aa r X i v : . [ m a t h . C V ] J a n SIGNED QUASIREGULAR CURVES
SUSANNA HEIKKILÄ
Abstract.
We define a subclass of quasiregular curves, called signedquasiregular curves, which contains holomorphic curves and quasiregularmappings. As our main result, we prove a growth theorem of Bonk-Heinonen type for signed quasiregular curves. To obtain our main result,we prove that signed quasiregular curves satisfy a weak reverse Hölderinequality and that this weak reverse Hölder inequality implies the mainresult. We also obtain higher integrability for signed quasiregular curves.Further, we prove a cohomological value distribution result for signedquasiregular curves by using our main result and equidistribution. Introduction
Our motivation for defining signed quasiregular curves comes from Liou-ville type growth results in conformal geometry. By the classical Liouville’stheorem, every bounded entire function C → C is constant. The same re-sult holds for quasiregular mappings R n → R n . Recall that a continuousmapping f : M → N between oriented Riemannian n -manifolds, n ≥ , is K -quasiregular for K ≥ if f belongs to the Sobolev space W ,n loc ( M, N ) andsatisfies the distortion inequality k Df k n ≤ KJ f almost everywhere in M , where k Df k is the operator norm of the differential Df of f and J f is the Jacobian determinant of f . We refer to [1] and [11]for the theory of quasiregular mappings.The aforementioned Liouville theorem for quasiregular maps R n → R n follows from the following growth bound: Given n ≥ and K ≥ thereexists a constant ε = ε ( n, K ) > so that every K -quasiregular mapping f : R n → R n satisfying lim | x |→∞ | x | − ε | f ( x ) | = 0 is constant ; see [11, Corollary III.1.13].For Riemannian manifolds this Euclidean result takes the following form(Bonk and Heinonen [2, Theorem 1.11]): Given n ≥ and K ≥ thereexists a constant ε = ε ( n, K ) > so that every nonconstant K -quasiregularmapping f : R n → N into a closed, connected, and oriented Riemannian n -manifold N , that is not a rational cohomology sphere, satisfies lim inf r →∞ r ε Z B n ( r ) J f > . Mathematics Subject Classification.
Primary 30C65; Secondary 32A30, 53C15,53C57.This work was supported in part by the Academy of Finland project
In this paper, we prove a version of this Bonk-Heinonen growth result forsigned quasiregular curves. For the definition of quasiregular curves, we givethe auxiliary definition of an n -volume form on an m -manifold for n ≤ m .We say that a smooth differential n -form ω ∈ Ω n ( N ) on a Riemannian m -manifold N is an n -volume form for n ≤ m if ω is closed and pointwisenonvanishing.A continuous mapping f : M → N between oriented Riemannian mani-folds, ≤ n = dim M ≤ dim N , is a K -quasiregular ω -curve for K ≥ andan n -volume form ω ∈ Ω n ( N ) if f belongs to the Sobolev space W ,n loc ( M, N ) and satisfies the distortion inequality ( k ω k ◦ f ) k Df k n ≤ K ( ⋆f ∗ ω ) almost everywhere in M , where k ω k is the pointwise comass norm of ω givenby k ω x k = max { | ω x ( v , . . . , v n ) | : v , . . . , v n unit vectors in T x N } for every x ∈ N and ( ⋆f ∗ ω ) is the function satisfying ( ⋆f ∗ ω )vol M = f ∗ ω .Quasiregular curves have similar properties as quasiregular mappings to someextent; see [7] and [9].To define the subclass of signed quasiregular curves, we introduce thefollowing terminology.Let N be a connected and oriented Riemannian m -manifold of dimension m ≥ and let C ∞ b ( N ) be the space of all smooth and bounded functions on N . For ℓ = 1 , . . . , m , we denote Z ℓb ( N ) the space of all smooth, bounded,and closed ℓ -forms.For ≤ n ≤ m , let A nb ( N ) be the C ∞ b ( N ) -algebra generated by products α ∧ β of forms α ∈ Z ℓb ( N ) and β ∈ Z n − ℓb ( N ) , that is, a smooth n -form ω ∈ Ω n ( N ) belongs to the space A nb ( N ) if there exists ϕ i ∈ C ∞ b ( N ) , α i ∈ Z ℓ i b ( N ) ,and β i ∈ Z n − ℓ i b ( N ) for i = 1 , . . . , j for which ω = j X i =1 ϕ i α i ∧ β i . For example, A nb ( R m ) = Ω nb ( R m ) . Definition.
A quasiregular ω -curve f : M → N is signed with respect tothe n -volume form ω ∈ Ω n ( N ) if ω = P ji =1 ϕ i α i ∧ β i ∈ A nb ( N ) , where ϕ i ∈ C ∞ b ( N ) , α i ∈ Z ℓ i b ( N ) , and β i ∈ Z n − ℓ i b ( N ) for i = 1 , . . . , j , and thealmost everywhere defined measurable functions ( ⋆f ∗ ( α i ∧ β i )) do not changesign for any i = 1 , . . . , j . In this case, we also say that the representation ω = P ji =1 ϕ i α i ∧ β i is f -signed .For example, holomorphic curves are signed with respect to the standardsymplectic form. In Section 2, we show that, if N is closed and not a rationalcohomology sphere, then vol N has a representation in A mb ( N ) which is signedwith respect to all quasiregular mappings.We focus on signed quasiregular curves R n → N but our methods workalso for quasiregular ω -curves R n → N with respect to an n -volume form ω in the algebra A nb ( N ) with constant coefficients. Formally, we say that an IGNED QUASIREGULAR CURVES 3 n -form ω = P ji =1 ϕ i α i ∧ β i ∈ A nb ( N ) has an R -linear representation if thefunctions ϕ i are constant functions for i = 1 , . . . , j .We are now ready to state the main result. Theorem 1.1.
Let f : R n → N be a nonconstant K -quasiregular ω -curve,where N is a connected and oriented Riemannian m -manifold, m ≥ n ≥ , K ≥ , and ω ∈ A nb ( N ) is an n -volume form satisfying inf N k ω k > .Suppose that either ω has an f -signed representation or an R -linear repre-sentation. Then there exists a constant ε = ε ( n, K, ω ) > for which (1.1) lim inf r →∞ r ε Z B n ( r ) f ∗ ω > . We say that a quasiregular ω -curve f : R n → N satisfying the growthcondition (1.1) has fast growth of order ε . In particular, Theorem 1.1 yieldsthe following corollary. Corollary 1.2.
A nonconstant signed quasiregular ω -curve f : R n → N intoa closed, connected and oriented Riemannian manifold N with respect to an n -volume form ω ∈ A nb ( N ) has fast growth. Theorem 1.1 yields the following result that may be viewed as a cohomo-logical value distribution result for signed quasiregular curves; see Section5.
Theorem 1.3.
Let f : R n → N be a nonconstant K -quasiregular ω -curve,where N is a connected and oriented Riemannian m -manifold, m ≥ n ≥ , K ≥ , and ω ∈ A nb ( N ) is an n -volume form satisfying inf N k ω k > . Suppose that either ω has an f -signed representation or an R -linearrepresentation. Then, for every ω ∈ Ω n ( N ) satisfying ω − ω = dτ for some τ ∈ Ω n − b ( N ) , there exists a set E ⊂ [1 , ∞ ) for which R E d rr < ∞ and lim inf r →∞ r / ∈ E r ε Z B n ( r ) f ∗ ω > , where ε = ε ( n, K, ω ) > . In particular, if ( ⋆f ∗ ω ) ≥ almost everywherein R n , then lim inf r →∞ r ε Z B n ( r ) f ∗ ω > . Remark 1.4.
Note that, since ω ∈ A nb ( N ) , the set { ω + dτ : τ ∈ Ω n − b ( N ) } contains the bounded de Rham cohomology class of the n -volume form ω ∈ Ω n ( N ) ; see e.g. [12]. In particular, if N is closed, then this set is exactly thede Rham cohomology class of ω . Corollary 1.5.
A nonconstant signed quasiregular ω -curve f : R n → N into a closed, connected and oriented Riemannian manifold N with respectto an n -volume form ω ∈ A nb ( N ) satisfies, for every ω ∈ Ω n ( N ) in the deRham cohomology class of ω , lim inf r →∞ r / ∈ E r ε Z B n ( r ) f ∗ ω > , where ε > and the exception set E has finite logarithmic measure. SUSANNA HEIKKILÄ
Theorem 1.1 immediately yields the following corollary for constant coef-ficient n -volume forms. We identify constant coefficient n -volume forms in R m with n -covectors in Λ n ( R m ) in the statement. Corollary 1.6.
Let f : R n → R m be a nonconstant K -quasiregular ω -curve,where m ≥ n ≥ , K ≥ , and ω ∈ Λ n ( R m ) is a nonzero covector. Then f has fast growth of order ε = ε ( n, K, ω ) > . If ω is an n -volume form that splits into ω = ω ∧ ω , where each ω i is abounded lower dimensional volume form, then every quasiregular ω -curve issigned. Thus, Theorem 1.1 yields the following corollary. Corollary 1.7.
Let f : R n → N be a nonconstant K -quasiregular ω -curve,where N is a connected and oriented Riemannian m -manifold, m ≥ n ≥ , K ≥ , and ω ∈ Ω n ( N ) is an n -volume form satisfying inf N k ω k > . Sup-pose that ω splits into ω = ω ∧ ω , where each ω i ∈ Ω n i b ( N ) is an n i -volumeform for n i ≥ . Then f has fast growth of order ε = ε ( n, K, ω , ω ) > . In particular, we have the following corollary for quasiregular curves intoproducts of closed manifolds.
Corollary 1.8.
For i = 1 , , let N i be a closed, connected and orientedRiemannian m i -manifold, m i ≥ n i ≥ , and let π i : N × N → N i be aprojection. Let also n = n + n and let ω ∈ Ω n ( N × N ) be an n -volumeform ω = π ∗ ω ∧ π ∗ ω , where each ω i ∈ Ω n i ( N i ) is an n i -volume form. Thenevery nonconstant signed quasiregular ω -curve f : R n → N × N has fastgrowth. Every quasiregular mapping is signed if the target manifold is closed andnot a rational cohomology sphere. Hence, we also obtain the growth resultof Bonk and Heinonen as a corollary of Theorem 1.1.
Corollary 1.9.
Let f : R n → N be a nonconstant K -quasiregular mapping,where N is a closed, connected, and oriented Riemannian n -manifold whichis not a rational cohomology sphere, n ≥ , and K ≥ . Then f has fastgrowth of order ε = ε ( n, K, vol N ) > . Method of the proof of Theorem 1.1; weak reverse Hölder inequal-ity.
Our main method of proof for Theorem 1.1 is a weak reverse Hölderinequality for signed quasiregular curves. The method of proof grew out ofthe observation that the growth result of Bonk and Heinonen follows almostimmediately from the higher integrability result of Prywes [10, Proposition2.5]. The connection between the weak reverse Hölder inequality and fastgrowth is formally stated in the following theorem.
Theorem 1.10.
Let f : R n → N be a nonconstant K -quasiregular ω -curve,where N is a connected and oriented Riemannian m -manifold, m ≥ n ≥ , K ≥ , and ω ∈ Ω n ( N ) is an n -volume form. If there exists constants p > and C p > such that ( ⋆f ∗ ω ) satisfies the weak reverse Hölder inequality (cid:12)(cid:12) B n (cid:0) r (cid:1)(cid:12)(cid:12) Z B n ( r )( ⋆f ∗ ω ) p ! p ≤ C p | B n ( r ) | Z B n ( r ) f ∗ ω for every r > , then f has fast growth of order ε = ε ( n, p ) > . IGNED QUASIREGULAR CURVES 5
Having Theorem 1.10 at our disposal, it suffices to prove the followingversion of Prywes’ higher integrability result for signed quasiregular curvesto obtain Theorem 1.1.
Theorem 1.11.
Let f : R n → N be a nonconstant K -quasiregular ω -curve,where N is a connected and oriented Riemannian m -manifold, m ≥ n ≥ , K ≥ , and ω ∈ A nb ( N ) is an n -volume form satisfying inf N k ω k > . Sup-pose that either ω has an f -signed representation or an R -linear representa-tion. Then there exists constants p = p ( n, K, ω ) > and C = C ( n, K, ω ) > for which the inequality (cid:12)(cid:12) B (cid:12)(cid:12) Z B ( ⋆f ∗ ω ) p ! p ≤ C | B | Z B f ∗ ω holds for every open ball B = B n ( x, r ) ⊂ R n , where B = B n (cid:0) x, r (cid:1) . As a byproduct of the method, we obtain a higher integrability resultsimilar to [7, Theorem 1.3].
Theorem 1.12.
Let f : R n → N be a nonconstant K -quasiregular ω -curve,where N is a connected and oriented Riemannian m -manifold, m ≥ n ≥ , K ≥ , and ω ∈ A nb ( N ) is an n -volume form satisfying inf N k ω k > .Suppose that either ω has an f -signed representation or an R -linear repre-sentation. Then f ∈ W ,q loc ( R n , N ) for some q = q ( n, K, ω ) > n . Organization of the article.
In Section 2, we construct a representation ofthe Riemannian volume form that is signed with respect to all quasiregularmappings. In Section 3, we prove Theorem 1.10. In Section 4, we proveTheorems 1.11, 1.12, and 1.1. In Section 5, we discuss the equidistributionof quasiregular curves and prove Theorem 1.3. In Section 6, we present afamily of examples of signed quasiregular curves.
Acknowledgements.
The author thanks her thesis advisor Pekka Pankkafor all his help improving the manuscript.2.
Signed representation of the Riemannian volume form
In this section, we describe how to construct a representation of the Rie-mannian volume form that is signed with respect to all quasiregular map-pings. The construction is based on the work of Prywes [10].Let N be a closed, connected, and oriented Riemannian n -manifold, n ≥ ,which is not a rational cohomology sphere. Let α ∈ Ω ℓ ( N ) and β ∈ Ω n − ℓ ( N ) be closed forms satisfying Z N α ∧ β = Z N vol N for some ≤ ℓ ≤ n − . Following the proof of [10, Lemma 2.3], there existsa positive function h ∈ C ∞ ( N ) , a smooth partition of unity { λ i } ji =1 on N ,and orientation preserving diffeomorphisms Φ i : N → N , i = 1 , . . . , j , forwhich vol N = h j X i =1 λ i Φ ∗ i ( α ∧ β ) = j X i =1 hλ i (Φ ∗ i α ∧ Φ ∗ i β ) . SUSANNA HEIKKILÄ
In our terminology, this is a representation of vol N in the algebra A nb ( N ) ,since hλ i ∈ C ∞ b ( N ) , Φ ∗ i α ∈ Z ℓb ( N ) , and Φ ∗ i β ∈ Z n − ℓb ( N ) for i = 1 , . . . , j .To obtain a representation that is signed with respect to all quasiregularmappings, we apply this construction to a Hodge pair. Lemma 2.1.
Let N be a closed, connected, and oriented Riemannian n -manifold, n ≥ , which is not a rational cohomology sphere. Then vol N hasa representation which is signed with respect to all quasiregular mappingsinto N .Proof. Since N is not a rational cohomology sphere, there exists ≤ ℓ ≤ n − for which H ℓ ( N ) = 0 . Let = c ∈ H ℓ ( N ) and let ξ c ∈ c be the harmonicrepresentative of c , that is, ξ c is the unique form in the de Rham cohomologyclass c such that dξ c = 0 and d ∗ ξ c = 0 . We may assume that Z N ξ c ∧ ⋆ξ c = Z N vol N . Then the volume form vol N has a representation(2.1) vol N = h j X i =1 λ i Φ ∗ i ( ξ c ∧ ⋆ξ c ) , where h ∈ C ∞ ( N ) is a positive function, { λ i } ji =1 is a smooth partition ofunity on N , and Φ i : N → N is an orientation preserving diffeomorphism forevery i = 1 , . . . , j .Let f : M → N be a K -quasiregular mapping, where M is an orientedRiemannian n -manifold and K ≥ . Then f ∗ (Φ ∗ i ( ξ c ∧ ⋆ξ c )) = f ∗ (Φ ∗ i ( h ξ c , ξ c i vol N )) = ( h ξ c , ξ c i ◦ Φ i ◦ f )( J Φ i ◦ f ) J f vol M = ( k ξ c k ◦ Φ i ◦ f )( J Φ i ◦ f ) J f vol M for every i = 1 , . . . , j . The function k ξ c k is nonnegative, each Jacobiandeterminant J Φ i is positive, and the Jacobian J f is nonnegative. Hence, therepresentation (2.1) is f -signed. (cid:3) Weak reverse Hölder inequality implies fast growth
In this section, we prove Theorem 1.10.
Proof of Theorem 1.10.
Since (cid:12)(cid:12) B n (cid:0) r (cid:1)(cid:12)(cid:12) Z B n ( r )( ⋆f ∗ ω ) p ! p ≤ C p | B n ( r ) | Z B n ( r ) f ∗ ω for every r > , we have that C − p | B n (1) | − p np Z B n ( r )( ⋆f ∗ ω ) p ! p ≤ R B n ( r ) f ∗ ωr ε for every r > for ε = n (cid:16) − p (cid:17) > . Since f is a nonconstant K -quasiregular ω -curve, we may choose r > for which Z B n ( r )( ⋆f ∗ ω ) p > . IGNED QUASIREGULAR CURVES 7
Then, for C = C − p | B n (1) | − p np > , we have the estimate R B n ( r ) f ∗ ωr ε ≥ C Z B n ( r )( ⋆f ∗ ω ) p ! p ≥ C Z B n ( r )( ⋆f ∗ ω ) p ! p for every r ≥ r . Thus lim inf r →∞ R B n ( r ) f ∗ ωr ε ≥ C Z B n ( r )( ⋆f ∗ ω ) p ! p . This proves the theorem. (cid:3) Higher integrability of signed quasiregular curves
Let f : R n → N be a quasiregular ω -curve. Then u = ( ⋆f ∗ ω ) is a locallyintegrable function and we may apply known results for locally integrablefunctions such as Gehring’s lemma; see e.g. [1, Theorem 4.2]. Hence, wemay prove Theorem 1.11 by proving the following result that is similar toresults for quasiregular mappings due to Prywes (see [10, Proposition 2.2]and [10, Lemma 2.4]). The proof uses techniques developed there. Proposition 4.1.
Let f : R n → N be a nonconstant K -quasiregular ω -curve,where N is a connected and oriented Riemannian m -manifold, m ≥ n ≥ , K ≥ , and ω ∈ A nb ( N ) is an n -volume form satisfying inf N k ω k > .Suppose that either ω has an f -signed representation or an R -linear repre-sentation. Then (cid:12)(cid:12) B (cid:12)(cid:12) Z B f ∗ ω ≤ C ( n, K, ω ) (cid:18) | B | Z B ( ⋆f ∗ ω ) nn +1 (cid:19) n +1 n for every open ball B ⊂ R n .Proof. Let F ( ω ) = inf j X i =1 k ϕ i k ∞ k α i k ∞ k β i k ∞ , where the infimum is taken over all f -signed and R -linear representations ω = P ji =1 ϕ i α i ∧ β i of ω in A nb ( N ) . Then < F ( ω ) < ∞ . Thus, thereexists either an f -signed or an R -linear representation ω = P ji =1 ϕ i α i ∧ β i for which j X i =1 k ϕ i k ∞ k α i k ∞ k β i k ∞ < F ( ω ) . For every i = 1 , . . . , j , the forms f ∗ α i and f ∗ β i are weakly closed sincethe forms α i and β i are closed. By writing α i ∧ β i = (( − ℓ i ( n − ℓ i ) β i ) ∧ α i if necessary, we may assume that for every i = 1 , . . . , j , α i ∈ Z ℓ i b ( N ) for ≤ ℓ i ≤ n . SUSANNA HEIKKILÄ
Let B ⊂ R n be an open ball. Then, by the Poincaré inequality fordifferential forms, there exists for every i = 1 , . . . , j a differential form τ i ∈ W ,p i (Λ ℓ i − B ) satisfying dτ i = f ∗ α i in the weak sense and(4.1) k τ i k nqin − qi ,B ≤ C ( n ) k f ∗ α i k q i ,B , where p i = nℓ i and q i = nℓ i nn +1 ; see Iwaniec and Lutoborski [5, Corollary 4.2].Let ψ ∈ C ∞ c ( R n ) be a nonnegative function such that ψ ( x ) = 1 for every x ∈ B , ψ ( x ) = 0 for every x ∈ R n \ B , and | dψ | ≤ r . By the nonnegativityof ψ ( ⋆f ∗ ω ) , we have that Z B f ∗ ω ≤ Z B ψf ∗ ω = j X i =1 Z B ψ ( ϕ i ◦ f ) f ∗ ( α i ∧ β i ) . Suppose first that the representation ω = P ji =1 ϕ i α i ∧ β i is R -linear. Thenthe functions ϕ i are constant functions ϕ i ≡ c i for every i = 1 , . . . , j and j X i =1 Z B ψ ( ϕ i ◦ f ) f ∗ ( α i ∧ β i ) = j X i =1 c i Z B ψf ∗ ( α i ∧ β i ) ≤ j X i =1 k ϕ i k ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z B ψf ∗ ( α i ∧ β i ) (cid:12)(cid:12)(cid:12)(cid:12) . Suppose now that the representation ω = P ji =1 ϕ i α i ∧ β i is f -signed.For every i = 1 , . . . , j , the function ( ⋆f ∗ ( α i ∧ β i )) is either nonnegative ornonpositive almost everywhere. If the function ( ⋆f ∗ ( α i ∧ β i )) is nonnegative,then Z B ψ ( ϕ i ◦ f ) f ∗ ( α i ∧ β i ) ≤ k ϕ i k ∞ Z B ψf ∗ ( α i ∧ β i )= k ϕ i k ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z B ψf ∗ ( α i ∧ β i ) (cid:12)(cid:12)(cid:12)(cid:12) . On the other hand, if the function ( ⋆f ∗ ( α i ∧ β i )) is nonpositive, then Z B ψ ( ϕ i ◦ f ) f ∗ ( α i ∧ β i ) = Z B ψ ( − ϕ i ◦ f )( − f ∗ ( α i ∧ β i )) ≤ k ϕ i k ∞ Z B ψ ( − f ∗ ( α i ∧ β i ))= k ϕ i k ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z B ψf ∗ ( α i ∧ β i ) (cid:12)(cid:12)(cid:12)(cid:12) . Thus j X i =1 Z B ψ ( ϕ i ◦ f ) f ∗ ( α i ∧ β i ) ≤ j X i =1 k ϕ i k ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z B ψf ∗ ( α i ∧ β i ) (cid:12)(cid:12)(cid:12)(cid:12) . Hence, it suffices to estimate the term (cid:12)(cid:12)(cid:12)(cid:12)Z B ψf ∗ ( α i ∧ β i ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z B dτ i ∧ ( ψf ∗ β i ) (cid:12)(cid:12)(cid:12)(cid:12) IGNED QUASIREGULAR CURVES 9 for every i = 1 , . . . , j . We obtain by integration by parts that (cid:12)(cid:12)(cid:12)(cid:12)Z B dτ i ∧ ( ψf ∗ β i ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z B τ i ∧ dψ ∧ f ∗ β i (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n ) Z B | τ i | | dψ | | f ∗ β i | for every i = 1 , . . . , j . Further, by the inequality | dψ | ≤ r , Hölder’s inequal-ity and inequality (4.1), we have the estimate C ( n ) Z B | τ i | | dψ | | f ∗ β i | ≤ C ( n ) r k τ i k nqin − qi ,B k f ∗ β i k nn − ℓi nn +1 ,B ≤ C ( n ) r k f ∗ α i k q i ,B k f ∗ β i k nn − ℓi nn +1 ,B for every i = 1 , . . . , j .Since f is a K -quasiregular ω -curve, we may estimate each term k f ∗ α i k q i ,B by k f ∗ α i k q i ,B = (cid:18)Z B | f ∗ α i | nℓi nn +1 (cid:19) ℓin n +1 n ≤ C ( n ) k α i k ∞ (cid:18)Z B k Df k n nn +1 (cid:19) ℓin n +1 n ≤ C ( n ) k α i k ∞ (cid:18) inf N k ω k (cid:19) − ℓin (cid:18)Z B (( k ω k ◦ f ) k Df k n ) nn +1 (cid:19) ℓin n +1 n ≤ C ( n ) k α i k ∞ (cid:18) inf N k ω k (cid:19) − ℓin K ℓin (cid:18)Z B ( ⋆f ∗ ω ) nn +1 (cid:19) ℓin n +1 n . Similarly, we get the estimate k f ∗ β i k nn − ℓi nn +1 ,B ≤ C ( n ) k β i k ∞ (cid:18) inf N k ω k (cid:19) ℓi − nn K n − ℓin (cid:18)Z B ( ⋆f ∗ ω ) nn +1 (cid:19) n − ℓin n +1 n for every i = 1 , . . . , j .Combining all the estimates, we arrive at (cid:12)(cid:12) B (cid:12)(cid:12) Z B f ∗ ω ≤ j X i =1 k ϕ i k ∞ k α i k ∞ k β i k ∞ ! (cid:18) inf N k ω k (cid:19) − C ( n ) r n +1 K (cid:18)Z B ( ⋆f ∗ ω ) nn +1 (cid:19) n +1 n < F ( ω ) C ( ω ) C ( n ) K (cid:18) r n Z B ( ⋆f ∗ ω ) nn +1 (cid:19) n +1 n = C ( n, K, ω ) (cid:18) | B | Z B ( ⋆f ∗ ω ) nn +1 (cid:19) n +1 n . This concludes the proof. (cid:3)
As mentioned, Theorem 1.11 now follows from Gehring’s lemma andProposition 4.1. Consequently, Theorem 1.12 is almost immediate.
Proof of Theorem 1.12.
By Theorem 1.11, ( ⋆f ∗ ω ) ∈ L p loc ( R n ) for some p = p ( n, K, ω ) > . Let q = np > n . Then, for every bounded domain U ⊂ R n , we have the estimate Z U k Df k q ≤ (cid:18) inf N k ω k (cid:19) − p Z U (( k ω k ◦ f ) k Df k n ) p ≤ (cid:18) inf N k ω k (cid:19) − p K p Z U ( ⋆f ∗ ω ) p < ∞ . Hence, k Df k ∈ L q loc ( R n ) and f ∈ W ,q loc ( R n , N ) . (cid:3) Next, we combine Theorems 1.11 and 1.10 to prove Theorem 1.1.
Proof of Theorem 1.1.
We have by Theorem 1.11 that there exists constants p = p ( n, K, ω ) > and C = C ( n, K, ω ) > for which the inequality (cid:12)(cid:12) B (cid:12)(cid:12) Z B ( ⋆f ∗ ω ) p ! p ≤ C | B | Z B f ∗ ω holds for every open ball B ⊂ R n . Then, by considering open balls B n ( r ) ⊂ R n for r > , the claim follows from Theorem 1.10. (cid:3) Equidistribution of quasiregular curves
We begin by proving an equidistribution theorem for quasiregular curves inthe spirit of Mattila and Rickman [6]. The following statement is analogousto result of Pankka [8, Theorem 4] and while the proof is essentially thesame, we present it for the reader’s convenience.
Theorem 5.1.
Let f : R n → N be a nonconstant K -quasiregular ω -curve,where N is a connected and oriented Riemannian m -manifold, m ≥ n ≥ , K ≥ , and ω ∈ Ω n ( N ) is an n -volume form satisfying inf N k ω k > . Suppose that the function A ω ,f : (0 , ∞ ) → [0 , ∞ ) , r R B n ( r ) f ∗ ω , isunbounded. Then, for every ω ∈ Ω n ( N ) satisfying ω − ω = dτ for some τ ∈ Ω n − b ( N ) , there exists a set E ⊂ [1 , ∞ ) for which R E d rr < ∞ and lim r →∞ r / ∈ E R B n ( r ) f ∗ ω R B n ( r ) f ∗ ω = 1 . Remark 5.2.
Theorem 5.1 is cohomological contrary to the result in [6]which is for measures. This important distinction is illustrated further inSection 6. There, we give an example of a volume form ω for which thereexists both a quasiregular ω -curve with nowhere dense image and a quasireg-ular ω -curve with dense image. Proof of Theorem 5.1.
Let ω ∈ Ω n ( N ) and τ ∈ Ω n − b ( N ) be forms such that ω − ω = dτ . Let n − n < δ < and let E ⊂ [1 , ∞ ) be the set of all radii r ≥ satisfying Z B n ( r ) f ∗ ω ! δ ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z S n − ( r ) f ∗ τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . IGNED QUASIREGULAR CURVES 11
Then Hölder’s inequality yields the estimate Z B n ( r ) f ∗ ω ! δ ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z S n − ( r ) f ∗ τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z S n − ( r ) | f ∗ τ | ≤ C ( n ) k τ k ∞ Z S n − ( r ) k Df k n − ≤ C ( n ) k τ k ∞ Z S n − ( r ) k Df k n ! n − n Z S n − ( r ) ! n ≤ C ( n ) k τ k ∞ r n − n (cid:18) inf N k ω k (cid:19) − nn Z S n − ( r ) ( k ω k ◦ f ) k Df k n ! n − n ≤ C ( n ) k τ k ∞ r n − n (cid:18) inf N k ω k (cid:19) − nn K n − n Z S n − ( r ) f ∗ ω ! n − n for almost every r ∈ E .Since ( ⋆f ∗ ω ) ∈ L ( R n ) , the function A ω ,f is differentiable almost ev-erywhere and ( A ω ,f ) ′ ( r ) = Z S n − ( r ) f ∗ ω for almost every r > ; see e.g. [3, Theorem 3.12] for details. Thus, by theprevious estimate, the function A ω ,f satisfies the differential inequality ( A ω ,f ( r )) δ nn − ≤ Cr ( A ω ,f ) ′ ( r ) for almost every r ∈ E , where C = C ( n, τ, ω , K ) . Since A ω ,f is unbounded,we may choose r > for which A ω ,f ( r ) > . Then, for every R > r ,the function − δ nn − ( A ω ,f ) − δ nn − is well-defined and nondecreasing on theinterval [ r , R ] . Hence Z E d rr ≤ log( r ) + lim R →∞ C Z Rr ( A ω ,f ) ′ ( r )( A ω ,f ( r )) δ nn − d r ≤ log( r ) + lim R →∞ C − δ nn − (cid:16) ( A ω ,f ( R )) − δ nn − − ( A ω ,f ( r )) − δ nn − (cid:17) = log( r ) + Cδ nn − − A ω ,f ( r )) − δ nn − < ∞ ; see [4, Theorem 18.14] for details for the second step.Finally, since δ < , we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R B n ( r ) f ∗ ω R B n ( r ) f ∗ ω − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R B n ( r ) f ∗ ( ω − dτ ) R B n ( r ) f ∗ ω − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R B n ( r ) f ∗ dτ R B n ( r ) f ∗ ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)R S n − ( r ) f ∗ τ (cid:12)(cid:12)(cid:12)R B n ( r ) f ∗ ω ≤ (cid:16)R B n ( r ) f ∗ ω (cid:17) δ R B n ( r ) f ∗ ω = ( A ω ,f ( r )) δ − → as r → ∞ , r / ∈ E . This concludes the proof. (cid:3) Having Theorem 5.1 at our disposal, we have that, if a quasiregular ω -curve has fast growth, then the growth condition is also satisfied in a largeset of radii for any form ω satisfying ω − ω = dτ for a bounded τ . Proposition 5.3.
Let f : R n → N be a nonconstant K -quasiregular ω -curve, where N is a connected and oriented Riemannian m -manifold, m ≥ n ≥ , K ≥ , and ω ∈ Ω n ( N ) is an n -volume form satisfying inf N k ω k > . Suppose that f has fast growth of order ε > . Then, for every ω ∈ Ω n ( N ) satisfying ω − ω = dτ for some τ ∈ Ω n − b ( N ) , there exists a set E ⊂ [1 , ∞ ) for which R E d rr < ∞ and lim inf r →∞ r / ∈ E r ε Z B n ( r ) f ∗ ω > . In particular, if ( ⋆f ∗ ω ) ≥ almost everywhere in R n , then lim inf r →∞ r ε Z B n ( r ) f ∗ ω > . Remark 5.4.
In the proof of Proposition 5.3, we use the following propertyof sets of finite logarithmic measure:
Given E ⊂ [1 , ∞ ) for which R E d rr < ∞ there exists i ∈ N having the property that for every i ≥ i and every t ∈ E ∩ [2 i , i +1 ] , (cid:2) t , t (cid:3) E . This property follows immediately from theobservation that R t t d rr = ln 2 . Proof of Proposition 5.3.
Let ω ∈ Ω n ( N ) and τ ∈ Ω n − b ( N ) be differentialforms satisfying ω − ω = dτ . The function A ω ,f : (0 , ∞ ) → [0 , ∞ ) , r R B n ( r ) f ∗ ω , is unbounded since f has fast growth of order ε . Hence, byTheorem 5.1, there exists a set E ⊂ [1 , ∞ ) satisfying R E d rr < ∞ and lim r →∞ r / ∈ E R B n ( r ) f ∗ ω R B n ( r ) f ∗ ω = 1 . Then lim inf r →∞ r / ∈ E r ε Z B n ( r ) f ∗ ω = lim inf r →∞ r / ∈ E R B n ( r ) f ∗ ω r ε R B n ( r ) f ∗ ω R B n ( r ) f ∗ ω ≥ lim inf r →∞ r / ∈ E R B n ( r ) f ∗ ω r ε ! lim inf r →∞ r / ∈ E R B n ( r ) f ∗ ω R B n ( r ) f ∗ ω ! ≥ lim inf r →∞ r ε Z B n ( r ) f ∗ ω ! > . Now suppose that ( ⋆f ∗ ω ) ≥ almost everywhere in R n . Suppose towardscontradiction that lim inf r →∞ R B n ( r ) f ∗ ωr ε = 0 . For every k ∈ N , we may choose r k ≥ k for which inf s ≥ r k R B n ( s ) f ∗ ωs ε < k . IGNED QUASIREGULAR CURVES 13
Hence R B n ( s k ) f ∗ ωs εk < k for some s k ≥ r k . In particular, lim k →∞ R B n ( s k ) f ∗ ωs εk = 0 . We have two cases. Suppose first that there exists k ∈ N so that s k / ∈ E for every k ≥ k . Then k →∞ R B n ( s k ) f ∗ ωs εk ≥ lim inf r →∞ r / ∈ E R B n ( r ) f ∗ ωr ε > , which is a contradiction. Suppose now that for every j ∈ N there exists k j ≥ j so that s k j ∈ E . Since s k j ≥ r k j ≥ k j for every j ∈ N , there exists j ∈ N so that for every j ≥ j there exists t j / ∈ E satisfying s kj ≤ t j < s k j .Then R B n ( t j ) f ∗ ωt εj ≤ ε R B n ( s kj ) f ∗ ωs εk j < ε k j for every j ≥ j . Hence lim j →∞ R B n ( t j ) f ∗ ωt εj = 0 . Again, this leads to a contradiction, since j →∞ R B n ( t j ) f ∗ ωt εj ≥ lim inf r →∞ r / ∈ E R B n ( r ) f ∗ ωr ε > . (cid:3) We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3.
Let ω ∈ Ω n ( N ) and τ ∈ Ω n − b ( N ) be differentialforms satisfying ω − ω = dτ . By Theorem 1.1, f has fast growth of order ε = ε ( n, K, ω ) > . Then, by Proposition 5.3, there exists a set E ⊂ [1 , ∞ ) for which R E d rr < ∞ and lim inf r →∞ r / ∈ E r ε Z B n ( r ) f ∗ ω > . In particular, if ( ⋆f ∗ ω ) ≥ almost everywhere in R n , then lim inf r →∞ r ε Z B n ( r ) f ∗ ω > . This completes the proof. (cid:3) A family of examples
In this section, we give an example of an n -volume form which admits bothquasiregular curves with dense image and quasiregular curves with nowheredense image. This complements the equidistribution result in Section 5 (The-orem 5.1).Let n ≥ and let T n +1 = S × · · · × S be the ( n + 1) -dimensionaltorus equipped with the product Riemannian metric. Let pr j : T n +1 → S be the j th projection ( e πix , . . . , e πix n +1 ) e πix j for every j = 1 , . . . , n and let π : R n +1 → T n +1 be the standard locally isometric covering map ( x , . . . , x n +1 ) ( e πix , . . . , e πix n +1 ) . Let vol S ∈ Ω ( S ) be the standardangular form and let ω = pr ∗ vol S ∧ · · · ∧ pr ∗ n vol S ∈ Ω n ( T n +1 ) . Clearly, ω is an n -volume form on T n +1 satisfying k ω k = 1 and π ∗ ω = dx ∧ · · · ∧ dx n ∈ Ω n ( R n +1 ) . Further, ω has an R -linear representation inthe algebra A nb ( T n +1 ) and every quasiregular ω -curve is signed with respectto ω .Let y = ( y , . . . , y n ) ∈ R n and let L y : R n → R n +1 be the mapping x ( x, x · y ) . Let also f y : R n → T n +1 be the composed mapping π ◦ L y ∈ C ∞ ( R n , T n +1 ) . Since k Df y k n = k D ( π ◦ L y ) k n = k (( Dπ ) ◦ L y ) DL y k n = k DL y k n ≤ (1 + | y | ) n and f ∗ y ω = ( π ◦ L y ) ∗ ω = L ∗ y ( dx ∧ · · · ∧ dx n ) = vol R n , the mapping f y is a (1 + | y | ) n -quasiregular ω -curve.Next we show that the image f y ( R n ) ⊂ T n +1 is nowhere dense if y ∈ Q n and dense if y ∈ ( R \ Q ) n . To that end, denote the Riemannian distancefunction on S by d and the Riemannian distance function on T n +1 by d n +1 . Rational case.
Suppose that y ∈ Q n . Recall that for a rational angle θ theset { e πikθ : k ∈ Z } ⊂ S is finite.Suppose towards contradiction that the closure of f y ( R n ) has nonemptyinterior. Then there exists a point v = ( v , . . . , v n +1 ) ∈ Q n × ( R \ Q ) forwhich π ( v ) = ( e πiv , . . . , e πiv n +1 ) ∈ T n +1 belongs to the interior of theclosure of f y ( R n ) . For every j = 1 , . . . , n , we have v j = p j q j for some p j ∈ Z and q j ∈ Z + . Let E = n e πik y q · · · · · e πik n ynqn : k , . . . , k n ∈ Z o ⊂ S . The set E is finite and e πiv n +1 / ∈ E . Let r = min { d ( z, e πiv n +1 ) : z ∈ E } > . Now for each t ∈ R either d ( e πit , e πiv n +1 ) > r or d ( e πit , E ) > r . Weshow that this leads to a contradiction.Let < δ < r be such that nδ (max ≤ j ≤ n | y j | ) < r . Since π ( v ) belongs tothe interior of the closure of f y ( R n ) , there exists a point x ∈ R n satisfying IGNED QUASIREGULAR CURVES 15 d n +1 ( f y ( x ) , π ( v )) < δ . Denote U = (cid:18) v − , v + 12 (cid:19) × · · · × (cid:18) v n +1 − , v n +1 + 12 (cid:19) ⊂ R n +1 . Let ℓ = ( ℓ , . . . , ℓ n +1 ) ∈ Z n +1 be so that L y ( x ) − ℓ ∈ U . Then d ( e πix · y , e πiv n +1 ) = | x · y − ℓ n +1 − v n +1 | ≤ | L y ( x ) − ℓ − v | = d n +1 ( f v ( x ) , π ( v )) < δ < r. Also d ( e πix · y , e πi ( q ℓ + p ) y q · · · · · e πi ( q n ℓ n + p n ) ynqn ) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x · y − n X j =1 ( q j ℓ j + p j ) y j q j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 x j y j − n X j =1 ℓ j y j + p j q j y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 ( x j − ℓ j − v j ) y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) max ≤ j ≤ n | y j | (cid:19) n X j =1 | x j − ℓ j − v j |≤ (cid:18) max ≤ j ≤ n | y j | (cid:19) n X j =1 | L y ( x ) − ℓ − v | < (cid:18) max ≤ j ≤ n | y j | (cid:19) nδ < r. Since e πi ( q ℓ + p ) y q · · · · · e πi ( q n ℓ n + p n ) ynqn ∈ E , we arrive at a contradiction. Irrational case.
Suppose that y ∈ ( R \ Q ) n . Recall that for an irrationalangle θ the set { e πikθ : k ∈ Z } is dense in S .Let v = ( v , . . . , v n ) ∈ R n +1 . It suffices to show that for every δ > thereexists x ∈ R n for which d n +1 ( f y ( x ) , π ( v )) < δ .Let δ > . The set { e πiky n : k ∈ Z } is dense in S , so we may choose k n ∈ Z for which d ( e πi ( − v n y n + v n +1 ) , e πik n y n ) < δ/n . Let h n ∈ Z be suchthat − v n y n + v n +1 − h n ∈ ( k n y n − / , k n y n + 1 / . Similarly, for every j =1 , . . . , n − , there exists k j , h j ∈ Z satisfying d ( e πi ( − v j y j ) , e πik j y j ) < δ/n and − v j y j − h j ∈ ( k j y j − / , k j y j + 1 / . Let x = ( v + k , . . . , v n + k n ) ∈ R n and k = ( k , . . . , k n , − P nj =1 h j ) ∈ Z n +1 . Then L y ( x ) − k − v = , . . . , , n X j =1 ( v j + k j ) y j + n X j =1 h j − v n +1 and d n +1 ( f y ( x ) , π ( v )) = d n +1 ( π ( L y ( x ) − k ) , π ( v )) ≤ | L y ( x ) − k − v | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 ( v j + k j ) y j + n X j =1 h j − v n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | v n y n − v n +1 + h n + k n y n | + n − X j =1 | v j y j + h j + k j y j | . Since | v n y n − v n +1 + h n + k n y n | = d ( e πi ( − v n y n + v n +1 ) , e πik n y n ) < δ/n and | v j y j + h j + k j y j | = d ( e πi ( − v j y j ) , e πik j y j ) < δ/n for j = 1 , . . . , n − , wehave that d n +1 ( f y ( x ) , π ( v )) < δ . References [1] B. Bojarski and T. Iwaniec. Analytical foundations of the theory of quasiconformalmappings in R n . Ann. Acad. Sci. Fenn. Ser. A I Math. , 8(2):257–324, 1983.[2] M. Bonk and J. Heinonen. Quasiregular mappings and cohomology.
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