aa r X i v : . [ m a t h . M G ] J u l DUALITY OF MODULI IN REGULAR TOROIDALMETRIC SPACES
ATTE LOHVANSUU
Abstract.
We generalize a result of Freedman and He [4, Th.2.5.], concerning the duality of moduli and capacities in solid tori,to sufficiently regular metric spaces. This is a continuation of thework of the author and K. Rajala [12] on the corresponding dualityin condensers. Introduction
Given a metric measure space (
T, d, µ ), with µ Borel-regular, and acollection Γ of paths in T , the p -modulus of Γ is the numbermod p Γ := inf ρ ˆ T ρ p dµ, where the infimum is taken over non-negative Borel-functions ρ thatsatisfy(1) ˆ γ ρ ds > γ ∈ Γ. The path modulus is a widely used toolin geometric function theory, especially in connection to quasiconformalmappings [7, 14, 15].In the 1960s, F. Gehring [6] and W. Ziemer [16] proved that themoduli of paths connecting two compact and connected sets in R n aredual to the moduli of surfaces that separate the two sets. The moduliof surface families are defined as above, but instead of condition (1) werequire ˆ S ρ d H n − > , where H n − denotes the ( n − G of any metric space, anddisjoint connected compact sets E, F ⊂ G , denote by Γ( E, F ; G ) thefamily of paths in G that intersect both E and F , and by Γ ∗ ( E, F ; G )the family of compact subsets of G that separate E and F . We saythat a set S separates E and F in G if E and F belong to differentcomponents of G − S . Triples ( E, F, G ) are called condensers . Let
Mathematics Subject Classification 2010:
Primary 30L10, Secondary 30C65,28A75, 51F99.The author was supported by the Vilho, Yrj¨o and Kalle V¨ais¨al¨a foundation. p ∗ = pp − be the dual exponent of 1 < p < ∞ . By Gehring and Ziemerwe then have(2) (mod p Γ( E, F ; G )) p (mod p ∗ Γ ∗ ( E, F ; G )) p ∗ = 1in R n with n > q -regular metric spaces that support a 1-Poincar´e inequality.In more detail, a special case of what is shown in [12] is(3) 1 C (mod q Γ( E, F ; G )) q (mod q ∗ Γ ∗ ( E, F ; G )) q ∗ C for some constant C that depends only on the data of the space, i.e.the constants that appear in the definitions (see Section 2) of Ahlforsregularity and the Poincar´e inequalities. Here E, F and G are as in (2),and the sets in Γ ∗ are equipped with the ( q − T homeomorphic to the solidtorus S × D . It is natural to ask if the duality results above remainvalid for the family of paths that go around the ’hole’ and the familyof surfaces which are bounded by meridians on the boundary torus. Itturns out that this is not the case. Freedman and He [4] studied con-formal moduli on riemannian tori in connection with their research ondivergence-free vector fields. They showed that the path-modulus canbe arbitrarily small compared to the corresponding surface modulus,even in the smooth setting. However, they managed to prove a dualityresult by replacing the path modulus with a certain capacity.Suppose now that T is equipped with a metric d and a Borel-regularmeasure µ , so that ( T, d, µ ) is Ahlfors q -regular. That is, there areconstants a, A > ar q µ ( B ) Ar q for all balls B with radius r < diam( T ).Following Freedman and He [4] we consider the degree 1 capacity instead of the path modulus. It is defined bycap p T := inf φ ˆ T Lip( φ ) p dµ, where the infimum is taken over pointwise Lipschitz constantsLip( φ )( x ) := lim sup r → sup y ∈ B ( x,r ) | φ ( x ) − φ ( y ) | r UALITY OF MODULI IN REGULAR TOROIDAL METRIC SPACES 3 of Lipschitz maps φ : T → S of degree 1. Loosely speaking, a mapis said to have degree 1 if it takes (oriented) loops which generate thecorresponding fundamental group to (oriented) generating loops in S .We assume S is equipped with a metric that makes it isometric to aeuclidean circle of length 1 equipped with its geodesic metric.The surface modulus mod p T is defined to be the p -modulus of alllevel sets of continuous functions of degree 1, see Section 2, equippedwith the ( q − THEOREM 1.1.
Let ( T, d, µ ) be a compact Ahlfors q -regular metricmeasure space that supports a weak -Poincar´e inequality. Suppose T is homeomorphic to the solid torus S × D . Let < p < ∞ . If cap p T is nonzero, then C (cap p T ) p (mod p ∗ T ) p ∗ C where C is a constant that depends only on the data of T . Moreover cap p T = 0 if and only if mod p ∗ T = ∞ . A similar result, with C = 1, was proved by Freedman and He [4,Th. 2.5] for smooth solid tori equipped with riemannian metrics.Theorem 1.1 is obtained from slightly more general statements. Theseare Theorems 2.2 and 2.3, and they correspond to the lower and upperbounds of the inequality in Theorem 1.1, respectively. The proof of thelower bound is essentially the same as the proof of the lower bound of(3) found in [12]. The main difficulty of the proof of Theorem 1.1 isthen the upper bound.In [12] the proof of the upper bound boils down to showing thatgiven any path γ that connects the two continua E and F , and aneighborhood N γ of | γ | , there is a function admissible for the modulusof surfaces separating E and F that is supported in N γ . This approachcannot be adopted in our current situation, since the paths have beenreplaced with Lipschitz maps. Instead, given any level set S of a mapof degree 1 and a neighborhood N S of S , we construct a Lipschitz mapof degree 1 that is constant outside N S . Note that this implies thatthe pointwise Lipschitz constant of this map can be assumed to besupported in N S . This approach seems to be new. It can be seen as adual to the one in [12], and as such it can in fact be used to reprove(3).Section 2 contains some definitions and the main results. Theorems2.2 and 2.3 are proved in Sections 3 and 4, respectively. Acknowledgement.
The author expresses his thanks to the anony-mous referee, whose comments led to several improvements.
ATTE LOHVANSUU Main results and definitions
For the rest of this text we fix a compact metric measure space(
T, d, µ ) that supports a weak 1-Poincar´e inequality. We also assumethat µ is doubling. In order to apply the theory of covering spaces lateron, we also have to assume that T is semilocally simply connected (localand global path connectedness follow from the 1-Poincar´e inequality [7,8.3.2]).We call a measure µ doubling if it is Borel-regular and there existsa constant C µ >
1, such that for every ball B = B ( x, r ) with radius r < diam( T ) 0 < µ (2 B ) < C µ µ ( B ) < ∞ . Here 2 B = B ( x, r ).Let M be a set of Borel-regular measures on T and let 1 p < ∞ .We define the p - modulus of M to bemod p M = inf ˆ T ρ p dµ, where the infimum is taken over all Borel measurable functions ρ : T → [0 , ∞ ] with(4) ˆ T ρ dν > ν ∈ M . Such functions are called admissible functions of M . Ifthere are no admissible functions we define the modulus to be infinite.If ρ is an admissible function for M − N where N has zero p -modulus,we say that ρ is p -weakly admissible for M . As a direct consequenceof the definitions we see that the p -modulus does not change if theinfimum is taken over only p -weakly admissible functions. If someproperty holds for all ν ∈ M − N we say that it holds for p - almostevery ν in M .Given a family Γ of paths in T , the path p -modulus of Γ is denotedand defined like the modulus of a family of measures, but instead of(4) it is required that ˆ γ ρ ds > γ ∈ Γ.A Borel function ρ : T → [0 , ∞ ] is an upper gradient of a function u : T → Y , where ( Y, d Y ) is a metric space, if(5) d Y ( u ( γ ( a )) , u ( γ ( b ))) ˆ γ ρ ds for all rectifiable paths γ : [ a, b ] → T . The target Y = [ −∞ , ∞ ] is alsoallowed, but with an additional requirement that the right-hand side of(5) has to equal ∞ whenever either | u ( γ ( a )) | = ∞ or | u ( γ ( b )) | = ∞ . Ifthe family of paths for which (5) fails has zero p -modulus, we say that UALITY OF MODULI IN REGULAR TOROIDAL METRIC SPACES 5 ρ is a p - weak upper gradient. The inequality (5) is called the uppergradient inequality for the pair ( u, ρ ) on γ .A p -integrable p -weak upper gradient ρ of u is minimal if for anyother p -integrable p -weak upper gradient ρ ′ of u we have ρ ρ ′ µ -almost everywhere. By [7, Theorem 6.3.20] minimal p -weak upper gra-dients exist whenever p -integrable upper gradients do.The space T is said to support a weak p - Poincar´e inequality withconstants C P and λ P if all balls in T have positive and finite measure,and − ˆ B | u − u B | dµ C P diam ( B ) (cid:18) − ˆ λ P B ρ p dµ (cid:19) p for all locally integrable functions u and all upper gradients ρ of u .Here u B = − ˆ B u dµ = 1 µ ( B ) ˆ B u dµ. In this paper we consider toroidal spaces, meaning that we assumethe fundamental group of T to be isomorphic to Z with respect toany basepoint. Fix a generator [ α x ] ∈ π ( T, x ). We say that a loop γ with basepoint x ∈ T is a degree 1 loop if it is loop-homotopic to α x = γ xx ∗ α x ∗ ← γ xx for some path γ xx that starts at x and ends at x . It can be shown that the equivalence class [ α x ] ∈ π ( T, x ) does notdepend on the choice of γ xx .For every continuous map f : T → R / Z there is a unique integerdeg f , called the degree of f , so that for every x ∈ T and every degree1 loop γ based at x the push-forward f ∗ γ = f ◦ γ is loop-homotopic to[ f ( x )] + deg f · β , where β : [0 , → R / Z is the path β ( t ) = [ t ].Now let 1 < p < ∞ . We define the degree 1 p -capacity of T to bethe number cap p T := inf ˆ T ρ pf dµ, where the infimum is taken over all Lipschitz maps f : T → R / Z with deg f = 1, and ρ f denotes the minimal p -weak upper gradient of f . Note that for Lipschitz maps the minimal upper gradient agreesalmost everywhere with the pointwise Lipschitz constant Lip( f ), see[3] and [7, 13.5.1]. We assume here and hereafter that R / Z is equippedwith the metric | [ x ] − [ y ] | = inf a ∈ Z | x + a − y | , where the equivalence classes of R / Z are denoted by brackets. Observethat with this metric R / Z is isometric to a 1-dimensional euclideansphere of total length 1 equipped with its intrinsic length metric.Denote by Γ ∗ the family of all level sets φ − [0] with finite codimension1 spherical Hausdorff measure, where φ : T → R / Z is a continuous mapof degree 1. The codimension 1 spherical Hausdorff measure is defined ATTE LOHVANSUU by H ( A ) := sup δ> H δ ( A ) , where H δ ( A ) := inf X i µ ( B i ) r i , and the infimum is taken over countable covers { B i } of A by balls withradii r i δ . By the Carath´eodory construction H is a Borel-regularmeasure. A simple application of a coarea estimate, see Proposition 3.1,shows that almost all level sets of Lipschitz maps have finite H -measure.On the other hand, the relative isoperimetric inequality (Lemma 4.6)shows that level sets of Lipschitz maps of degree 1 must have nonzero H -measure.As a dual counterpart to cap p T we consider the surface modulus ofΓ ∗ . We abbreviate(6) mod p ∗ T = mod p ∗ {H S | S ∈ Γ ∗ } . The definitions of cap p T and mod p ∗ T are rather trivial if Lipschitzmaps of degree 1 do not exist. Although path-connected topologicalspaces with fundamental groups isomorphic to Z can fail to admit mapsof nonzero degree, it seems to be unknown whether the existence of sucha map is implied by the additional structure of ( T, d, µ ). To make lifeeasier we simply assume that there exists at least one Lipschitz map f : T → R / Z of degree 1.Let us gather all of the assumptions into one place for clarity andfuture reference. Assumptions 2.1.
The metric measure space (
T, d, µ ) is doubling andsupports a weak 1-Poincar´e inequality. The space T is compact andsemilocally simply connected. The fundamental group of T with re-spect to any basepoint is isomorphic to Z and there exists at least oneLipschitz map φ : T → R / Z of degree 1.With these assumptions our main results are the following THEOREM 2.2.
Let < p < ∞ . If cap p T > , then C (cap p T ) p (mod p ∗ T ) p ∗ , where the constant C depends only on the data of T . If cap p T = 0 ,then mod p ∗ T = ∞ . THEOREM 2.3.
Let < p < ∞ . If mod p ∗ Γ ∗ < ∞ , then (cap p T ) p (mod p ∗ T ) p ∗ C, UALITY OF MODULI IN REGULAR TOROIDAL METRIC SPACES 7 where the constant C depends only on the data of T . If mod p ∗ T = ∞ ,then cap p T = 0 . We say that a constant
C > T , denoted C = C ( T ), if it depends only on the constants C µ , C P and λ P appearingin the definitions of doubling measures and Poincar´e inequalities. Thesame symbol C will be used for various different constants.If we let the metric measure space ( T, d, µ ) be as in Theorem 1.1, itsatisfies Assumptions 2.1. The existence of Lipschitz maps of degree1 follows from Proposition 4.5. In Ahlfors q -regular spaces the H -measure is comparable to the ( q − C -quasiconvex for some C = C ( T ). This means that the change of metrics ( T, d ) → ( T, d ′ ) is C -biLipschitz, when d ′ is the intrinsic length metric induced by d . Itfollows that we may assume without any loss of generality that d isthe length metric. It is then implied by compactness that ( T, d ) is infact geodesic. Note that in geodesic spaces we can choose λ P = 1. Forthese facts see Theorem 8.3.2 and Remark 9.1.19 in [7].3. Proof of Theorem 2.2
The proof of Theorem 2.2 is exactly the same as the proof of Theorem3.1 in [12], but with a different coarea estimate.
Proposition 3.1.
Let u : T → R / Z be Lipschitz and let ρ be a p -integrable upper gradient of u in T . Let g : T → [0 , ∞ ] be a p ∗ -integrableBorel function. Then (7) ˆ ∗ R / Z ˆ u − ( t ) g d H dt C ˆ T gρ dµ for some C = C ( T ) . Proposition 3.1 follows by applying [12, Prop. 4.1] in small enoughballs.
Proof of Theorem 2.2.
First assume that cap p T >
0. If mod p ∗ T = ∞ ,there is nothing to prove. Otherwise let g ∈ L p ∗ ( T ) be admissible formod p ∗ T . Let u : T → R / Z be Lipschitz with degree 1 and note that u must be surjective. Let ρ be an upper gradient of u . We may assumethat ρ is p -integrable. Note that by (7) H ( u − ( t )) < ∞ for almost ATTE LOHVANSUU every t . Proposition 3.1 and H¨older’s inequality give1 ˆ ∗ R / Z ˆ u − ( t ) g d H dt C ˆ T gρ dµ C (cid:18) ˆ T g p ∗ dµ (cid:19) p ∗ (cid:18) ˆ T ρ p dµ (cid:19) p . The lower bound follows by taking infima over admissible functions g and ρ . The same argument would lead to a contradiction if mod p ∗ T was finite when cap p T = 0. (cid:3) Proof of Theorem 2.3
Theorem 2.3 follows, once we have shown that there is a non-negativeBorel function ρ defined on T , such thatcap p T = ˆ T ρ p dµ, and that(8) cap p T C ( T ) ˆ S M C ( T ) /n ( ρ p − ) d H for all S ∈ Γ ∗ and all large enough n , depending on S . Here M r for r > n → ∞ and applying the general Fuglede’s lemma [5, Theorem 3] wefind that(9) cap p T C ( T ) ˆ S ρ p − d H for mod p ∗ -almost every S . Now suppose mod p ∗ T < ∞ . If cap p T =0, there is nothing to prove. Otherwise it follows from (9) that thefunction C ( T )cap p T ρ p − is weakly admissible for mod p ∗ T . Thus(mod p ∗ T ) /p ∗ C ( T )cap p T (cid:18) ˆ T ρ p ∗ ( p − dµ (cid:19) /p ∗ = C ( T )(cap p T ) − /p . The same calculation shows that mod p ∗ T must be finite if cap p T isnonzero. This proves Theorem 2.3. The rest of this section is focusedon finding ρ and proving (8).Let us begin by constructing ρ . We would like to apply the usualmethod of constructing minimizers for capacities or moduli. Thismethod would consist of picking a minimizing sequence ( φ i ) i of Lip-schitz maps of degree 1 and their upper gradients ( ρ i ) i , applying weakcompactness properties of L p -spaces and Mazur’s lemma to find a sub-sequence of convex combinations of ρ i that converges strongly to somelimit ρ , and finally showing that ρ is an upper gradient of a Lipschitzmap of degree 1. The obvious flaw with this method is that it is not UALITY OF MODULI IN REGULAR TOROIDAL METRIC SPACES 9 clear whether the proposed minimizer ρ or the convex combinationsof the functions ρ i are upper gradients of Lipschitz maps of degree 1.To fix this, we replace the collection of upper gradients of degree 1Lipschitz maps by a slightly larger collection F and show in Proposi-tion 4.3 that the capacity does not change if we take the infimum overfunctions of F instead. The collection F is defined using the universalcover ( ˜ T , π ) of T , and consists of those non-negative Borel functions ρ on T for which the function ρ ◦ π is an upper gradient of a Newto-nian map, which satisfies an analogue of the degree 1 -property. SeeSubsection 4.2 for the definition of Newtonian maps. Once we haveset the proper definition of F , it is easy to see that it is convex, andby applying the proofs of existing compactness results on Newtonianspaces we show in Proposition 4.4 that the limit ρ is a member of F as well.4.1. Universal cover and lifts.
We denote the universal cover of T by ( ˜ T , π ). The metric ˜ d on ˜ T is defined as the path metric induced bypulling back the length functional of T with π . This means that givenpoints ˜ x, ˜ y ∈ ˜ T we define˜ d (˜ x, ˜ y ) = inf γ ℓ ( π ◦ γ ) , where the infimum is taken over all paths in ˜ T that connect ˜ x and ˜ y ,and ℓ ( π ◦ γ ) is the length of the path π ◦ γ . With this metric π becomesa local isometry.We equip ˜ T with the Borel-regular measure ˜ µ that satisfies˜ µ ( A ) := ˆ π ( A ) N ( x, π, A ) dµ ( x ) , for all Borel sets A ⊂ ˜ T . Here N ( x, π, A ) denotes the cardinality of π − ( x ) ∩ A . The area formula ˆ ˜ T f d ˜ µ = ˆ T X y ∈ π − ( x ) f ( y ) dµ ( x )holds for every integrable Borel-function f .Denote by τ : ˜ T → ˜ T the unique deck transform that satisfies τ (˜ γ (0)) = ˜ γ (1) , for all lifts ˜ γ : [0 , → ˜ T of all degree 1 loops γ : [0 , → T . Withthe additional metric and measure theoretic structure the classic liftingtheorems imply the following. Lemma 4.1.
Suppose f : T → R / Z is a Lipschitz map of degree andlet ρ be one of its upper gradients. There exists a function ˜ f : ˜ T → R ,called the lift of f , that satisfies the following properties. (1) [ ˜ f ] = f ◦ π . In particular ˜ f is locally Lipschitz (2) ρ ◦ π is an upper gradient of ˜ f (3) ˜ f ◦ τ − ˜ f = 1 .Moreover, if ˜ f ′ is another lift that satisfies the properties above, thenthere is a k ∈ Z such that ˜ f ′ = ˜ f ◦ τ k = ˜ f + k . Claim (2) follows from the identity ˆ γ ρ ◦ π ds = ˆ π ◦ γ ρ ds, which holds for every rectifiable path γ in ˜ T .Conversely, we have the following. Lemma 4.2.
For every locally Lipschitz g : ˜ T → R with g ◦ τ − g = 1 there is a Lipschitz map f : T → R / Z of degree 1, that satisfies [ g ] = f ◦ π . Moreover, if ρ f is the minimal p -weak upper gradient of f in T ,then ρ f ◦ π is the minimal p -weak upper gradient of g in ˜ T .Proof. We define f locally by f = [ g ◦ π − ] . Then f is well defined due to the property g ◦ τ − g = 1. It is certainlylocally Lipschitz, has degree 1, and satisfies [ g ] = f ◦ π .It remains to show the relation between the upper gradients. Givenany x ∈ ˜ T there is a ball B ′ that contains x and on which π is anisometry onto B = π ( B ′ ). Clearly ρ ◦ π | − B ′ is a p -weak upper gradientof f in B whenever ρ is a p -weak upper gradient of g in B ′ . Thus, if ρ f ◦ π is a p -weak upper gradient of g in ˜ T , it must be the minimal one.Now let γ : [0 , → B ′ be a rectifiable path, so that the upper gra-dient inequality holds for the pair ( f, ρ f ) on every subpath of π ◦ γ .Almost every path in B ′ is such a path, since ρ f is a p -weak uppergradient of f , and as an isometry π | B ′ preserves all path moduli. Con-tinuity of g implies that we can decompose γ into γ = γ ∗ · · · ∗ γ k , sothat γ i = γ | [ t i ,t i +1 ] and(10) | g ( γ i ( t i +1 )) − g ( γ i ( t i )) | = | [ g ( γ i ( t i +1 ))] − [ g ( γ i ( t i ))] | for all i = 1 , . . . , k . On these subpaths we have ˆ γ i ρ f ◦ π ds = ˆ π ◦ γ i ρ f ds > | f ( π ( γ i ( t i +1 ))) − f ( π ( γ i ( t i ))) | = | [ g ( γ i ( t i +1 ))] − [ g ( γ i ( t i ))] | . UALITY OF MODULI IN REGULAR TOROIDAL METRIC SPACES 11
Combining this with triangle inequality and (10) yields | g ( γ (1)) − g ( γ (0)) | ˆ γ ρ f ◦ π ds. Given an open set U ⊂ ˜ T , denote the set of all paths in U on whichthe upper gradient inequality fails for the pair ( g, ρ f ◦ π ) by Γ U . Weneed to show that mod p Γ ˜ T = 0. Cover ˜ T by countably many balls B ′ i ,on which π is an isometry onto π ( B ′ i ). Note that if the upper gradientinequality fails for the pair ( g, ρ f ◦ π ) on some path η , it must fail onsome subpath of η that is contained in one of the balls B ′ i . In otherwords, for every path in the collection Γ ˜ T there is a subpath in one ofthe collections Γ B ′ i . Nowmod p Γ ˜ T mod p [ i Γ B ′ i ! X i mod p Γ B ′ i = 0 , since the first part of the proof shows that mod p Γ B ′ i = 0 for all i . (cid:3) Minimizers.
Motivated by Lemmas 4.1 and 4.2 we find an alter-native definition for the capacity.We say that a function f : ˜ T → R belongs to the Newtonian space N ,p ( ˜ T ) if f is p -integrable and admits a p -weak upper gradient thatis also p -integrable. See [7, Chapter 7] or [1, Chapter 5] for furtherproperties of these spaces. We say that f ∈ N ,ploc ( ˜ T ) if f | U ∈ N ,p ( U )for every open U ⊂⊂ ˜ T (note that ˜ T is proper). The space N ,p ( U ) isequipped with the seminorm k f k N ,p ( U ) := k f k L p ( U ) + inf ρ k ρ k L p ( U ) , where the infimum is taken over all p -weak upper gradients ρ of f in U .Let F be the collection of all positive Borel functions ρ on T , forwhich ρ ◦ π is a p -weak upper gradient of some f ∈ N ,ploc ( ˜ T ) with f ◦ τ − f = 1 almost everywhere. Definecap F p T := inf ρ ∈F ˆ T ρ p dµ. Note that by Lemma 4.1 every upper gradient of a map admissible forcap p T belongs to F . Thereforecap F p T cap p T. The reverse inequality is also valid, but requires a bit more work.
Proposition 4.3. cap p T = cap F p T Proof.
We must first show that locally Lipschitz functions of degree 1are dense in the space of degree 1 functions of N ,ploc ( ˜ T ). Here havingdegree 1 means satisfying the property f ◦ τ − f = 1 almost everywhere.A result by Bj¨orn and Bj¨orn [2, Th. 8.4.] shows that locally Lipschitzfunctions are dense in N ,ploc ( ˜ T ). A simple modification of the proof ofthis result shows that the approximating locally Lipschitz maps can bechosen to be of degree 1 whenever the limit is of degree 1. We providethe main points of this modification.Following the proof of Theorem 8.4 of [2], we start by choosing forevery x ∈ ˜ T a ball B x centered at x , so that • the 1-Poincar´e inequality and the doubling property hold within B x , in the sense of [2] • the covering map π is an isometry on B x .Let U x := π − ( π ( B x )). The space T is compact, so there is a finitesubcollection { U x i } mi =1 that covers ˜ T . Write B j = B x j and U j = U x j .Note that U j can be written as a disjoint union U j = S k ∈ Z τ k B j . Wedenote cU j = S k ∈ Z τ k ( cB j ) for any c >
0. For each j pick a Lipschitzfunction ψ ′ j : B j → R that satisfies χ B j ψ ′ j χ B j . Extend these toLipschitz functions ψ j : ˜ T → R first by defining ψ j | τ k (2 B j ) := ψ ′ j ◦ τ − k in2 U j and then extending as zero to the rest of ˜ T . Next, define Lipschitzmaps ϕ j : ˜ T → R recursively with ϕ = ψ and for j > ϕ j = ψ j · − j − X k =1 ϕ k ! . Then P ik =1 ϕ k = 1 in U i and ϕ j = 0 in U i for all j > i . Therefore { ( ϕ j , U j ) } j is a partition of unity.Now let f ∈ N ,ploc ( ˜ T ) be a degree 1 map, f ◦ τ − f = 1. Let ε > v j : 2 B j → R with k f − v j k N ,p (2 B j ) ε L j , where L j is the Lipschitz constant of ϕ j . Extend v j to 2 U j with v j | τ k (2 B j ) = k + v j ◦ τ − k . Then v j ◦ τ − v j = 1, and for all k k f − v j k N ,p ( τ k (2 B j )) ε L j . UALITY OF MODULI IN REGULAR TOROIDAL METRIC SPACES 13
As in [2] we get(11) k ϕ j ( f − v j ) k pN ,p ( τ k (2 B j )) ε p . The function v := P mj =1 ϕ j v j is locally Lipschitz, and satisfies the de-gree 1 property v ◦ τ − v = 1.Now (11) gives k v − f k N ,p ( U ) C ( U ) ε, for any domain U ⊂⊂ ˜ T . This proves the density of degree 1 locallyLipschitz functions in the space of N ,ploc ( ˜ T )-functions of degree 1.Now if we let ρ ∈ F , the function ρ ◦ π is a p -weak upper gradient ofsome f ∈ N ,ploc ( ˜ T ), and we find a sequence of locally Lipschitz functions( v j ) of degree 1, such that k v j − f k N ,p ( U ) j →∞ −→ . for every U ⊂⊂ ˜ T . Let w j be the Lipschitz projections of v j , given byLemma 4.2. Then the minimal upper gradients satisfy ρ v j = ρ w j ◦ π .Now(12) k ρ v j − ρ f k L p ( B i ) j →∞ −→ i . Let A = π ( B ) and for 1 j m − A j +1 := π ( B j +1 ) − S ji =1 A j . Let π j : B j ∩ π − ( A j ) → A j be the restriction of π and define ρ ′ f := P j χ A j ρ f ◦ π − j . The Borel sets A j are disjoint andcover T , so a quick calculation shows that(13) k ρ ′ f k pL p ( T ) k ρ k pL p ( T ) , since by definition of ρ we have ρ f ρ ◦ π almost everywhere. Finally,note that k ρ w j − ρ ′ f k pL p ( T ) = m X j =1 k ρ w j − ρ ′ f k pL p ( A j ) m X j =1 k ρ v j − ρ f k pL p ( B j ) , and thus (12) implies(14) lim j →∞ k ρ w j k pL p ( T ) = k ρ ′ f k pL p ( T ) , since there are only finitely many sets A j . Combining (13) and (14)yields cap p T k ρ k pL p ( T ) , which finishes the proof. (cid:3) Proposition 4.4.
There is a minimizer ρ ∈ F , i.e. cap p T = cap F p T = ˆ T ρ p dµ. Moreover, for any other p -integrable ρ ∈ F (15) cap p T ˆ T ρ p − ρ dµ, where equality holds if and only if ρ = ρ almost everywhere.Proof. Note that F is convex. Once we know the existence of a min-imizer, the proof of the variation inequality (15) is standard. See forexample [12, Lemma 5.2.]. Uniqueness of the minimizer follows fromthe convexity of F and the uniform convexity of L p ( T ).We now show the existence of a minimizer. First recall that we haveassumed in Assumptions 2.1 that there exists at least one Lipschitzmap of degree 1. It follows that cap p T is finite. Let ( f i ) i be a sequenceof locally Lipschitz maps f i : T → R / Z of degree 1, so that for each i the function ρ i is an upper gradient of f i , andcap p T = lim i →∞ ˆ T ρ pi dµ. We claim that the lifts ˜ f i of the maps f i can be chosen so that thesequence ( ˜ f i ) is L p -bounded in any bounded domain of ˜ T .To this end, note that the length of any loop-homotopically non-trivial loop γ must satisfy(16) ℓ ( γ ) > c for some c >
0. This is implied by the existence of Lipschitz maps ofdegree 1.Let { x i } Ni =1 be a c -net in T , where c is the constant from (16). Notethat by the net property of { x i } any two balls B i := B ( x i , c ) are con-nected by a chain of balls of the same form. By a chain we mean asequence of balls, in which adjacent ones have nonempty intersection.The same chaining property holds for the balls 2 B i , but now addition-ally we find that the connecting chains (2 B i k ) k can be chosen so thatfor each k there is a ball B ′ k ⊂ B i k ∩ B i k +1 of radius c/ B i are evenly covered. In fact, π isan isometry when restricted to any component of π − (2 B i ). Fix acomponent ˜ B of π − ( B ). Set V = ˜ B . For k > V k recursively by adding components of π − ( B i ) for suitable B i . At UALITY OF MODULI IN REGULAR TOROIDAL METRIC SPACES 15 step k + 1 we choose exactly one component of π − ( B i ), call it ˜ B i , tobe added to V k if and only if π − ( B i ) intersects V k and there are nocomponents of π − ( B i ) that are contained in V k .After at most N steps no new balls can be added. Let V = V N . Itfollows from the construction that V is a bounded domain on which π is surjective. It may happen that the previous construction does notdefine ˜ B i for all B i . If so, just let ˜ B i be a component of π − ( B i ) thatis contained in V . Thus V = N [ i =1 ˜ B i . Denote by 2 ˜ B i the component of π − (2 B i ) that contains ˜ B i .By adding integers if necessary, we may now fix the lifts ˜ f i by re-quiring(17) 0 ( ˜ f i ) B < . If j = 1, by construction there is a chain (2 ˜ B j k ) lk =1 with j = 1, j l = j and l N , so that for every 1 k < l there is a ball ˜ B ′ k ⊂ B j k ∩ B j k +1 of radius c/
8. Let m := min { µ ( B i ) } >
0. By the Poincar´e inequalityand the doubling condition | ( ˜ f i ) B jk − ( ˜ f i ) ˜ B ′ k | C − ˆ B jk | ˜ f i − ( ˜ f i ) B jk | d ˜ µ C k ρ i k pL p (2 B jk ) , where C = C ( T, p, m, c ), and the same calculation shows | ( ˜ f i ) B jk +1 − ( ˜ f i ) ˜ B ′ k | C k ρ i k pL p (2 B jk +1 ) as well. Thus by the triangle inequality and (17) | ( ˜ f i ) B j | CN k ρ i k pL p ( T ) + 1 . Now by the Sobolev-Poincar´e inequality, see [7, Thm. 9.1.2], and thelocal isometry of π ˆ B j | ˜ f i − ( ˜ f i ) B j | p d ˜ µ C ( T, p, m, c ) k ρ i k L p ( T ) . It follows that the sequence ( ˜ f i ) i is bounded in L p (2 ˜ B j ).Since V is covered by finitely many balls 2 B j , we find that bothsequences ( ˜ f i ) i and ( ρ i ◦ π ) i are bounded in L p ( V ), and also in every L p ( W k ), where W k := k [ l = − k τ l V. Note that W = V . Now by extracting enough subsequences we mayassume that ( ˜ f i ) i and ( ρ i ◦ π ) i converge weakly to functions ˜ f and˜ ρ in L p ( W ). By Lemma 3.1 of [10] there exist sequences of convexcombinations ( ˜ f k ) and ( ˜ ρ k ) of the functions ˜ f i and ρ i ◦ π , respectively,that converge strongly to ˜ f and ˜ ρ . Moreover ˜ ρ is a p -weak uppergradient of ˜ f in W .This allows us to define sequences ( ˜ f k +1 i ) and ( ˜ ρ k +1 i ) recursively to bethe sequences in L p ( W k +1 ) that are obtained by applying the argumentabove on W k +1 instead of W and on sequences ( ˜ f ki ) and ( ˜ ρ ki ) i insteadof ( ˜ f i ) i and ( ρ i ◦ π ) i . Let ˜ f k +1 and ˜ ρ k +1 be the corresponding limitsin L p ( W k +1 ). It follows that ˜ f k +1 | Ω k = ˜ f k and ˜ ρ k +1 | Ω k = ˜ ρ k . Define ˜ f and ˜ ρ : ˜ T → R by setting ˜ f | W k = ˜ f k and ˜ ρ | W k = ˜ ρ k . It is immediatethat ˜ ρ is a p -weak upper gradient of ˜ f .Consider the diagonal sequences ( ˜ f jj ) j and ( ˜ ρ jj ) j . These maps arestill convex combinations of the functions ˜ f i and ρ i ◦ π , respectively. Itfollows that these sequences converge to ˜ f and ˜ ρ in L ploc ( ˜ T ). Moreover˜ f ◦ τ − ˜ f = 1 and ˜ ρ ◦ τ − ˜ ρ = 0 almost everywhere, since these holdeverywhere for all maps in the respective sequences. The latter equalityallows us to define ρ by projecting ˜ ρ . Therefore ρ ∈ F andcap F p T = ˆ T ρ p dµ, since ( ˜ ρ jj ) is still a minimizing sequence, due to convexity of F . (cid:3) Competing admissible maps.
Now that the minimizer ρ hasbeen found, the proof of Theorem 2.3 is only missing the proof of (8).Recall that (8) says that for all S ∈ Γ ∗ cap p T C ( T ) ˆ S M C ( T ) /n ( ρ p − ) d H , where M denotes the Hardy-Littlewood maximal operator. Given an S ∈ Γ ∗ we construct suitable Lipschitz maps of degree 1 that are con-stant outside a small neighborhood of S . Then we can apply the varia-tion inequality (15) of Proposition 4.4 on the upper gradients of theseLipschitz maps to conclude (8).In this subsection we construct these Lipschitz maps. It turns outthat the same construction can be used to obtain Lipschitz maps ofdegree 1 out of general (continuous) maps of any nonzero degree. Weonly need to consider maps of positive degree by composing with theantipodal map of R / Z if necessary. UALITY OF MODULI IN REGULAR TOROIDAL METRIC SPACES 17
To simplify the notation, we omit some parentheses and write forexample φ − [0] and π ˜ φ − (0) instead of φ − ([0]) and π ( ˜ φ − (0)) fromnow on. Proposition 4.5.
Let φ : T → R / Z be a continuous map of nonzeropositive degree. There is a number N = N ( φ ) , such that for all n > N there is a finite pairwise disjoint collection of balls { B i } of radius /n in T , such that for all i H ( φ − [0] ∩ B i ) > C ( T ) nµ ( B i ) and such that the Borel function ρ = n X i χ B i is an upper gradient of a Lipschitz map ψ : T → R / Z of degree 1.Proof of (8) assuming Proposition 4.5. Let S ∈ Γ ∗ . Then S = φ − [0]for some degree 1 map φ . Let { B i } be the collection of balls and let ρ be the Borel function that is obtained by applying Proposition 4.5 forsome large enough n . Now H ( S ∩ B i ) > C ( T ) nµ ( B i )for all i . Applying this along with the variation inequality (15) ofCorollary 4.4, the doubling property of µ and the definition of theHardy-Littlewood maximal operator givescap p T ˆ T ρρ p − dµ C ( T ) X i nµ ( B i ) − ˆ B i ρ p − dµ C ( T ) X i H ( S ∩ B i ) inf x ∈ B i M C ( T ) /n ( ρ p − )( x ) C ( T ) ˆ S M C ( T ) /n ( ρ p − ) d H , which is exactly (8). (cid:3) The rest of the section is focused on proving Proposition 4.5. Let φ : T → R / Z be a continuous map of nonzero positive degree. Let x ∈ φ − [0], ˜ x ∈ π − ( x ) and let ˜ φ : ˜ T → R be the lift of φ thatsatisfies ˜ φ (˜ x ) = 0. Compactness of T implies that(18) δ := min (cid:26) d ( π ˜ φ − ( ±
18 ) , π ˜ φ − (0)) , d ( π ˜ φ − ( ±
14 ) , π ˜ φ − ( ±
18 )) (cid:27) is strictly positive. Denote U + = π ˜ φ − (0 , /
4) and U − = π ˜ φ − ( − / , S = π ˜ φ − (0). Observe that S ⊂ φ − [0], and if deg φ = 1,then S = φ − [0].For our intents and purposes the relative isoperimetric inequalitytakes the following form. Lemma 4.6. (Relative isoperimetric inequality)There are constants C = C ( T ) and λ = λ ( T ) > such that min (cid:26) µ ( B ∩ U + ) µ ( B ) , µ ( B ∩ U − ) µ ( B ) (cid:27) C rµ ( λB ) H ( S ∩ λB ) for all balls B = B ( x, r ) for which λB ⊂ π ˜ φ − ( − / , / . This formulation is essentially the same as the one used in [12,Lemma 5.1], which is just an application of Theorems 6.2 and 1.1 of[11]. The same proof is valid here as well. Note that restricting theballs to φ − ( − / , /
4) ensures that ∂U + ∩ λB ⊂ S ∩ λB .Denote by Γ the set of all paths γ that connect π ˜ φ − ( − /
8) to π ˜ φ − (1 /
8) inside π ˜ φ − ( − / , / Corollary 4.7.
For every n > δ and γ ∈ Γ there is a ball B nγ that iscentered on γ , has radius n and satisfies H ( S ∩ B nγ ) > Cnµ ( B nγ ) for some constant C = C ( T ) .Proof. The proof is essentially contained in the discussion followingLemma 5.1 in [12]. We sketch the idea here for completeness. Givena path γ : [0 , → T of Γ, we consider the balls B t := B ( γ ( t ) , λn ),where λ is as in Lemma 4.6. We may assume that | γ | is contained in π ˜ φ − [ − / , / δ each B t is con-tained in π ˜ φ − ( − / , / t µ ( U + ∩ B t ) µ ( B t )vanishes when t is near 0 and is equal to 1 when t is near 1. Pick t := sup { t ∈ (0 , | Φ( t ) / } and choose B nγ := 2 λB t . The lower bound on the measure of theboundary is then given by the relative isoperimetric inequality. (cid:3) Now let F n be the collection of balls B nγ that arise from the paths inΓ as in Corollary 4.7 with n fixed. Apply the 5 r covering theorem on UALITY OF MODULI IN REGULAR TOROIDAL METRIC SPACES 19 F n to find a pairwise disjoint subcollection G n with the property [ B ∈F n B ⊂ [ B ∈G n B. Note that G n must be finite due to the compactness of T . Write G n = { B i } Ni =1 . Define a positive Borel function ρ : T → R with ρ := n N X i =1 χ B i . Let Ω be the open set that consists of the points that can be connectedto π ˜ φ − ( − /
8) by a rectifiable path inside π ˜ φ − ( − / , / ψ : T → R inside Ω with˜ ψ ( x ) := inf γ x ˆ γ x ρ ds, where the infimum is taken over all rectifiable paths γ x that connect π ˜ φ − ( − /
8) to x inside π ˜ φ − ( − / , / ψ as zero to the restof T . Finally, the desired competing admissible map ψ : T → R / Z isdefined by ψ ( x ) := [min { , ˜ ψ ( x ) } ] . Lemma 4.8.
The mapping ψ is Lipschitz and ρ is one of its uppergradients.Proof. It is straightforward to prove that ρ is an upper gradient of both˜ ψ and min { , ˜ ψ } in Ω, see e.g. [1, Lemma 5.25]. Let γ be a rectifiablepath in T that connects two points x, y ∈ T . The upper gradientinequality for the pair ( ψ, ρ ) on γ is immediate if x, y ∈ Ω and | γ | ⊂ Ω,or if ψ ( x ) = ψ ( y ).In order to prove the upper gradient inequality in the other possiblesituations we need to show that ˜ ψ > π ˜ φ − (1 / , / ∩ Ω. To thisend, let η be a rectifiable path that connects π ˜ φ − ( − /
8) to a point x ∈ π ˜ φ − (1 / , /
4) inside π ˜ φ − ( − / , / η has a subpath η ′ ∈ Γ. Let B nη ′ ∈ F n be the ball obtained by applying Corollary 4.7on η ′ . Now ˆ η ρ ds > ˆ | η ′ | ρ d H > n N X i =1 H ( | η ′ | ∩ B i ) > n H ( | η ′ | ∩ B nη ′ ) > , since B nη ′ is covered by the balls 5 B i . This holds for every connectingpath η , which implies that ˜ ψ ( x ) > x, y ∈ Ω with ˜ ψ ( x ) , ˜ ψ ( y ) ∈ (0 ,
1) and | γ | 6⊂ Ω. Notethat min { , ˜ ψ } equals 0 in π ˜ φ − ( − / , − / ∩ Ω, since ρ vanishes there. This means that there exist subpaths γ = γ | [0 ,t ] and γ = γ | [ t , of γ that satisfy | γ | ∪ | γ | ⊂ Ω and ψ ( γ ( t )) = ψ ( γ ( t )) = [0]. Therefore | ψ ( x ) − ψ ( y ) | | ψ ( x ) − ψ ( γ ( t )) | + | ψ ( γ ( t )) − ψ ( y ) | ˆ γ ρ ds + ˆ γ ρ ds ˆ γ ρ ds. The same argument can be applied in the case of x ∈ Ω, y Ω. Weomit the details.The upper gradient inequality implies that ψ is Lipschitz, since T isgeodesic and ρ is bounded. (cid:3) Degree of ψ . In this subsection we prove that deg ψ = 1.Pick a rectifiable degree 1 loop γ and a point a ∈ (1 / , / γ are on π ˜ φ − ( a ). Since T isgeodesic and semilocally simply connected, we may assume that γ hasfinite length. This, and moving the starting point if necessary, allowsus to decompose γ into(19) γ = ( γ ∗ η ) ∗ · · · ∗ ( γ k ∗ η k ) , so that each γ i intersects π ˜ φ − ( a ) precisely at the endpoints, and noneof the paths η i intersect π ˜ φ − ( − a ).For the next lemma we denote for brevity ζ := min { , ˜ ψ } . Lemma 4.9.
Let η : [0 , → Ω be a rectifiable path. Suppose thatthe endpoints of ζ ∗ η belong to { , } . Then ψ ∗ η is loop-homotopic to ζ ∗ η (1) − ζ ∗ η (0) times the standard generator of π ( R / Z , [0]) .Proof. If the starting point is 0 and the end point is 1, the homotopyis given by H : [0 , → R / Z , H ( s, t ) = [ sζ ∗ η ( t ) + (1 − s ) t ] . It is straightforward to check all the requirements. The other cases aresimilar. (cid:3)
Corollary 4.10.
The paths ψ ∗ η i and φ ∗ η i are loop-contractible.Proof. The endpoints of the path η i must be in the set π ˜ φ − ( a ). Since γ has finite length, η i can be decomposed into η i = η i ∗ · · · ∗ η li , where the endpoints of each η ji are in π ˜ φ − ( a ), and if | η ji | 6⊂ Ω, thenthere are no other intersections with π ˜ φ − ( a ). UALITY OF MODULI IN REGULAR TOROIDAL METRIC SPACES 21
Now if η ji is contained in Ω, Lemma 4.9 implies that it is loop-contractible. Otherwise ψ ∗ η ji is already a constant path. Therefore ψ ∗ η i is loop-contractible as well. The path φ ∗ η i cannot be surjective,so it is loop-contractible. (cid:3) Let α : R / Z → R / Z be the isomorphism α [ x ] = [ x − a ]. Note that α ∗ φ ∗ γ i and ψ ∗ γ i are all loops with the same basepoint [0].Denote the domain of γ i by [ a i , b i ]. Let γ ′ i : [ a i , b i ] → R be the uniquelift of α ∗ φ ∗ γ i for which γ ′ i ( a i ) = 0. Further decompose each γ i into γ i = γ i ∗ γ i ∗ γ i , where γ i and γ i intersect π ˜ φ − ( ± a ) exactly at their endpoints. Lemma 4.11.
The lifted path γ ′ i intersects integer multiples of deg φ exactly at its endpoints. In particular γ ′ i ( b i ) = ± deg φ or γ ′ i ( b i ) = 0 .Moreover, γ i (respectively γ i ) is contained in Ω if and only if γ ′ i isnegative in a neighborhood of a i ( γ ′ i γ ′ ( b i ) in a neighborhood of b i ).Proof. Let ˜ γ i : [ a i , b i ] → ˜ T be the lift of γ i that satisfies ˜ φ ∗ ˜ γ i ( a i ) = a .Then due to uniqueness of lifts we have γ ′ i = ˜ φ ∗ ˜ γ i − a . Since ˜ φ is alift of φ , we have ˜ φ ◦ τ k = k · deg φ + ˜ φ for any integer k . It followsthat γ ′ i ( t ) = k · deg φ if and only if ˜ φ ( τ − k (˜ γ i ( t ))) = a , which can becombined with the lifting property π ∗ ˜ γ i = γ i to conclude that γ ′ i ( t )equals an integer multiple of deg φ if and only if γ i ( t ) ∈ π ˜ φ − ( a ). Byconstruction the latter happens if and only if t equals either endpointof [ a i , b i ]. This proves the first assertion of the lemma.The definitions of γ i , γ i and Ω imply that these paths are containedin Ω if and only if they are contained in π ˜ φ − [ − a, a ]. Therefore γ i is contained in Ω if and only if the part of ˜ γ i corresponding to γ i iscontained in ˜ φ − ( k · deg φ + [ − a, a ]) for some fixed integer k . This k must be 0, since we chose ˜ φ ∗ ˜ γ i ( a i ) = a . Thus γ ′ i = ˜ φ ∗ ˜ γ i − a is negativein a neighborhood of a i if and only if γ i is contained in Ω. The path γ i can be treated similarly. (cid:3) Corollary 4.12.
The paths α ∗ φ ∗ γ i and deg φ · ψ ∗ γ i are loop-homotopic.Proof. We need to check four different cases, corresponding to γ i and γ i being or not being contained in Ω. The proofs are essentially thesame, so we write down only one of them.Assume that γ i is not contained in Ω but γ i is. Then ψ ∗ γ i is aconstant path, and ψ ∗ γ i is loop-homotopic to the standard generatorby Lemma 4.9. Arguing exactly as in the proof of Corollary 4.10, we see that ψ ∗ γ i is loop-contractible. Therefore ψ ∗ γ i is loop-homotopic tothe standard generator.By Lemma 4.11 the lift γ ′ i satisfies γ ′ i ( a i ) = 0 and γ ′ i ( b i ) = ± deg φ or γ ′ i ( b i ) = 0. We also find that γ ′ i is positive in a neighborhood of a i , and less than γ ′ i ( b i ) in a neighborhood of b i . Combining these gives γ ′ i ( b i ) = deg φ , which means precisely that α ∗ φ ∗ γ i is loop-homotopic todeg φ times the standard generator. (cid:3) Applying Corollaries 4.10 and 4.12 to the decomposition (19) yields α ∗ φ ∗ γ ≃ deg φ · ψ ∗ γ. Now by applying the identity deg ( α ◦ φ ) = deg φ , we see that ψ ∗ γ isloop-homotopic to the standard generator. Therefore deg ψ = 1 andthe proof of Proposition 4.5 is finished. References [1] Anders Bj¨orn and Jana Bj¨orn.
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Department of Mathematics and Statistics, University of Jyv¨askyl¨a,P.O. Box 35 (MaD), FI-40014, University of Jyv¨askyl¨a, Finland.