Hedlund-Metrics and the Stable Norm

Hedlund Metrics and the Stable Norm Madeleine Jotz Se ction de Math ´ ematiques, Ec ole Polyte chnique F ´ ed´ er ale de L ausanne, 1015 L ausanne, Switzerland. Abstract The real homology of a compact Riemannian manifold M is naturally endow ed with the stable norm. The stable norm on H 1 ( M , R ) arises from the Riemannian length functional by homogenizatio n . It is difficult and in teresting to decide which norms on the finite- dimensional vector space H 1 ( M , R ) are stable nor ms of a Riemannia n metric o n M . If the dimension o f M is at lea st three, I. Bab enk o a nd F. Ba lac heff prov ed in [1] that every po lyhedral no rm ball in H 1 ( M , R ), whose vertices ar e rational with resp ect to the lattice of in teger classes in H 1 ( M , R ), is the stable norm ball of a Riemannian metric on M . This metric can even b e chosen to b e co nf o r mally equiv alent to any g iv en metric. In [1], the stable norm induced by the constructed metric is computed by comparing the metric with a p olyhedral one. Her e w e pr esen t an alterna tive co nstruction for the metr ic, which remains in the geometric fra m ework of smo oth Riema nn ia n metrics. Key wor ds: Riemannian metrics, stable norm, p olytopes. 2008 MSC: 53C22, 5 3C38, 5 8 A10, 58 F1 7, 53B2 1 1. Introduction On every compact Riemannian manifold M the r eal homolog y vector spaces H m ( M ; R ) are endow ed with a natural norm k · k s , called stable norm . This concept app eared for the first time in F ederer [4] and was named stable norm in Gromov [5]. T he stable no r m on H 1 ( M ; R ) aris es directly fro m the Riemannian metric on the manifold M . The following equality fo r an integral class v ∈ H 1 ( M ; R ) (see [5]) k v k s := inf { n − 1 L ( γ ) | γ is a closed curve repres e n ting n v, n ∈ N } allows a descriptio n of this ob ject that is ge o metrically very intu itive: the stable norm describ es the geometry of the Abelia n covering ¯ M of M from a p oin t of view fro m which fundamental doma ins lo ok arbitra rily s m a ll. Knowing the unit ba ll of this norm, one can decide on e x istence and prop erties o f some o f the minimal geo desics re la tiv e to the Riemannian Ab elian covering of the manifold; these are curves in M who se lifts to the Riemannia n Ab elian cov ering minimize ar c length betw een each tw o of their po in ts. Email addr ess: madeleine.jotz @epfl.ch (Madeleine Jotz) Pr e p rint submitted to Elsevi er Novemb er 19, 2021 Bangert has presented in [3] a Riemannian metric o n the 3-torus T 3 , suc h that the unit ball of the induced stable norm on H 1 ( T 3 ; R ) ≃ R 3 is a symmetric o ctahedro n. F urthermore, Bab enko and Balacheff have shown in [1] that, given a compact Riema nnian manifold ( M , ρ ) of dimension grea ter than 2, for every centrally symmetric a nd co nvex po lytop e in H 1 ( M ; R ) with nonempty in terior , such that the directions of its vertices are ratio na l, there is a Riemannian metric on M that is confor ma l to ρ and induces the given po lytop e as unit ball o f the stable norm. Here w e prop ose an alternative Riemannian metric, satisfying the sa me conditions. O ur construction is a g eneralizatio n of the Hedlund metric in Ba ngert [3]. The idea, tha t can b e alrea dy found in the o riginal pap er of Hedlund [6] and is a ls o us ed in [1], is to constr uct a metric tha t is “small” in tubular neighbor ho o ds of disjoint clo sed curves repres e nting the vertices of the p olytop e, and muc h “bigger” everywhere else. The convexit y prop er ties of the p olytop e play a decisive r ole in our computation o f the sta ble nor m induced by the Hedlund metric . Bangert and Hedlund use such metrics in order to illustra te their results o n minimal geo desics. Here we fo cuse only on the pro of of the theorem of Babenko and Balacheff [1]. In fact, if we wan ted to show results on minimal geo desics, we would need to sp ecify the definitio n of the Hedlund metric we give here. A discussio n of the minima l geo desics for such metrics (with a dditio na l assumptions ) was made in Jotz [7 ]. Outline of t he p ap er:. in the next section the construction o f tubular neighborho o ds of curves will be recalled. There a lemma on existence of representatives for coho mology classes with “go o d” prop erties o n the tubular neighborho o d will be stated. In the fol- lowing section the cons tr uction of the Riemannian metr ic will be given and the fo r mula for the corr esp onding stable no rm will b e computed. Notations:. In the following M will denote a compact smo oth manifold with dim M ≥ 3 and ρ a Riemannian metric on M . Let ¯ M denote the Ab elian cov ering o f M . More precisely ¯ M is the sub covering o f the universal covering whose g r oup of deck tra ns forma- tions is the set H 1 ( M ; Z ) R of integer classes in H 1 ( M ; R ). W e denote b y p : ¯ M → M the cov ering map and by ¯ ρ := p ∗ ρ the pull-back metric. If h : π 1 ( M ) → H 1 ( M ; Z ) denotes the Hurewicz homomor phism ([see 9]) and T the to rsion subg roup of H 1 ( M ; Z ), then the Abelia n covering can b e describ ed a s the quo tient manifold of the actio n of the no rmal subgroup h − 1 ( T ) ⊆ π 1 ( M ) of the fundamental gr oup on the univ ersa l cover ˜ M of M . Hence the op era tion Φ : H 1 ( M ; Z ) R × ¯ M → ¯ M ( v , m ) 7→ Φ( v, m ) =: m + v of H 1 ( M ; Z ) R on ¯ M is ab elian and tor sionfree (that is why we choos e to use this +- notation). The de Rham c o homolog y vector space H 1 dR ( M ) is isomo r phic to the dual of H 1 ( M ; R ) [8, de Rham theor em]. In the following, we will use this iso morphism without mentioning it. Given a Riemannia n metric g on M , we will write g ∗ for its dual metric. The spac e of 1-forms on M (resp ectively on ¯ M ) will b e denoted by Ω 1 ( M ) (res pe c tively Ω 1 ( ¯ M )). W e will denote by k · k x (or also simply k · k ) the norm on T x M induced by the considered metric on M (we will also us e this notation for the norm on T ¯ x ¯ M , ¯ x ∈ ¯ M induced fr om the corres p o nding metric on ¯ M ). F or a c ur ve γ : I → M , L ( γ ) will b e the length induced 2 from the given metric on M and fo r a curve ¯ γ : I → ¯ M , ¯ L ( ¯ γ ) the length induced from the co r resp onding p erio dic metric on ¯ M . Given a p olytop e P , we will ca ll the s et { P k i =1 α i v i | α i ≥ 0 } the c one over the fac e S of the p olytop e, wher e v 1 , . . . , v k are the vertices of P lying in this face (i.e. S = { P k i =1 α i v i | α i ≥ 0 and P k i =1 α i = 1 } ). An integer class v in H 1 ( M ; Z ) R will be ca lled indivisible if the equatio n v = n · v ′ , n ∈ Z and v ′ ∈ H 1 ( M ; Z ) R yields n = ± 1 . A cknow le dgment:. I would like to thank Prof. Victor Bangert who sup ervised my diploma thesis a nd g av e me muc h advice for this pap er. I a m a ls o very grateful that he g ave me the po ssibility to stay at the Universit y of F reiburg during a few months a fter my diploma . I also thank the re fer ees for ma ny useful comments. 2. T ubular nei g h b o rho o ds of curv es, adapted one-forms T ubular neighb orho o ds and semi-ge o desic c o or dinates.. Let γ : [0 , 1] → M be a r egular simple closed curve. In the following, such a cur ve will b e c a lled admissible . W e can write γ : S 1 → M and assume the curve γ is par ametrized prop o r tionally to a rc leng th. F or  > 0 let V  (Γ) denote the bundle of balls o f radius  in the normal bundle π : N Γ → Γ o f the em b edded submanifold Γ := γ ( S 1 ) in M . Analogously , if I ⊆ S 1 is an interv al, then V  ( γ ( I )) = V  (Γ) ∩ π − 1 ( γ ( I )). W e choos e  > 0 small enough such that the normal exp onential map E res tricted to V  (Γ) is a diffeomorphism o nto an o pe n neighborho o d U  (Γ) ⊆ M of Γ (and similar ly U  ( γ ( I )) = E ( V  ( γ ( I ))) ). Suc h a n op en set U  (Γ) is called the tu bular neighb orho o d (of r adius  ) of Γ. Cho ose an orthog onal frame ( E 1 , . . . , E m ) o n U ⊆ M op en, such that for all x = γ ( t ) in Γ ∩ U , E 1 | x = ˙ γ ( t ) and, consequently , ( E 2 | x , . . . , E m | x ) forms a basis for N x Γ. Assume the op en set U is such that U  (Γ) ∩ U = U  ( γ ( I )) for an op en interv al I ⊆ S 1 . The diffeomorphism ϕ : U  ( γ ( I )) → I × B m − 1  ⊆ R m x 7→ ( s ( x ) , ϕ 2 ( x ) , . . . , ϕ m ( x )) , where ϕ j ( x ) and s ( x ) ar e such that E − 1 ( x ) = m X j =2 ϕ j ( x ) · E j | γ ( s ( x )) ∈ V  , will b e called a semige o desic chart for U  (Γ). A pa rticularity of this chart is that ∂ ϕ 1 | x = ˙ γ ( t ) and, for j = 2 , . . . , m , ∂ ϕ j | x = E j | x holds for all x = γ ( t ) ∈ Γ ∩ U (note that Γ ∩ U = γ ( I ) ). The map s is defined globally on U  (Γ) and we hav e the identit y ds | γ ( t ) ( ˙ γ ( t )) = d dt s ◦ γ ( t ) = d dt t = 1 (1) for all t in S 1 . 3 Let γ 1 , . . . , γ N be disjoint admissible lo ops and choo se  > 0 so that the co ns truction ab ov e is p os sible for a ll the curves γ 1 , . . . , γ N simult a ne o usly . Choo se fur ther more ε with  > ε > 0 such that the tubular neighborho o ds with radius ε o f the curves are disjoint. Set Γ j = γ j ( S 1 ), Γ = ∪ N j =1 Γ j , and U ε (Γ) := ∪ N j =1 U ε (Γ j ). Then there exists a bump- function ζ on M for the tubular neigh b or ho o ds, i.e., ζ is a smo oth function such that the following holds: ζ ( y ) =  1 , y ∈ U ε (Γ) 0 , y ∈ M \ U  (Γ) . (2) “Go o d” one-forms.. Cho ose a connected fundamental do ma in F 0 for the a ction of H 1 ( M ; Z ) R on ¯ M . Deno te by ¯ γ j the lift of γ j to ¯ M such tha t ¯ γ j (0) ∈ F 0 (note that γ j is here con- sidered as a smo oth 1-per io dic curve γ j : R → M ). W rite ¯ Γ i = ¯ γ i ( R ) and U  ( ¯ Γ i ) the corres p o nding lift to ¯ M of U  (Γ i ). Hence U  ( ¯ Γ i ) is the tubular neighbor ho o d of radius ρ of ¯ Γ i . Thus the no tio n of a semigeo desic chart for U  ( ¯ Γ i ) makes als o sense here , a nd ¯ s i : U  ( ¯ Γ i ) → R exists with ¯ s i ( ¯ γ i ( t )) = t for all t ∈ R . Since the cov ering map p : ¯ M → M is a lo cal isometry , ¯ x ∈ e x p ¯ M ( N ¯ γ i ( t ) ¯ Γ i ) ⇔ p ( ¯ x ) ∈ exp M ( N p ◦ ¯ γ i ( t ) Γ i ) holds for all ¯ x ∈ U  ( ¯ Γ i ) and ( p ∗ ds i ) | U  ( ¯ Γ i ) = d ¯ s i . (3) Define L i = ¯ Γ i + H 1 ( M ; Z ) R and U  ( L i ) = U  ( ¯ Γ i ) + H 1 ( M ; Z ) R , as w ell as L = ∪ N j =1 L j and U  ( L ) = ∪ N j =1 U  ( L j ). Cho os e ε with 0 < ε <  and define U ε ( ¯ Γ i ), U ε ( L i ) and U ε ( L ) as ab ove. The connected comp onents of L will b e called lines in the following. In the following, a regula r simple closed cur ve will be calle d an admissible curve. Prop ositi on 2. 1 L et v 1 , . . . , v N b e indivisi ble inte ger classes in H 1 ( M ; Z ) R , t hat sp an H 1 ( M ; R ) as a r e al ve ctor sp ac e. L et γ 1 , . . . , γ N b e disjoint admissible r epr esentatives of those classes, and U ε (Γ 1 ) , . . . , U ε (Γ N ) disjoi nt tubular neighb orho o ds of these curves. F urthermor e let λ ∈ H 1 dR ( M ) b e an arbitr ary c ohomolo gy class. Then ther e exists a one-form ω r epr esenting λ such that: ω | x = λ ( v i ) ds i | x for x ∈ U ε (Γ i ) , i = 1 , . . . , N . Proof: F or j = 1 , . . . , N , the function ¯ s j is defined on U  ( ¯ Γ j ). Set ¯ s j = 0 on U  ( ¯ Γ i ) fo r i 6 = j and define: s λ : U  ( L ) → R x = x 0 + v 0 7→ N X i =1 λ ( v i ) ¯ s i ( x 0 ) + λ ( v 0 ) . Doing so , each elemen t U  ( L j ) is wr itten x = x 0 + v 0 with x 0 ∈ U  ( ¯ Γ j ) ∩ F 0 and v 0 ∈ H 1 ( M ; Z ) R . F or x ∈ U  ( ¯ Γ j ) ∩ F 0 holds: s λ ( x ) = λ ( v j ) ¯ s j ( x ). Thus, with the definition o f s λ , for v = z · v j with z ∈ Z : s λ ( x + v ) = λ ( v j ) ¯ s j ( x ) + λ ( v ) = λ ( v j ) · ( ¯ s j ( x ) + z ) (3) = λ ( v j ) · ¯ s j ( x + v ) . 4 This leads to s λ | U  ( ¯ Γ j ) = λ ( v j ) ¯ s j , and analog ously: s λ | U  ( ¯ Γ j )+ v = λ ( v j ) ¯ s j ◦ Φ( − v , · )+ λ ( v ). Thu s, s λ is a smo oth function. Cho ose an arbitrary representativ e ω ′ for λ . Since ω ′ is closed, the 1-fo r m ˜ p ∗ ω ′ ∈ Ω 1 ( ˜ M ) is a lso clos ed, wher e ˜ p : ˜ M → M is the universal cov ering of M . Since each closed 1-form o n ˜ M is exact, there ex ists ˜ f ∈ C ∞ ( ˜ M ) such that ˜ p ∗ ω ′ = d ˜ f . One can show easily that ˜ f is inv aria nt under the action o f h − 1 ( T ) on ˜ M and des c e nds to ¯ f ∈ C ∞ ( ¯ M ), i.e., ˜ f = ¯ f ◦ q where q : ˜ M → ˜ M /h − 1 ( T ) = ¯ M is the pro jection. W e hav e p ◦ q = ˜ p a nd q ∗ d ¯ f = d ˜ f = ˜ p ∗ ω ′ = q ∗ ( p ∗ ω ′ ) and hence d ¯ f = p ∗ ω ′ . Let ¯ g := s λ − ¯ f | U  ( L ) : U  ( L ) → R . A computation shows that for a ll x ∈ U  ( L ) and v ∈ H 1 ( M ; Z ) R , we have ¯ g ( x + v ) = ¯ g ( x ) and the existence of g : U  (Γ) → R with ¯ g = g ◦ p follows. The map g is smo oth and we have on U  ( L ): p ∗ dg = d ¯ g = N X i =1 λ ( v i ) d ¯ s i − d ¯ f = p ∗ N X i =1 λ ( v i ) ds i − ω ′ ! . Since p is a surjective lo cal diffeomorphism, the equality dg = P N i =1 λ ( v i ) ds i − ω ′ follows. Define now the smo oth 1-fo r m ω := g dζ + (1 − ζ ) ω ′ + ζ N X i =1 λ ( v i ) ds i with ζ as in (2). Using the fact that ω ′ is clo sed on U  (Γ) and the pro pe r ties of ζ , one can eas ily verify that ω is smo oth a nd clo sed. F urthermo re, for x ∈ U ε (Γ j ): ω | x = g ( x ) dζ | x + (1 − ζ ( x )) ω ′ | x + ζ ( x ) · N X i =1 λ ( v i ) ds i | x = λ ( v j ) ds j | x , as cla imed. W e get [ ω ]( v j ) = Z γ j ω = λ ( v j ) Z 1 0 ds j | γ j ( t ) ( ˙ γ j ( t )) dt ( 1 ) = λ ( v j ) for j = 1 , . . . , N . With s pa n { v 1 , . . . , v N } = H 1 ( M ; R ), this yields that ω is a representa- tive for λ .  In the follo wing, such a representativ e ω will b e called a go o d r epr esent ative of λ with resp ect to the family { v 1 , . . . , v N } . 3. He dlund metrics Let P be a centrally symmetric and conv ex polyto pe in H 1 ( M ; R ) with nonempt y int er ior, such that the directions of its vertices are ra tional. Suc h a p olyto pe will be called admissible . W e call ˜ V P = { ˜ v 1 , . . . , ˜ v N , − ˜ v 1 , . . . , − ˜ v N } the set of vertices of P . Let v 1 , . . . , v N be indivisible integer clas ses suc h that v i = ε i ˜ v i with ε i > 0, i = 1 , . . . , N . Define V P := { v 1 , . . . , v N , − v 1 , . . . , − v N } and let J i be the subse t o f V P con- sisting of the indivis ible in teger class e s corresp onding to the vertices belonging to the i -th face S i of P . In order to s implify the notatio n, w e assume without loss o f generality 5 that J 1 = { v 1 , . . . , v k } for an integer k ≤ N . The norm | · | o n H 1 ( M ; R ), who se unit ball is P , is given a s follows (for vectors lying in the cone ov er the face S 1 ): v = k X j =1 α j ˜ v j with k X j =1 α j = 1 and all α j ≥ 0 ⇒ | v | = 1 (4) or g enerally v = k X j =1 α j ˜ v j with all α j ≥ 0 ⇒ | v | = k X j =1 α j and likewise for every o ther face of P . Since P is conv ex, for each face S i of P ex ists a n element λ i of H 1 dR ( M ) ≃ H 1 ( M , R ) such that λ i ( ˜ v j )  = 1 , v j = ε j ˜ v j ∈ J i < 1 , v j = ε j ˜ v j 6∈ J i (i.e. λ i ≡ 1 on the plane defined by the face S i and λ i is sma ller on the res t of the po lytop e). Now, since P is symmetr ic , − λ i is the 1-form c o rresp o nding to − S i and we get in fact: − 1 < λ i ( ˜ v j ) < 1 for ± v j 6∈ J i . (5) W e get an alter native definition for the norm: v ∈ k M j =1 R ≥ 0 · v j ⇒ | v | = λ 1 ( v ) , (6) and likewise for every o ther face of P . The metrics defined below will be ca lle d H e d lund metrics since such a metric first app ears in Hedlund’s pap er [6] in the ca se M = T 3 : Definition 3.1 L et P admissible p olytop e with vertic es { ˜ v 1 , . . . , ˜ v N , − ˜ v 1 , . . . , − ˜ v N } . L et v 1 , . . . , v N ∈ H 1 ( M , Z ) R b e the indivisible inte ger classes such that ε i ˜ v i = v i for some ε i > 0 , i = 1 , . . . , N . Cho ose disjo int admissible curves γ 1 , . . . , γ N r epr esenting the classes v 1 , . . . , v N . F or e ach fac e S i of P , let η i b e a go o d r epr esent ative of λ i with r esp e ct to the family { v 1 , . . . , v N } . A Hedlund metric asso ciate d to P on ( M , ρ ) is a Rie mann ian metric g that is c onformal to ρ and such that its dual metric g ∗ satisfies: (H 1) g ∗ γ i ( t ) ( ds i | γ i ( t ) , ds i | γ i ( t ) ) = max x ∈ U ε (Γ i ) g ∗ x ( ds i | x , ds i | x ) = 1 ε 2 i for al l t ∈ [0 , 1] and g ∗ x ( ds i | x , ds i | x ) < 1 ε 2 i for x ∈ U ε (Γ i ) \ Γ i and al l i ∈ { 1 , . . . , N } . (H 2) g ∗ x ( η i | x , η i | x ) ≤ 1 for al l i = 1 , . . . , N and x 6∈ U ε (Γ) . Remark that for o rientable co mpact surfaces of p ositive gen us, it is not p oss ible to choo se disjoint lo ops repr esenting the v ertices of the p olytop e. In fact, it is shown in Banger t [3] that in the c a se of the 2-torus , the stable no rm induced b y a Riemannian metric on T 2 has alwa ys a s trictly co nv ex unit ball. Y et, Massa r t shows in [1 0] that this is not true 6 in general: the stable no r m induced by a smo o th Finsler metric on a clos ed, orientable surface has neither to b e strictly conv ex, nor smo oth. F or a non-orientable surface, the analogo n to Theorem 3.5 ca n b e found in Balacheff and Mas sart [2]: they show that if M is a clo sed non-o rientable sur fa ce equipp ed with a Riemannia n metric , then there exists in every conformal class a metric on M whose stable norm has a po lyhedron as its unit ball. Existenc e and pr op erties of su ch a metric. Prop ositi on 3. 2 On every c omp act Riemannian manifold ( M , ρ ) with dim M ≥ 3 and for every admissible P in H 1 ( M , R ) ther e exists a He d lu nd metric asso ciate d t o P on ( M , ρ ) . Proof: Giv en the admissible po lytop e P , choos e disjoint admiss ible curves γ 1 , . . . , γ N representing the indivisible integer cla sses v 1 , . . . , v N corres p o nding to its vertices ˜ v 1 , . . . , ˜ v N . Let ε 1 , . . . , ε N be the co efficients as in Definition 3 .1. F or each fac e S i of P , i = 1 , . . . , l , let η i be a go o d representativ e for λ i . Set Ω := max j =1 ,...,l x ∈ M \ U ε (Γ) ρ ∗ x ( η j | x , η j | x ) and Ω i := ma x { max j =1 ,...,l x ∈ U  (Γ i ) ρ ∗ x ( η j | x , η j | x ) ρ ∗ x ( ds i | x , ds i | x ) , ε 2 i } for i = 1 , . . . , N . Define: h i : U  (Γ i ) → (0 , ∞ ) x 7→ 1 ε 2 i ρ ∗ x ( ds i | x , ds i | x ) · exp( − C i · ℓ ( x ) 2 ) where C i := ln  Ω i ε 2 i  · 1 ε 2 > 0 and ℓ ( x ) is the dis ta nce from to x to its “pr o jection” γ i ( s i ( x )) ∈ Γ i . Define the s mo oth function F : M → (0 , ∞ ) by F ( x ) = ζ ( x ) · b X i =1 h i ( x ) + (1 − ζ ( x )) · 1 Ω , where ζ is a smo oth bump function as in (2 ). It is then easy to verify that the metric g defined by g ∗ x = F ( x ) ρ ∗ x for all x ∈ M is a Hedlund metr ic a sso ciated to P .  7 Prop ositi on 3. 3 It r esult s imme diately fr om D efinition 3.1 and fr om the pr op erties of an admissible p olytop e that k η i k ∗ := ma x x ∈ M   η i | x   ∗ x = 1 (7) for e ach fac e S i of P . Proof: Here ag ain, we a s sume that i = 1. The ar g uments a r e the same for every other face of P . Outside o f U ε (Γ), Definition 3.1 yields   η 1 | x   ∗ x ≤ 1 . With   η 1 | x   ∗ x =                    ε j   ds j | x   ∗ x = 1 , x ∈ Γ j and j = 1 , . . . k ε j   ds j | x   ∗ x < 1 , x ∈ U ε (Γ j ) \ Γ j and j = 1 , . . . k | λ 1 ( v j ) | ·   ds j | x   ∗ x = ε j | λ 1 ( ˜ v j ) | · 1 ε j ( 5 ) < 1 , x ∈ U ε (Γ j ) and j > k , this pr oves the statement.  F or the pr o of of the following lemma, we need to co mpute the lengths of the chosen admissible curve γ 1 , . . . , γ N relative to the new metric. Cho o se x = γ i ( t ) ∈ Γ i and a semi-geo desic chart ϕ around x . Recall the constructio n of such a chart; the matrix representing ρ re la tive to the or thogonal basis ( ˙ γ i ( t ) , ∂ ϕ 2 | x , . . . , ∂ ϕ m | x ) of T x M is diag onal. Hence, beca use g is conformal to ρ , the ma tr ix representing g r e lative to this basis is diagonal, to o. Since the cov ectors ( ds i | x , dϕ 2 | x , . . . , dϕ m | x ) for m a dual basis o f T ∗ x M , we obtain g x ( ˙ γ i ( t ) , ˙ γ i ( t )) = 1 g ∗ x ( ds i | x , ds i | x ) , using the fact that the matr ice representing g x in the basis ( ˙ γ i ( t ) , ∂ ϕ 2 | x , . . . , ∂ ϕ m | x ) is inv erse to the matrix r e presenting g ∗ x in the dual basis. But b ecause o f (H 1) in Definition 3.1, we hav e g ∗ x ( ds i | x , ds i | x ) = 1 ε 2 i . Hence, this leads to: L ( γ i ) = Z 1 0 ε i dt = ε i . (8) It is po ssible to show that γ i is even the shortest curve r e pr esenting v i : Assume, without loss o f gene r ality , that v i ∈ J 1 and choos e an arbitrary c ur ve c : [0 , 1] → M repre senting v i . W e hav e λ 1 ( v i ) = ε i and henc e ε i = Z c η 1 = Z 1 0 η 1 | c ( t ) ( ˙ c ( t )) dt ≤ Z 1 0 k η 1 | c ( t ) k ∗ k ˙ c ( t ) k dt (7) ≤ Z 1 0 1 · k ˙ c ( t ) k dt = L ( c ) . Lemma 3.4 Th er e is a c onstant C = C ( M , P ) such that for e ach fac e S i of P , every w ∈ L v ∈ J i N · v and every x ∈ ¯ M , the distanc e fr om x to x + w is b ounde d ab owe by λ i ( w ) + C . 8 Proof: Recall the definitions of γ i , Γ i , ¯ γ i , ¯ Γ i , i = 1 , . . . , N , L and F 0 . Define D := max 1 ≤ i,j ≤ N min x ∈ Γ i y ∈ Γ j d ( x, y ) , diam( M ) := max x,y ∈ M d ( x, y ) and cho ose a real p os itive num b er e such that e > max i =1 ,...,N ε i . Let C := 2 · diam( M ) + κ · ( D + e ) (9) where d is the distanc e on M induced from the Hedlund metric g and κ = κ ( P ) is the maximal num b er of vertices lying on a common face o f P . Without los s of generality , w e a ssume that w ∈ L v ∈ J 1 N · v , i.e. we can write w = P k i =1 n i v i with n 1 , . . . , n k ∈ N . W e giv e a path fr o m x to x + w that has length bo unded ab ov e by λ 1 ( w ) + C = P k i =1 ε i n i + C . Assume that x ∈ F 0 (otherwise, if x ∈ F 0 + u with u ∈ H 1 ( M ; Z ) R , we can replac e the path with sta rtp oint x − u as constructed b elow with its image under Φ u ). W e join x with x + w by a path tha t runs as much as p ossible in L with “changes o f lines” that ar e as short as p ossible : Cho ose i 1 ∈ { j | 1 ≤ j ≤ k , n j 6 = 0 } such tha t the p oint x 1 in L ∩ F 0 with minimal distance fr om x lies in ¯ Γ i 1 . Let τ 1 be the c orresp o nding g eo desic segment from x to x 1 with minimal length. This length ¯ L ( γ 1 ) is smalle r than diam( M ). Let c 1 be the se gment of ¯ γ i 1 connecting x 1 and x 1 + n i 1 v i 1 . This segment has length equal to ¯ L ( c 1 ) = n i 1 · L ( γ i 1 ) (8) = n i 1 · ε i 1 . Now cho ose i 2 ∈ { j | 1 ≤ j ≤ k , n j 6 = 0 } \ i 1 and x 2 ∈ ¯ Γ i 2 + n i 1 v i 1 such that x 2 is the po int of ( L \ ¯ Γ i 1 ) ∩ ( F 0 + n i 1 v i 1 ) having minimal distance from ¯ Γ i 1 ∩ ( F 0 + n i 1 v i 1 ). Let x ′ 1 be the p oint in ¯ Γ i 1 ∩ ( F 0 + n i 1 v i 1 ) at this minimal dis tance from x 2 . Let c ′ 1 be the s ection of ¯ γ i 1 connecting x 1 and x ′ 1 ; the length of c ′ 1 lies in [ n i 1 · ε i 1 − e, n i 1 · ε i 1 + e ]. Let τ 2 be the minimal geo desic s egment joining x ′ 1 and x 2 , it has length sma ller tha n D . Now contin ue in this way; choose i 3 ∈ { j | 1 ≤ j ≤ k , n j 6 = 0 } \ { i 1 , i 2 } and x 3 ∈ ¯ Γ i 3 + n i 1 v i 1 + n i 2 v i 2 such that x 3 is the po int of ( L \ ( ¯ Γ i 1 ∪ ¯ Γ i 2 )) ∩ ( F 0 + n i 1 v i 1 + n i 2 v i 2 ) ha ving minimal distance from ¯ Γ i 2 ∩ ( F 0 + n i 1 v i 1 + n i 2 v i 2 ). Let x ′ 2 be the p o int in ¯ Γ i 2 ∩ ( F 0 + n i 1 v i 1 + n i 2 v i 2 ) at this minimal distance from x 3 . The curve c ′ 2 joining x 2 and x ′ 2 on ¯ Γ i 2 + n i 1 v i 1 has length smaller tha n n i 2 · ε i 2 + e . If n j 6 = 0 for j = 1 , . . . , k , o ur path w ill b e the c omp osition γ := τ 1 ∗ c ′ 1 ∗ τ 2 ∗ c ′ 2 ∗ · · · ∗ c ′ k ∗ τ i k +1 where τ k +1 is the path joining the la st p o int in L ∩ ( F 0 + P k i =1 n i v i ) with minimal distance from x + w to x + w and has length smaller than diam( M ). Summing all the lengths of those segments we get ¯ L ( γ ) ≤ diam( M ) + n i 1 · ε i 1 + e + D + n i 2 · ε i 2 + e + D + · · · + n i k · ε i k + e + diam( M ) = λ 1 ( w ) + k · e + k · D + 2 · dia m( M ) ≤ λ 1 ( w ) + C . Finally , if n j = 0 for some j ∈ { 1 , . . . , k } , we need to make fewer changes of lines, and the inequa lit y can be shown the sa me wa y .  9 The stable norm and the main t he or em.. In the introduction of this pap er, we gave the definition of the sta ble norm induced from a Riemannia n metric g on M . Here we give a wa y to compute the stable norm of a vector lying in H 1 ( M ; Z ) R : Define f : H 1 ( M ; Z ) R → R ≥ 0 v 7→ inf { L ( γ ) | γ closed curve representing v } and f n : n − 1 H 1 ( M ; Z ) R → R ≥ 0 , f n ( v ) = n − 1 f ( nv ). In Bangert [3 ] it is sho wn that f n conv erges uniformly on compact s e ts to the stable no r m k · k s . Es p e cially , we hav e: if ( v n ) n ∈ N is a sequence in H 1 ( M ; Z ) R with lim n →∞ v n n = v ∈ H 1 ( M ; R ) (relative to the standard top ology on the vector s pace H 1 ( M ; R ) ≃ R b ), then we hav e for the norm of v : k v k s = lim n →∞ f ( v n ) n . If ¯ d is the distance o n ¯ M induced fro m p ∗ g , we hav e for v ∈ H 1 ( M ; Z ) R : f ( v ) = inf x ∈ ¯ M ¯ d ( x, x + v ) = min x ∈ F 0 ¯ d ( x, x + v ) bec ause p ∗ g is a p erio dic metric and the clos ure of F 0 is a compact s et. With lim n →∞ nv n = v , this yields : k v k s = lim n →∞ f ( nv ) n = lim n →∞ min x ∈ F 0 ¯ d ( x, x + nv ) n . Theorem 3. 5 The p olytop e P is the unit b al l of the stable norm on H 1 ( M ; R ) induc e d by an arbitr ary He d lu nd metric asso ciate d to P on M . Note that by Definition 3 .1, the Hedlund metric is chosen in the confor mal c la ss of the given Riemannian metric ρ on M . Proof: Let g b e a Hedlund-metric asso c ia ted to P . W e show that fo r each w ∈ L k j =1 N · v j , the stable nor m of w is given by k w k s = λ 1 ( w ). The pro of o f this works a nalogous ly for every other face of P . Consequently , this holds for all vectors in H 1 ( M ; R ) that can be written as linear combinations of the vectors v 1 , . . . , v N with rational co efficients, and then, by co nt inuit y , this holds for all vectors in H 1 ( M ; R ). Let x b e an ar bitrary p oint in F 0 and let n ∈ N . Let γ : [0 , 1] → ¯ M b e an arbitra ry path from x to x + nw . W e have λ 1 ( nw ) = Z γ η 1 = Z 1 0 η 1 | γ ( t ) ( ˙ γ ( t )) dt ≤ Z 1 0 k η 1 | γ ( t ) k ∗ k ˙ γ ( t ) k dt (7) ≤ Z 1 0 1 · k ˙ γ ( t ) k dt = ¯ L ( γ ) With this and Lemma 3.4 we g et λ 1 ( n · w ) ≤ ¯ d ( x, x + nw ) ≤ λ 1 ( n · w ) + C. 10 Thu s λ 1 ( n · w ) ≤ min x ∈ F 0 ¯ d ( x, x + nw ) ≤ λ 1 ( n · w ) + C, and λ 1 ( w ) ≤ min x ∈ F 0 ¯ d ( x, x + nw ) n ≤ λ 1 ( w ) + C n . 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