ. Then(i)
ifh0=∞, the set ξ is of p-parabolic type,(ii)
ifh0<∞, the set ξ is of p-hyperbolic type.Proof.
Choose0<t1<t2<h0 such that K⊂Bh(t1). We need to estimate the p-capacity of the condenser (Bh(t1),ℳ∖Bh(t2);ℳ). We have
(3.32)capp(B¯h(t1),M∖Bh(t2);M)=J(t2-t1)p-1,
where
(3.33)J=∫Σh(t)|∇h|p-1dHMn-1
is a quantity independent of t>h(K)=sup{h(m):m∈K}. Indeed, for the variational problem (2.1) we choose the function φ0, φ0(m)=0 for m∈Bh(t1),
(3.34)φ0(m)=h(m)-t1t2-t1,m∈Bh(t2)∖Bh(t1)
and φ0(m)=1 for m∈ℳ∖Bh(t2). Using the Kronrod-Federer formula [7, Theorem 3.2.22], we get
(3.35)capp(Bh(t1),M∖Bh(t2);M)≤∫M|∇φ0|p*1M≤1(t2-t1)p∫t1<h(m)<t2|∇h(m)|p*1M=∫t1t2dt∫Σh(t)|∇h(m)|p-1dHMn-1.
Because the special exhaustion function satisfies (2.13) and the boundary condition (a2), one obtains for arbitrary τ1,τ2, h(K)<τ1<τ2<h0(3.36)∫Σh(t2)|∇h|p-1dHMn-1-∫Σh(t1)|∇h|p-1dHMn-1=∫Σh(t2)|∇h|p-2〈∇h,ν〉dHMn-1-∫Σh(t1)|∇h|p-2〈∇h,ν〉dHMn-1=∫t1<h(m)<t2div⁡M(|∇h|p-2∇h)*1M=0.
Thus we have established the inequality
(3.37)capp(Bh(t1),M∖Bh(t2);M)≤J(t2-t1)p-1.
By the conditions, imposed on the special exhaustion function, the functionφ0 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).
Ifh0=∞, then letting t2→∞ in (3.32) we conclude the parabolicity of the type of ξ. Let h0<∞. Consider an exhaustion {𝒰k} and choose t0>0 such that the h-ball Bh(t0) contains the compact set K.
Settk=supm∈∂𝒰kh(m). Then for tk>t0 we have (3.38)capp(U¯k0,Uk;M)≥capp(Bh(t0),Bh(tk);M)=J(tk-t0)p-1,
hence
(3.39)lim⁡inf⁡k→∞capp(U¯k0,Uk;M)≥J(h0-t0)p-1>0,
and the boundary set ξ is of p-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 class𝒲𝒯2 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

Letℳ be an n-dimensional Riemannian manifold with nonempty boundary ∂ℳ. We will fix a closed differential form w, degw=k, 1≤k≤n, w∈Llocp(ℳ) of class 𝒲𝒯1 and the complementary closed form θ, degθ=n-k, θ∈Llocq(ℳ), satisfying the condition (1.4). We assume that there exists a differential form Z∈Wp,loc1 with continuous coefficients for which dZ=w.Leth:ℳ→(0,h0) be an exhaustion function of ℳ. As  mentioned before we let Bh(τ) be an h-ball and Σh(τ) an h-sphere.


### 4.2. Dirichlet Condition with Zero Boundary Values

We will say that the formZ∈Wp,loc1(with continuous coefficients and such that dZ=w) satisfies Dirichlet's condition with zero boundary values on ∂ℳ if for every differential form v∈Llocq(ℳ), degv=n-k, and for almost every τ∈(0,h0)(4.1)∫Bh(τ)w∧v+(-1)k-1∫Bh(τ)Z∧dv=∫Σh(τ)Z∧v.In particular, the formZ satisfies the boundary condition (4.1), if its coefficients are continuous and if its support does not intersect with ∂ℳ, that is (4.2)supp⁡Z∩∂M=∅,wheresupp⁡Z={m∈M:Z(m)≠0}¯.Ifℳ is compact then (4.1) takes the form
(4.3)∫Mw∧v+(-1)k-1∫MZ∧dv=0.


### 4.3. Neumann Condition with Zero Boundary Values

We will say that a formZ satisfies Neumann's condition with zero boundary values, if for every differential form v∈Wp,loc1(ℳ), degv=k-1, and for almost every τ∈(0,h0)(4.4)∫Bh(τ)dv∧θ=∫Σh(τ)v∧θ.Ifℳ is compact then (4.4) takes the form (4.5)∫Mdv∧θ=0.


### 4.4. Mixed Zero Boundary Condition

We will say that a formZ satisfies mixed zero boundary condition if for an arbitrary function ϕ∈C1(ℳ) and for almost every τ∈(0,h0) we have (4.6)∫Bh(τ)ϕw∧θ+(-1)n-1∫Bh(τ)Z∧θ∧dϕ=∫Σh(τ)ϕZ∧θ.Ifℳ is compact then (4.6) takes the form (4.7)∫Mϕw∧θ+(-1)n-1∫MZ∧θ∧dϕ=0.We assume that the form(4.8)Z∈C2(int⁡M)∩C1(∂M)
has the property (4.1). On the basis of Stokes' formula (the standard Stokes formula with generalized derivatives) we conclude that for almost every τ∈(0,h0)(4.9)∫Bh(τ)dZ∧v+(-1)k-1∫Bh(τ)Z∧dv=∫∂Bh(τ)Z∧v
holds. Therefore we get (4.10)∫∂Bh(τ)∖Σh(τ)Z∧v=0,∀v∈Wq,loc⁡1(M).
This implies that the restriction of Z onto the boundary ∂ℳ is the zero form, that is, (4.11)Z|∂M(m)=0ateverypointm∈∂M.We next clarify the geometric meaning of the condition (4.11). We assume that m∈∂ℳ is a point where the boundary ∂ℳ has a tangent plane Tm(∂ℳ) and that the form Z satisfies the regularity condition (4.8) in some neighborhood of the point m.Proposition 4.1.
If a formZ is simple at a point m∈ℳ, then the condition (4.11) is fulfilled if and only if the form Z is of the form
(4.12)Z=ω∧dxn,
where ω is a form, degω=degZ-1.Proof.
We give an orthonormal system of coordinatesx1,…,xn at the point m such that the hyperplane Tm(∂ℳ) is given by the equation xn=0. Let degZ=l. Because the form Z is simple, we can represent it as follows
(4.13)Z=(∑i=1la1,idxi+a1,ndxn)∧⋯∧(∑i=1lal,idxi+al,ndxn),
where ai,j=ai,j(m) are some constants. The condition (4.11) can now be rewritten as follows
(4.14)(∑i=1la1,idxi)∧⋯∧(∑i=1lal,idxi)=0,
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 (4.15)Z,v∈C2(int⁡M)∩C1(∂M).
By Stokes' formula we have for almost every τ∈(0,h0)(4.16)∫∂Bh(τ)v∧θ=∫Bh(τ)dv∧θ+(-1)k-1∫Bh(τ)v∧dθ.
Because the form θ is closed, condition (4.4) gives (4.17)∫Σh(τ)v∧θ=0∀v∈Wp,loc⁡1.
Therefore (4.18)θ|∂M(m)=0
at every point m∈∂ℳ.Exactly in the same way we verify that the mixed zero boundary condition (4.6) is equivalent to the condition (4.19)Z∧θ|∂M(m)=0
at every point m∈∂ℳ.Consider the case of quasilinear equations (2.9). Let m∈∂ℳ be a regular point and let x1,…,xn be local coordinates in a neighborhood of this point. We have (4.20)θ=*∑i=1nAi(m,∇f(m))dxi=∑i=1n(-1)i-1Ai(m,∇f(m))dx1∧⋯∧dxî∧⋯∧dxn.
We set Z=f. In the case (4.1) we choose v=ϕθ where θ is an arbitrary locally Lipschitz function. We obtain (4.21)∫Bh(τ)ϕdf∧θ+∫Bh(τ)fd(ϕθ)=∫Σh(τ)ϕfθ
and further
(4.22)∫Bh(τ)∑i=1n(ϕf)xiAi(m,∇f)*1=∫Σh(τ)ϕfθ,∀ϕ.
This condition characterizes generalized solutions of (2.9) with zero Dirichlet boundary condition on ∂ℳ.On the other side, choosing in the case of the Neumann condition (4.4) for v an arbitrary locally Lipschitz function ϕ we get for almost every τ∈(0,h)(4.23)∫Bh(τ)〈∇ϕ,A(m,∇f)〉*1=∫Σh(τ)ϕ〈A(m,∇f),ν〉dHMn-1
which characterizes generalized solutions of (2.9) with zero Neumann boundary conditions on ∂ℳ.It is easy to see that at every point of the boundary we have(4.24)(-1)i-1dx1∧⋯∧dxî∧⋯∧dxn|∂M=cos⁡(ν,xi)dHMn-1,
where (ν,xi) is the angle between the inner normal vector ν to ∂ℳ and the direction 0xi; dℋℳn-1 is the element of surface area on ℳ.Thus, at a regular boundary point, the condition (4.18) is equivalent to the requirement (4.25)〈A(m,∇f(m)),ν〉=0.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 for𝒲𝒯-Forms

Letℳ be a compact Riemannian manifold with nonempty boundary, dimℳ=n, and let v∈Llocp, degw=k, 1≤k≤n, be a differential form of class 𝒲𝒯1 on ℳ. Let θ, degθ=n-k, be a form complementary to the form w.Theorem 4.2.
Suppose that there exists a differential formZ∈Wp,loc1(ℳ), dZ=w on ℳ. If either (4.3) or (4.5) holds, then θ≡0 on ℳ.Proof.
We assume that (4.3) holds and set v=θ. Then (4.3) yields
(4.26)∫Mw∧θ=0.
Because
(4.27)(-1)k(n-k)(-1)k(n-k)*(w∧θ)=(-1)k(n-k)*-1(w∧θ)=*-1(w∧(-1)k(n-k)θ)=*-1(w∧*(*θ))=〈w,*θ〉,
we get
(4.28)∫Mw∧θ=∫M*(w∧θ)*1=∫M〈w,*θ〉*1.
Using (1.5) we deduce
(4.29)0=∫Mw∧θ≥ν0∫M|θ|q*1.
We assume that the boundary condition (4.5) holds. Choose v=Z. Then (4.5) gives(4.30)∫Mw∧θ=0.
As above, we arrive at the inequality (4.29). This inequality implies that θ≡0 on ℳ.In order to illustrate Theorem4.2 we consider the example of generalized solutions f∈Wp,loc1(ℳ) of (2.9) under the condition
(4.31)ν0|A(m,ξ)|p≤〈ξ,A(m,ξ)〉
for all ξ∈Tm(ℳ) with the constants p>1 and ν0>0.SettingZ=f we getCorollary 4.3.
Suppose that the manifoldℳ is compact and the boundary ∂ℳ is not empty. If the function f satisfies the condition (4.22) or (4.23), then f≡const on ℳ.


## 4.1. Boundary Conditions

Letℳ be an n-dimensional Riemannian manifold with nonempty boundary ∂ℳ. We will fix a closed differential form w, degw=k, 1≤k≤n, w∈Llocp(ℳ) of class 𝒲𝒯1 and the complementary closed form θ, degθ=n-k, θ∈Llocq(ℳ), satisfying the condition (1.4). We assume that there exists a differential form Z∈Wp,loc1 with continuous coefficients for which dZ=w.Leth:ℳ→(0,h0) be an exhaustion function of ℳ. As  mentioned before we let Bh(τ) be an h-ball and Σh(τ) an h-sphere.


## 4.2. Dirichlet Condition with Zero Boundary Values

We will say that the formZ∈Wp,loc1(with continuous coefficients and such that dZ=w) satisfies Dirichlet's condition with zero boundary values on ∂ℳ if for every differential form v∈Llocq(ℳ), degv=n-k, and for almost every τ∈(0,h0)(4.1)∫Bh(τ)w∧v+(-1)k-1∫Bh(τ)Z∧dv=∫Σh(τ)Z∧v.In particular, the formZ satisfies the boundary condition (4.1), if its coefficients are continuous and if its support does not intersect with ∂ℳ, that is (4.2)supp⁡Z∩∂M=∅,wheresupp⁡Z={m∈M:Z(m)≠0}¯.Ifℳ is compact then (4.1) takes the form
(4.3)∫Mw∧v+(-1)k-1∫MZ∧dv=0.


## 4.3. Neumann Condition with Zero Boundary Values

We will say that a formZ satisfies Neumann's condition with zero boundary values, if for every differential form v∈Wp,loc1(ℳ), degv=k-1, and for almost every τ∈(0,h0)(4.4)∫Bh(τ)dv∧θ=∫Σh(τ)v∧θ.Ifℳ is compact then (4.4) takes the form (4.5)∫Mdv∧θ=0.


## 4.4. Mixed Zero Boundary Condition

We will say that a formZ satisfies mixed zero boundary condition if for an arbitrary function ϕ∈C1(ℳ) and for almost every τ∈(0,h0) we have (4.6)∫Bh(τ)ϕw∧θ+(-1)n-1∫Bh(τ)Z∧θ∧dϕ=∫Σh(τ)ϕZ∧θ.Ifℳ is compact then (4.6) takes the form (4.7)∫Mϕw∧θ+(-1)n-1∫MZ∧θ∧dϕ=0.We assume that the form(4.8)Z∈C2(int⁡M)∩C1(∂M)
has the property (4.1). On the basis of Stokes' formula (the standard Stokes formula with generalized derivatives) we conclude that for almost every τ∈(0,h0)(4.9)∫Bh(τ)dZ∧v+(-1)k-1∫Bh(τ)Z∧dv=∫∂Bh(τ)Z∧v
holds. Therefore we get (4.10)∫∂Bh(τ)∖Σh(τ)Z∧v=0,∀v∈Wq,loc⁡1(M).
This implies that the restriction of Z onto the boundary ∂ℳ is the zero form, that is, (4.11)Z|∂M(m)=0ateverypointm∈∂M.We next clarify the geometric meaning of the condition (4.11). We assume that m∈∂ℳ is a point where the boundary ∂ℳ has a tangent plane Tm(∂ℳ) and that the form Z satisfies the regularity condition (4.8) in some neighborhood of the point m.Proposition 4.1.
If a formZ is simple at a point m∈ℳ, then the condition (4.11) is fulfilled if and only if the form Z is of the form
(4.12)Z=ω∧dxn,
where ω is a form, degω=degZ-1.Proof.
We give an orthonormal system of coordinatesx1,…,xn at the point m such that the hyperplane Tm(∂ℳ) is given by the equation xn=0. Let degZ=l. Because the form Z is simple, we can represent it as follows
(4.13)Z=(∑i=1la1,idxi+a1,ndxn)∧⋯∧(∑i=1lal,idxi+al,ndxn),
where ai,j=ai,j(m) are some constants. The condition (4.11) can now be rewritten as follows
(4.14)(∑i=1la1,idxi)∧⋯∧(∑i=1lal,idxi)=0,
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 (4.15)Z,v∈C2(int⁡M)∩C1(∂M).
By Stokes' formula we have for almost every τ∈(0,h0)(4.16)∫∂Bh(τ)v∧θ=∫Bh(τ)dv∧θ+(-1)k-1∫Bh(τ)v∧dθ.
Because the form θ is closed, condition (4.4) gives (4.17)∫Σh(τ)v∧θ=0∀v∈Wp,loc⁡1.
Therefore (4.18)θ|∂M(m)=0
at every point m∈∂ℳ.Exactly in the same way we verify that the mixed zero boundary condition (4.6) is equivalent to the condition (4.19)Z∧θ|∂M(m)=0
at every point m∈∂ℳ.Consider the case of quasilinear equations (2.9). Let m∈∂ℳ be a regular point and let x1,…,xn be local coordinates in a neighborhood of this point. We have (4.20)θ=*∑i=1nAi(m,∇f(m))dxi=∑i=1n(-1)i-1Ai(m,∇f(m))dx1∧⋯∧dxî∧⋯∧dxn.
We set Z=f. In the case (4.1) we choose v=ϕθ where θ is an arbitrary locally Lipschitz function. We obtain (4.21)∫Bh(τ)ϕdf∧θ+∫Bh(τ)fd(ϕθ)=∫Σh(τ)ϕfθ
and further
(4.22)∫Bh(τ)∑i=1n(ϕf)xiAi(m,∇f)*1=∫Σh(τ)ϕfθ,∀ϕ.
This condition characterizes generalized solutions of (2.9) with zero Dirichlet boundary condition on ∂ℳ.On the other side, choosing in the case of the Neumann condition (4.4) for v an arbitrary locally Lipschitz function ϕ we get for almost every τ∈(0,h)(4.23)∫Bh(τ)〈∇ϕ,A(m,∇f)〉*1=∫Σh(τ)ϕ〈A(m,∇f),ν〉dHMn-1
which characterizes generalized solutions of (2.9) with zero Neumann boundary conditions on ∂ℳ.It is easy to see that at every point of the boundary we have(4.24)(-1)i-1dx1∧⋯∧dxî∧⋯∧dxn|∂M=cos⁡(ν,xi)dHMn-1,
where (ν,xi) is the angle between the inner normal vector ν to ∂ℳ and the direction 0xi; dℋℳn-1 is the element of surface area on ℳ.Thus, at a regular boundary point, the condition (4.18) is equivalent to the requirement (4.25)〈A(m,∇f(m)),ν〉=0.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 for𝒲𝒯-Forms

Letℳ be a compact Riemannian manifold with nonempty boundary, dimℳ=n, and let v∈Llocp, degw=k, 1≤k≤n, be a differential form of class 𝒲𝒯1 on ℳ. Let θ, degθ=n-k, be a form complementary to the form w.Theorem 4.2.
Suppose that there exists a differential formZ∈Wp,loc1(ℳ), dZ=w on ℳ. If either (4.3) or (4.5) holds, then θ≡0 on ℳ.Proof.
We assume that (4.3) holds and set v=θ. Then (4.3) yields
(4.26)∫Mw∧θ=0.
Because
(4.27)(-1)k(n-k)(-1)k(n-k)*(w∧θ)=(-1)k(n-k)*-1(w∧θ)=*-1(w∧(-1)k(n-k)θ)=*-1(w∧*(*θ))=〈w,*θ〉,
we get
(4.28)∫Mw∧θ=∫M*(w∧θ)*1=∫M〈w,*θ〉*1.
Using (1.5) we deduce
(4.29)0=∫Mw∧θ≥ν0∫M|θ|q*1.
We assume that the boundary condition (4.5) holds. Choose v=Z. Then (4.5) gives(4.30)∫Mw∧θ=0.
As above, we arrive at the inequality (4.29). This inequality implies that θ≡0 on ℳ.In order to illustrate Theorem4.2 we consider the example of generalized solutions f∈Wp,loc1(ℳ) of (2.9) under the condition
(4.31)ν0|A(m,ξ)|p≤〈ξ,A(m,ξ)〉
for all ξ∈Tm(ℳ) with the constants p>1 and ν0>0.SettingZ=f we getCorollary 4.3.
Suppose that the manifoldℳ is compact and the boundary ∂ℳ is not empty. If the function f satisfies the condition (4.22) or (4.23), then f≡const on ℳ.


## 5. Estimates for Energy Integral: Applications

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


### 5.1. Basic Theorem

Letℳ be a noncompact Riemannian manifold, dimℳ=n. We consider a class ℱ of differential forms Z∈Wp,loc1(ℳ), degZ=k-1, such that the form dZ=w satisfies the conditions (1.1) and is in the class 𝒲𝒯2. Let θ∈Llocq be a form satisfying the condition (1.4), complementary to w.If the boundary∂ℳ is nonempty then we will 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 (4.1), (4.2), (4.4), and (4.6). We will denote by ℱB(ℳ) the set of forms Z, dZ∈𝒲𝒯2, satisfying the boundary condition B on ℳ. In particular, below we will operate with the classes of the ℱD, ℱ0, ℱN, and ℱDN forms corresponding to the boundary conditions (4.1), (4.2), (4.4), and (4.6), respectively.We fix a locally Lipschitz exhaustion functionh:ℳ→(0,h0), 0<h0≤∞. Let τ∈(0,h0) and let Bh(τ) be an h-ball, and Σh(τ) its boundary sphere as before.We introduce a characteristicε(τ) setting (5.1)ε(τ;FB)=inf⁡∫Σh(τ)|w|p|∇h|-1dHMn-1|∫Σh(τ)〈Z,*θ〉dHMn-1|,
where the infimum is taken over all Z∈ℱB(ℳ), Z≠0.Some estimates of (5.1) are given in [8, 9].Under these circumstances we have the following theorem.Theorem 5.1.
Suppose that the formZ∈ℱB(ℳ) satisfies one of the boundary condition (4.1), (4.4), or (4.6). Then for almost all τ∈(0,h0) and for an arbitrary τ0 the following relation holds
(5.2)ddτ(I(τ)exp⁡{-ν1∫τ0τε(t;FB)dt})≥0,
where
(5.3)I(τ)=∫Bh(τ)|w|p*1.
In particular, for allτ1<τ2 we have (5.4)I(τ1)≤I(τ2)exp⁡{-ν1∫τ1τ2ε(t)dt}.Proof.
The Kronrod-Federer formula yields(5.5)I(τ)=∫0τdt∫Σh(τ)|w|pdHMn-1|∇h|
and, in particular, the function I(τ) is absolutely continuous on closed intervals of (0,h0). Now it is enough to prove the inequality
(5.6)ddτI(τ)≥ν1I(τ)ε(τ).
From (5.5) we have for almost every τ∈(0,h0)(5.7)ddτI(τ)=∫Σh(τ)|w|pdHMn-1|∇h|.
By (1.6) we obtain(5.8)I(τ)=∫Bh(τ)|w|p*1≤ν1-1∫Bh(τ)〈w,*θ〉*1=ν1-1∫Bh(τ)w∧θ=ν1-1∫Bh(τ)dZ∧θ.
However, the form Z is weakly closed and satisfies one of the conditions (4.1), (4.2), or (4.4). Therefore for a.e. τ∈(0,h0),
(5.9)∫Bh(τ)dZ∧θ=∫Σh(τ)Z∧θ.
Thus we get
(5.10)I(τ)≤ν1-1∫Σh(τ)〈Z,*θ〉dHMn-1.
Further from (5.1) it follows that(5.11)∫Σh(τ)|w|pdHMn-1|∇h|≥ε(τ;FB)|∫Σh(τ)〈Z,*θ〉dHMn-1|.
Combining the above inequalities we obtain(5.12)I(τ)≤ν1-1ε(τ;FB)∫Σh(τ)|w|pdHMn-1|∇h|.
This inequality together with the equality (5.7) yields
(5.13)I(τ)≤ν1-1ε(τ;FB)ddτI(τ).
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 byℱ(Bh(τ)) the set of all differential forms
(5.14)Z0∈C1(Bh(τ)),deg⁡Z0=k-1,dZ0=0,
such that for almost every τ∈(0,h0) and for an arbitrary Lipschitz function ϕ the following formula holds (5.15)∫Σh(τ)ϕZ0∧θ=∫Bh(τ)dϕ∧Z0∧θ.Theorem 5.2.
If the differential formZ∈ℱB(ℳ), dZ∈𝒲𝒯2, satisfies the boundary condition (4.1), (4.4), or (4.6), then for all τ1<τ2<h0 and for an arbitrary form Z0∈ℱ(Bh(τ2)) the following relation holds
(5.16)ν1∫Bh(τ1)|dZ|p*1≤pτ2-τ1∫Bh(τ2)∖Bh(τ1)|∇h||(Z-Z0)∧θ|*1.Proof.
We consider the function(5.17)ϕ(m)={1form∈Bh(τ1),τ2-h(m)τ2-τ1form∈Bh(τ2)∖Bh(τ1),0form∈M∖Bh(τ2).
Suppose that the form Z satisfies the condition (4.1). Setting in (4.1) v=(ϕ)pZ∧θ we get
(5.18)∫Bh(τ2)(ϕ)pw∧θ+(-1)k-1∫Bh(τ2)Z∧d(ϕ)p∧θ=0,
or
(5.19)∫Bh(τ2)(ϕ)p〈w,*θ〉*1=(-1)kp∫Bh(τ2)(ϕ)p-1Z∧dϕ∧θ.
The function(ϕ)p is locally Lipschitz on B¯h(τ2) and ϕ|Σh(τ2)=0. Thus by (5.15) we get (5.20)∫Bh(τ2)dϕp∧Z0∧θ=∫Bh(τ2)d(ϕ)p∧Z0∧θ+∫Bh(τ2)(ϕ)pdZ0∧θ+∫Bh(τ2)(ϕ)pZ0∧dθ=∫Σh(τ2)(ϕ)pZ0∧θ=0.
Hence we arrive at the relation(5.21)∫Bh(τ2)(ϕ)p〈w,*θ〉*1=(-1)kp∫Bh(τ2)(ϕ)p-1(Z-Z0)∧dϕ∧θ,
which by (1.6) yields
(5.22)ν1∫Bh(τ2)(ϕ)p|w|p*1≤pτ2-τ1∫Bh(τ2)(ϕ)p-1|(Z-Z0)∧θ||∇h|*1.
Observing thatϕ(m)=1 for m∈Bh(τ1) and ϕ(m)=0 for m∈ℳ∖Bh(τ2) we obtain(5.23)ν1∫Bh(τ1)|w|p*1≤pτ2-τ1∫Bh(τ2)∖Bh(τ1)(ϕ)p-1|∇h||(Z-Z0)∧θ|*1.
Because |ϕ|≤1, the inequality (5.16) follows.
Let the formZ satisfy the condition (4.4). We choose v=(ϕ)pZ and observe that(5.24)v|Σh(τ2)=0.
Then we get
(5.25)∫Bh(τ2)(ϕ)pw∧θ=-∫Bh(τ2)d(ϕ)p∧Z∧θ=(-1)kp∫Bh(τ2)(ϕ)p-1Z∧dϕ∧θ.
Further details of the proof in this case are similar to those carried out above.
We assume that the formZ satisfies the mixed boundary condition (4.6). Observing that(5.26)(ϕ)p|Σh(τ2)=0,
we get
(5.27)∫Bh(τ2)(ϕ)pw∧θ=(-1)n∫Bh(τ2)Z∧θ∧d(ϕ)p=(-1)n-kp∫Bh(τ2)(ϕ)p-1Z∧dϕ∧θ.
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 formθ of dZ=w. Such an estimate is given in the next theorem.Theorem 5.3.
If the formZ, dZ∈𝒲𝒯2(ℳ), satisfies on ∂ℳ one of the boundary conditions (4.1), (4.4), or (4.6), then for all 0<τ1<τ2<h0 and for an arbitrary form Z0∈ℱ(Bh(τ2)) we have
(5.28)∫Bh(τ1)|dZ|p*1≤(pν2(τ2-τ1)ν1)p∫Bh(τ2)∖Bh(τ1)|∇h|p|Z-Z0|p*1.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
(5.29)∫M(ϕ)p-1|∇h||(Z-Z0)|∧θ*1≤∫M(ϕ)p-1|∇h||Z-Z0||θ|*1≤ν2∫M(ϕ)p-1|∇h||Z-Z0||w|p-1*1≤ν2(∫M|∇h|p|Z-Z0|p*1)1/p(∫Mϕp|w|p*1)(p-1)/p.
From (5.22) we get
(5.30)(ν1ν2)p∫M|w|p*1≤(pτ2-τ1)p∫M|∇h|p|Z-Z0|p*1.
Using the facts that ϕ=1 on Bh(τ1) and ϕ=0 on ℳ∖Bh(τ2) we easily obtain  (5.28).


### 5.2. Phragmén-Lindelöf Theorem

Letℳ be an n-dimensional noncompact Riemannian manifold with or without boundary and let w∈𝒲𝒯2 be a differential form as in (1.1), degw=k, and θ its complementary form as in (1.4).We assume that there exists a differential formZ∈Wp,loc1 with dZ=w. If the boundary ∂ℳ is nonempty, then we will assume that Z satisfies the boundary condition of Dirichlet (4.1), Neumann's condition (4.4), or the condition (4.6).We fix a locally Lipschitz exhaustion functionh:ℳ→(0,h0), 0<h0≤∞. Let, as above (5.31)I(τ;Z)=∫Bh(τ)|dZ|p*1,
and let (5.32)μ(τ;Z)=inf⁡∫τ<h(m)<τ+1|∇h||(Z-Z0)∧θ|*1,m(τ;Z)=inf⁡∫τ<h(m)<τ+1|∇h|p|Z-Z0|p*1,
where the infimum is taken over all closed forms Z0, satisfying conditions (5.14), (5.15) on Bh(τ).The following theorem exhibits a generalization of the classical Phragmén-Lindelöf principle for holomorphic functions.Theorem 5.4.
Suppose that the formZ, dZ∈𝒲𝒯2(ℳ), satisfies one of the boundary conditions (4.1), (4.4) or (4.6). The following alternatives hold: either the form dZ=0 a.e. on the manifold ℳ, or for all τ0∈(0,h0) we have
(5.33)lim⁡inf⁡τ→h0I(τ;Z)exp⁡{-ν1∫τ0τε(t;FB)dt}>0;(5.34)lim⁡inf⁡τ→h0μ(τ;Z)exp⁡{-ν1∫τ0τε(t;FB)dt}>0,(5.35)lim⁡inf⁡τ→h0m(τ;Z)exp⁡{-ν1∫τ0τε(t;FB)dt}>0.Proof.
The property (5.33) follows readily from (5.4) and is presented here only for the sake of completeness.
By (5.4) and (5.16), for a.e. τ∈(τ0,h0) we have(5.36)I(τ0)≤I(τ)exp⁡{-ν1∫τ0τε(t)dt}≤pν1-1∫Bh(τ+1)∖Bh(τ)|∇h||(Z-Z0)∧θ|*1exp⁡{-ν1∫τ0τε(t)dt}.
Therefore we get
(5.37)I(τ0)≤pν1-1μ(τ;Z)exp⁡{-ν1∫τ0τε(t)dt}.
Analogously, using (5.28) we get (5.38)I(τ0)≤(ν2ν1)pppm(τ;Z)exp⁡{-ν1∫τ0τε(t)dt}.
If we now assume that the formw≢0, then I(τ0)>0 for some τ0∈(0,h0). From this there easily follow (5.34) and (5.35).


### 5.3. 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 class𝒲𝒯2 defined on the manifold ℳ and the rate of growth of their energy integrals.Letℳ be an n-dimensional noncompact Riemannian manifold with or without boundary. We fix a locally Lipschitz exhaustion function h:ℳ→(0,h0), 0<h0≤∞, of the manifold ℳ.We assume that there areL≥1 mutually disjoint domains 𝒪1,𝒪2,…,𝒪L on ℳ such that 𝒪i∩∂ℳ=∅ if the boundary ∂ℳ is nonempty. We also assume that on each domain 𝒪i   there is given a differential form Zi with continuous coefficients and the properties:(i)
degZi=k-1, dZ≢0,(ii)
dZi=w∈𝒲𝒯2 with structure constants p,ν1,ν2, independent of i=1,2,…,L,(iii)
Zi satisfies on ∂𝒪i the zero boundary condition (4.2).We define a formZ on ℳ by setting Z|𝒪i=Zi and Z=0 on ℳ∖⋃i=1L𝒪i.According to Theorem4.2 each of the domains 𝒪i has a noncompact closure. Then by Theorem 5.4 the "narrower" the intersection of the domains 𝒪i with h-spheres Σh(t) for t→∞, the higher is the rate of growth of the form Z. Below we will consider the Denjoy-Carleman-Ahlfors theorem as a statement on the connection between the number L of mutually disjoint domains 𝒪i on ℳ0 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 will 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 Rn see [10, 11] for the history of the problem.We next introduce some necessary notation. We consider an open subsetD⊂ℳ with a noncompact closure and we assume that the restriction of the form Z to D satisfies condition (4.2).The functionh|D:D→(0,∞) is an exhaustion function of D. We fix an h-ball Bh(τ). Considering the variational problem (5.1) for the class of forms Z, satisfying the boundary condition (4.2) on D1 we define the characteristic (5.39)ε(t;D)=ε(t;F0)
where ℱ0 is defined in Section 5.1 and in (4.1).Following [12] we introduce the N-mean (5.40)E(t;N)=inf⁡1N∑k=1Nε(t;Dk)
where the infimum is taken over all decompositions of D into N nonintersecting open sets D1,D2,…,DN with noncompact closures.We record the following simple result.Lemma 5.5.
LetD1⊂D2 be arbitrary open subsets of ℳ with noncompact closures. Then
(5.41)ε(t;D2)≤ε(t;D1),(5.42)ε(t;M)≤E(t;N),N≥1.Proof.
It is enough to observe that each pair of formsZ,Z0 admissible for the variational problem (5.1) for the set D1 is also admissible for this problem for the set D2.
From (5.41) we get (5.42).We next derive a more general assertion about the monotonicity ofN-means.Lemma 5.6.
For arbitraryN>1 we have
(5.43)E(t;N+1)≥E(t;N).Proof.
We consider an arbitrary family of open subsets{Dk}, k=1,2,…,N+1, admissible for the infimum in (5.40). It is not difficult to see that
(5.44)1N+1∑k=1N+1ε(t;Dk)=1N+1∑k=1N+1(1N∑j=1,j≠kN+1ε(t;Dj)).
Because
(5.45)1N∑j=1,j≠kN+1ε(t;Dk)≥E(t;N),
we see that
(5.46)1N+1∑k=1N+1ε(t;Dk)≥E(t;N)
and the lemma is proved.The next theorem provides a solution to the aforementioned problem concerning the connection between the numberL of finite forms on ℳ, and the rate of growth of the total energy of these forms or the sum of their Lp-norms on an h-ball Bh(τ).Theorem 5.7.
Suppose that the manifoldℳ satisfies the properties listed in the beginning of this subsection and that for some N=1,2,…(5.47)∫h0h0E(t,N)dt=∞.
If the differential form Z, dZ∈𝒲𝒯2(ℳ), is such that
(5.48)lim⁡inf⁡τ→h0∫h(m)<τ|dZ|p*1exp⁡{-ν1∫τ0τE(t;N)dt}=0,
or
(5.49)lim⁡inf⁡τ′,τ′′→h00<τ′<τ′′<h0⁡∫τ′<h(m)<τ′′|∇h||Z∧θ|*1exp⁡{-ν1∫τ0τ′E(t;N)dt}=0,
or
(5.50)lim⁡inf⁡τ′,τ′′→h00<τ′<τ′′<h0⁡∫τ′<h(m)<τ′′|∇h|p|Z|p*1exp⁡{-ν1∫τ0τ′E(t;N)dt}=0,
then L<N.Proof.
We assume that there existsN mutually nonintersecting domains 𝒪1,𝒪2,…,𝒪N on the set ℳ0 and the forms Zi defined on 𝒪i with above properties. We denote by
(5.51)d(Ok)=inf⁡m∈Okh(m),d=max⁡1≤k≤Nd(Ok).
Fixτ0>d. Using the inequality (5.4) from Theorem 5.1 for an arbitrary k=1,2,…,N and a.e. 0<τ0<τ′<h0 we have (5.52)Ik(τ0)exp⁡{ν1∫τ0τ′εk(t)dt}≤Ik(τ′),
where
(5.53)Ik(τ′)=∫Ok∩Bh(τ′)|dZk|p*1,εk(τ′)=ε(τ′;Ok).
Adding these inequalities, we get
(5.54)min⁡1≤k≤NIk(τ0)∑k=1Nexp⁡{ν1∫τ0τ′εk(t)dt}≤I(τ′),
where
(5.55)I(τ′)=∫Bh(τ′)|dZ|p*1.
Applying the arithmetic-geometric mean inequality
(5.56)1N∑k=1Nexp⁡{ν1∫τ0τ′εk(t)dt}≥∏k=1Nexp⁡{ν1N∫τ0τ′εk(t)dt}
we get
(5.57)min⁡1≤k≤NIk(τ0)Nexp⁡{ν1N∫τ0τ′1N∑k=1Nεk(t)dt}≤I(τ′).
The domains𝒪1,𝒪2,…,𝒪N are nonintersecting. Therefore for all τ0<t<h0 we have (5.58)1N∑k=1Nεk(t)≥E(t;N).
The preceding inequality gives now
(5.59)min⁡1≤k≤NIk(τ0)Nexp⁡{ν1∫τ0τ′E(t;N)dt}≤I(τ′).
For the estimation of the integralI(τ′) we use the inequalities (5.16) and (5.28) and obtain (5.60)min⁡1≤k≤NIk(τ0)≤C1∫τ′<h(m)<τ′′|∇h||Z∧θ|*1exp⁡{-ν1∫τ0τ′E(t;N)dt}
or
(5.61)min⁡1≤k≤NIk(τ0)≤C2∫τ′<h(m)<τ′′|∇h|p|Z|p*1exp⁡{-ν1∫τ0τ′E(t;N)dt},
where
(5.62)C1=pτ′′-τ′,C2=(pν2(τ′′-τ′)ν1).
On the basis of the conditions (5.47)–(5.50) imposed on the form Z, for some k, 1≤k≤N, we have Ik(τ0)=0. Thus dZk(m)≡0 on 𝒪k∩Bh(τ0). Because we have chosen τ0>d arbitrarily, we can conclude that at least on one of the components 𝒪k, dZk≡0. A contradiction.


## 5.1. Basic Theorem

Letℳ be a noncompact Riemannian manifold, dimℳ=n. We consider a class ℱ of differential forms Z∈Wp,loc1(ℳ), degZ=k-1, such that the form dZ=w satisfies the conditions (1.1) and is in the class 𝒲𝒯2. Let θ∈Llocq be a form satisfying the condition (1.4), complementary to w.If the boundary∂ℳ is nonempty then we will 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 (4.1), (4.2), (4.4), and (4.6). We will denote by ℱB(ℳ) the set of forms Z, dZ∈𝒲𝒯2, satisfying the boundary condition B on ℳ. In particular, below we will operate with the classes of the ℱD, ℱ0, ℱN, and ℱDN forms corresponding to the boundary conditions (4.1), (4.2), (4.4), and (4.6), respectively.We fix a locally Lipschitz exhaustion functionh:ℳ→(0,h0), 0<h0≤∞. Let τ∈(0,h0) and let Bh(τ) be an h-ball, and Σh(τ) its boundary sphere as before.We introduce a characteristicε(τ) setting (5.1)ε(τ;FB)=inf⁡∫Σh(τ)|w|p|∇h|-1dHMn-1|∫Σh(τ)〈Z,*θ〉dHMn-1|,
where the infimum is taken over all Z∈ℱB(ℳ), Z≠0.Some estimates of (5.1) are given in [8, 9].Under these circumstances we have the following theorem.Theorem 5.1.
Suppose that the formZ∈ℱB(ℳ) satisfies one of the boundary condition (4.1), (4.4), or (4.6). Then for almost all τ∈(0,h0) and for an arbitrary τ0 the following relation holds
(5.2)ddτ(I(τ)exp⁡{-ν1∫τ0τε(t;FB)dt})≥0,
where
(5.3)I(τ)=∫Bh(τ)|w|p*1.
In particular, for allτ1<τ2 we have (5.4)I(τ1)≤I(τ2)exp⁡{-ν1∫τ1τ2ε(t)dt}.Proof.
The Kronrod-Federer formula yields(5.5)I(τ)=∫0τdt∫Σh(τ)|w|pdHMn-1|∇h|
and, in particular, the function I(τ) is absolutely continuous on closed intervals of (0,h0). Now it is enough to prove the inequality
(5.6)ddτI(τ)≥ν1I(τ)ε(τ).
From (5.5) we have for almost every τ∈(0,h0)(5.7)ddτI(τ)=∫Σh(τ)|w|pdHMn-1|∇h|.
By (1.6) we obtain(5.8)I(τ)=∫Bh(τ)|w|p*1≤ν1-1∫Bh(τ)〈w,*θ〉*1=ν1-1∫Bh(τ)w∧θ=ν1-1∫Bh(τ)dZ∧θ.
However, the form Z is weakly closed and satisfies one of the conditions (4.1), (4.2), or (4.4). Therefore for a.e. τ∈(0,h0),
(5.9)∫Bh(τ)dZ∧θ=∫Σh(τ)Z∧θ.
Thus we get
(5.10)I(τ)≤ν1-1∫Σh(τ)〈Z,*θ〉dHMn-1.
Further from (5.1) it follows that(5.11)∫Σh(τ)|w|pdHMn-1|∇h|≥ε(τ;FB)|∫Σh(τ)〈Z,*θ〉dHMn-1|.
Combining the above inequalities we obtain(5.12)I(τ)≤ν1-1ε(τ;FB)∫Σh(τ)|w|pdHMn-1|∇h|.
This inequality together with the equality (5.7) yields
(5.13)I(τ)≤ν1-1ε(τ;FB)ddτI(τ).
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 byℱ(Bh(τ)) the set of all differential forms
(5.14)Z0∈C1(Bh(τ)),deg⁡Z0=k-1,dZ0=0,
such that for almost every τ∈(0,h0) and for an arbitrary Lipschitz function ϕ the following formula holds (5.15)∫Σh(τ)ϕZ0∧θ=∫Bh(τ)dϕ∧Z0∧θ.Theorem 5.2.
If the differential formZ∈ℱB(ℳ), dZ∈𝒲𝒯2, satisfies the boundary condition (4.1), (4.4), or (4.6), then for all τ1<τ2<h0 and for an arbitrary form Z0∈ℱ(Bh(τ2)) the following relation holds
(5.16)ν1∫Bh(τ1)|dZ|p*1≤pτ2-τ1∫Bh(τ2)∖Bh(τ1)|∇h||(Z-Z0)∧θ|*1.Proof.
We consider the function(5.17)ϕ(m)={1form∈Bh(τ1),τ2-h(m)τ2-τ1form∈Bh(τ2)∖Bh(τ1),0form∈M∖Bh(τ2).
Suppose that the form Z satisfies the condition (4.1). Setting in (4.1) v=(ϕ)pZ∧θ we get
(5.18)∫Bh(τ2)(ϕ)pw∧θ+(-1)k-1∫Bh(τ2)Z∧d(ϕ)p∧θ=0,
or
(5.19)∫Bh(τ2)(ϕ)p〈w,*θ〉*1=(-1)kp∫Bh(τ2)(ϕ)p-1Z∧dϕ∧θ.
The function(ϕ)p is locally Lipschitz on B¯h(τ2) and ϕ|Σh(τ2)=0. Thus by (5.15) we get (5.20)∫Bh(τ2)dϕp∧Z0∧θ=∫Bh(τ2)d(ϕ)p∧Z0∧θ+∫Bh(τ2)(ϕ)pdZ0∧θ+∫Bh(τ2)(ϕ)pZ0∧dθ=∫Σh(τ2)(ϕ)pZ0∧θ=0.
Hence we arrive at the relation(5.21)∫Bh(τ2)(ϕ)p〈w,*θ〉*1=(-1)kp∫Bh(τ2)(ϕ)p-1(Z-Z0)∧dϕ∧θ,
which by (1.6) yields
(5.22)ν1∫Bh(τ2)(ϕ)p|w|p*1≤pτ2-τ1∫Bh(τ2)(ϕ)p-1|(Z-Z0)∧θ||∇h|*1.
Observing thatϕ(m)=1 for m∈Bh(τ1) and ϕ(m)=0 for m∈ℳ∖Bh(τ2) we obtain(5.23)ν1∫Bh(τ1)|w|p*1≤pτ2-τ1∫Bh(τ2)∖Bh(τ1)(ϕ)p-1|∇h||(Z-Z0)∧θ|*1.
Because |ϕ|≤1, the inequality (5.16) follows.
Let the formZ satisfy the condition (4.4). We choose v=(ϕ)pZ and observe that(5.24)v|Σh(τ2)=0.
Then we get
(5.25)∫Bh(τ2)(ϕ)pw∧θ=-∫Bh(τ2)d(ϕ)p∧Z∧θ=(-1)kp∫Bh(τ2)(ϕ)p-1Z∧dϕ∧θ.
Further details of the proof in this case are similar to those carried out above.
We assume that the formZ satisfies the mixed boundary condition (4.6). Observing that(5.26)(ϕ)p|Σh(τ2)=0,
we get
(5.27)∫Bh(τ2)(ϕ)pw∧θ=(-1)n∫Bh(τ2)Z∧θ∧d(ϕ)p=(-1)n-kp∫Bh(τ2)(ϕ)p-1Z∧dϕ∧θ.
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 formθ of dZ=w. Such an estimate is given in the next theorem.Theorem 5.3.
If the formZ, dZ∈𝒲𝒯2(ℳ), satisfies on ∂ℳ one of the boundary conditions (4.1), (4.4), or (4.6), then for all 0<τ1<τ2<h0 and for an arbitrary form Z0∈ℱ(Bh(τ2)) we have
(5.28)∫Bh(τ1)|dZ|p*1≤(pν2(τ2-τ1)ν1)p∫Bh(τ2)∖Bh(τ1)|∇h|p|Z-Z0|p*1.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
(5.29)∫M(ϕ)p-1|∇h||(Z-Z0)|∧θ*1≤∫M(ϕ)p-1|∇h||Z-Z0||θ|*1≤ν2∫M(ϕ)p-1|∇h||Z-Z0||w|p-1*1≤ν2(∫M|∇h|p|Z-Z0|p*1)1/p(∫Mϕp|w|p*1)(p-1)/p.
From (5.22) we get
(5.30)(ν1ν2)p∫M|w|p*1≤(pτ2-τ1)p∫M|∇h|p|Z-Z0|p*1.
Using the facts that ϕ=1 on Bh(τ1) and ϕ=0 on ℳ∖Bh(τ2) we easily obtain  (5.28).


## 5.2. Phragmén-Lindelöf Theorem

Letℳ be an n-dimensional noncompact Riemannian manifold with or without boundary and let w∈𝒲𝒯2 be a differential form as in (1.1), degw=k, and θ its complementary form as in (1.4).We assume that there exists a differential formZ∈Wp,loc1 with dZ=w. If the boundary ∂ℳ is nonempty, then we will assume that Z satisfies the boundary condition of Dirichlet (4.1), Neumann's condition (4.4), or the condition (4.6).We fix a locally Lipschitz exhaustion functionh:ℳ→(0,h0), 0<h0≤∞. Let, as above (5.31)I(τ;Z)=∫Bh(τ)|dZ|p*1,
and let (5.32)μ(τ;Z)=inf⁡∫τ<h(m)<τ+1|∇h||(Z-Z0)∧θ|*1,m(τ;Z)=inf⁡∫τ<h(m)<τ+1|∇h|p|Z-Z0|p*1,
where the infimum is taken over all closed forms Z0, satisfying conditions (5.14), (5.15) on Bh(τ).The following theorem exhibits a generalization of the classical Phragmén-Lindelöf principle for holomorphic functions.Theorem 5.4.
Suppose that the formZ, dZ∈𝒲𝒯2(ℳ), satisfies one of the boundary conditions (4.1), (4.4) or (4.6). The following alternatives hold: either the form dZ=0 a.e. on the manifold ℳ, or for all τ0∈(0,h0) we have
(5.33)lim⁡inf⁡τ→h0I(τ;Z)exp⁡{-ν1∫τ0τε(t;FB)dt}>0;(5.34)lim⁡inf⁡τ→h0μ(τ;Z)exp⁡{-ν1∫τ0τε(t;FB)dt}>0,(5.35)lim⁡inf⁡τ→h0m(τ;Z)exp⁡{-ν1∫τ0τε(t;FB)dt}>0.Proof.
The property (5.33) follows readily from (5.4) and is presented here only for the sake of completeness.
By (5.4) and (5.16), for a.e. τ∈(τ0,h0) we have(5.36)I(τ0)≤I(τ)exp⁡{-ν1∫τ0τε(t)dt}≤pν1-1∫Bh(τ+1)∖Bh(τ)|∇h||(Z-Z0)∧θ|*1exp⁡{-ν1∫τ0τε(t)dt}.
Therefore we get
(5.37)I(τ0)≤pν1-1μ(τ;Z)exp⁡{-ν1∫τ0τε(t)dt}.
Analogously, using (5.28) we get (5.38)I(τ0)≤(ν2ν1)pppm(τ;Z)exp⁡{-ν1∫τ0τε(t)dt}.
If we now assume that the formw≢0, then I(τ0)>0 for some τ0∈(0,h0). From this there easily follow (5.34) and (5.35).


## 5.3. 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 class𝒲𝒯2 defined on the manifold ℳ and the rate of growth of their energy integrals.Letℳ be an n-dimensional noncompact Riemannian manifold with or without boundary. We fix a locally Lipschitz exhaustion function h:ℳ→(0,h0), 0<h0≤∞, of the manifold ℳ.We assume that there areL≥1 mutually disjoint domains 𝒪1,𝒪2,…,𝒪L on ℳ such that 𝒪i∩∂ℳ=∅ if the boundary ∂ℳ is nonempty. We also assume that on each domain 𝒪i   there is given a differential form Zi with continuous coefficients and the properties:(i)
degZi=k-1, dZ≢0,(ii)
dZi=w∈𝒲𝒯2 with structure constants p,ν1,ν2, independent of i=1,2,…,L,(iii)
Zi satisfies on ∂𝒪i the zero boundary condition (4.2).We define a formZ on ℳ by setting Z|𝒪i=Zi and Z=0 on ℳ∖⋃i=1L𝒪i.According to Theorem4.2 each of the domains 𝒪i has a noncompact closure. Then by Theorem 5.4 the "narrower" the intersection of the domains 𝒪i with h-spheres Σh(t) for t→∞, the higher is the rate of growth of the form Z. Below we will consider the Denjoy-Carleman-Ahlfors theorem as a statement on the connection between the number L of mutually disjoint domains 𝒪i on ℳ0 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 will 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 Rn see [10, 11] for the history of the problem.We next introduce some necessary notation. We consider an open subsetD⊂ℳ with a noncompact closure and we assume that the restriction of the form Z to D satisfies condition (4.2).The functionh|D:D→(0,∞) is an exhaustion function of D. We fix an h-ball Bh(τ). Considering the variational problem (5.1) for the class of forms Z, satisfying the boundary condition (4.2) on D1 we define the characteristic (5.39)ε(t;D)=ε(t;F0)
where ℱ0 is defined in Section 5.1 and in (4.1).Following [12] we introduce the N-mean (5.40)E(t;N)=inf⁡1N∑k=1Nε(t;Dk)
where the infimum is taken over all decompositions of D into N nonintersecting open sets D1,D2,…,DN with noncompact closures.We record the following simple result.Lemma 5.5.
LetD1⊂D2 be arbitrary open subsets of ℳ with noncompact closures. Then
(5.41)ε(t;D2)≤ε(t;D1),(5.42)ε(t;M)≤E(t;N),N≥1.Proof.
It is enough to observe that each pair of formsZ,Z0 admissible for the variational problem (5.1) for the set D1 is also admissible for this problem for the set D2.
From (5.41) we get (5.42).We next derive a more general assertion about the monotonicity ofN-means.Lemma 5.6.
For arbitraryN>1 we have
(5.43)E(t;N+1)≥E(t;N).Proof.
We consider an arbitrary family of open subsets{Dk}, k=1,2,…,N+1, admissible for the infimum in (5.40). It is not difficult to see that
(5.44)1N+1∑k=1N+1ε(t;Dk)=1N+1∑k=1N+1(1N∑j=1,j≠kN+1ε(t;Dj)).
Because
(5.45)1N∑j=1,j≠kN+1ε(t;Dk)≥E(t;N),
we see that
(5.46)1N+1∑k=1N+1ε(t;Dk)≥E(t;N)
and the lemma is proved.The next theorem provides a solution to the aforementioned problem concerning the connection between the numberL of finite forms on ℳ, and the rate of growth of the total energy of these forms or the sum of their Lp-norms on an h-ball Bh(τ).Theorem 5.7.
Suppose that the manifoldℳ satisfies the properties listed in the beginning of this subsection and that for some N=1,2,…(5.47)∫h0h0E(t,N)dt=∞.
If the differential form Z, dZ∈𝒲𝒯2(ℳ), is such that
(5.48)lim⁡inf⁡τ→h0∫h(m)<τ|dZ|p*1exp⁡{-ν1∫τ0τE(t;N)dt}=0,
or
(5.49)lim⁡inf⁡τ′,τ′′→h00<τ′<τ′′<h0⁡∫τ′<h(m)<τ′′|∇h||Z∧θ|*1exp⁡{-ν1∫τ0τ′E(t;N)dt}=0,
or
(5.50)lim⁡inf⁡τ′,τ′′→h00<τ′<τ′′<h0⁡∫τ′<h(m)<τ′′|∇h|p|Z|p*1exp⁡{-ν1∫τ0τ′E(t;N)dt}=0,
then L<N.Proof.
We assume that there existsN mutually nonintersecting domains 𝒪1,𝒪2,…,𝒪N on the set ℳ0 and the forms Zi defined on 𝒪i with above properties. We denote by
(5.51)d(Ok)=inf⁡m∈Okh(m),d=max⁡1≤k≤Nd(Ok).
Fixτ0>d. Using the inequality (5.4) from Theorem 5.1 for an arbitrary k=1,2,…,N and a.e. 0<τ0<τ′<h0 we have (5.52)Ik(τ0)exp⁡{ν1∫τ0τ′εk(t)dt}≤Ik(τ′),
where
(5.53)Ik(τ′)=∫Ok∩Bh(τ′)|dZk|p*1,εk(τ′)=ε(τ′;Ok).
Adding these inequalities, we get
(5.54)min⁡1≤k≤NIk(τ0)∑k=1Nexp⁡{ν1∫τ0τ′εk(t)dt}≤I(τ′),
where
(5.55)I(τ′)=∫Bh(τ′)|dZ|p*1.
Applying the arithmetic-geometric mean inequality
(5.56)1N∑k=1Nexp⁡{ν1∫τ0τ′εk(t)dt}≥∏k=1Nexp⁡{ν1N∫τ0τ′εk(t)dt}
we get
(5.57)min⁡1≤k≤NIk(τ0)Nexp⁡{ν1N∫τ0τ′1N∑k=1Nεk(t)dt}≤I(τ′).
The domains𝒪1,𝒪2,…,𝒪N are nonintersecting. Therefore for all τ0<t<h0 we have (5.58)1N∑k=1Nεk(t)≥E(t;N).
The preceding inequality gives now
(5.59)min⁡1≤k≤NIk(τ0)Nexp⁡{ν1∫τ0τ′E(t;N)dt}≤I(τ′).
For the estimation of the integralI(τ′) we use the inequalities (5.16) and (5.28) and obtain (5.60)min⁡1≤k≤NIk(τ0)≤C1∫τ′<h(m)<τ′′|∇h||Z∧θ|*1exp⁡{-ν1∫τ0τ′E(t;N)dt}
or
(5.61)min⁡1≤k≤NIk(τ0)≤C2∫τ′<h(m)<τ′′|∇h|p|Z|p*1exp⁡{-ν1∫τ0τ′E(t;N)dt},
where
(5.62)C1=pτ′′-τ′,C2=(pν2(τ′′-τ′)ν1).
On the basis of the conditions (5.47)–(5.50) imposed on the form Z, for some k, 1≤k≤N, we have Ik(τ0)=0. Thus dZk(m)≡0 on 𝒪k∩Bh(τ0). Because we have chosen τ0>d arbitrarily, we can conclude that at least on one of the components 𝒪k, dZk≡0. A contradiction.



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*Source: 646392-2010-12-01.xml*