Metric currents and the Poincar\'e inequality

We show that a complete doubling metric space $(X,d,\mu)$ supports a weak $1$-Poincar\'e inequality if and only if it admits a pencil of curves (PC) joining any pair of points $s,t \in X$. This notion was introduced by S. Semmes in the 90's, and has been previously known to be a sufficient condition for the weak $1$-Poincar\'e inequality. Our argument passes through the intermediate notion of a generalised pencil of curves (GPC). A GPC joining $s$ and $t$ is a normal $1$-current $T$, in the sense of Ambrosio and Kirchheim, with boundary $\partial T = \delta_{t} - \delta_{s}$, support contained in a ball of radius $\sim d(s,t)$ around $\{s,t\}$, and satisfying $\|T\| \ll \mu$, with $$\frac{d\|T\|}{d\mu}(y) \lesssim \frac{d(s,y)}{\mu(B(s,d(s,y)))} + \frac{d(t,y)}{\mu(B(y,d(t,y)))}.$$ We show that the $1$-Poincar\'e inequality implies the existence of GPCs joining any pair of points in $X$. Then, we deduce the existence of PCs from a recent decomposition result for normal $1$-currents due to Paolini and Stepanov.

In short, GPCs are a relaxed version of Semmes' pencils of curves, introduced in the 90's. The construction of GPCs is based on the max flow -min cut theorem in graph theory.

INTRODUCTION
Let (X, d, µ) be a complete metric space, where µ is a doubling locally finite Borel measure. It is known, see for example [5,7], that plenty of analysis can be conducted on (X, d, µ) whenever the weak p-Poincaré inequality is satisfied for some C, p, λ ≥ 1, for all locally integrable Borel functions u : X → R, and for all upper gradients ρ of u. So, it is worthwhile to find necessary and sufficient conditions for the validity of (1.1). One well-known sufficient condition is the existence of pencils of curves, introduced by Semmes [11] in the 90's. To motivate the results in the present paper, we first discuss Semmes' condition in some detail; our definition is the one given in Section 14.2 in [7], where the setting is somewhat more general than in Semmes' original work [11]. Definition 1.2 (Pencils of curves). The space (X, d, µ) admits pencils of curves if there exists a constant C 0 ≥ 1 with the following property. For all distinct s, t ∈ X there is a family Γ s,t of rectifiable curves γ ⊂ B(s, C 0 d(s, t)), each joining s to t and satisfying H 1 (γ) ≤ C 0 d(s, t), and a probability measure α s,t on Γ s,t such that Γs,tˆγ g dH 1 dα s,t (γ) ≤ C 0ˆB (s,C 0 d(s,t)) g(y) Θ(s, d(s, y)) + g(y) Θ(t, d(t, y)) dµ(y).
It is not too hard to see that a doubling space (X, d, µ) admitting pencils of curves satisfies the Poincaré inequality (1.1) with exponent p = 1; this is discussed briefly after [11,Definition 14.2.4], and we also include (most of) the details after (4.11) in the present paper. We do not know if the converse holds. There are spaces which satisfy (1.1) for some p > 1 but not for p = 1, so the p-Poincaré inequality for p > 1 cannot imply the existence of pencils of curves. However, we are not aware of spaces satisfying the 1-Poincaré inequality but not admitting pencils of curves.
Of course, Semmes in [11] gives sufficient conditions for finding pencils of curves: his Standard Assumptions (see [11,Theorem 1.11] and above) include the space (X, d, µ) to be an orientable topological n-manifold, with µ = H n . Moreover, X has to be locally contractible (for more precise statements, see [11,Definition 1.7] or [11,Definition 1.15], but also the discussion in [11,Remark A.35]). These assumptions are certainly not necessary for a space (X, d, µ) to admit pencils of curves or support a Poincaré inequality; notably, the Laakso spaces [9] have pencils of curves, hence satisfy (1.1) with p = 1, but are generally not integer-dimensional.
The main purpose of the present paper is to introduce a relaxed condition which is close in spirit to pencils of curves, and can be used -by virtually duplicating the classical proof -to deduce the 1-Poincaré inequality. Conversely, we are able to show that the condition is implied by the weak 1-Poincaré inequality. Definition 1.3 (Generalised pencils of curves). The space (X, d, µ) admits generalised pencils of curves (GPC in short) if there exists a constant C 0 ≥ 1 with the following property. For all distinct s, t ∈ X, there exists a normal 1-current T on X (in the sense of Ambrosio and Kirchheim) satisfying the following three properties: for µ-a.e. y ∈ X.
As already mentioned above, the main result is the following: Let (X, d, µ) be complete and doubling. Then X satisfies (1.1) with p = 1 if and only if X admits generalised pencils of curves.
The structure of the paper is the following. In Section 3, we briefly recall the definition of, and some basic concepts related to, the metric currents of Ambrosio and Kirchheim. Then, in Section 4, we prove the easy "if" implication of Theorem 1.4, mostly using classical methods in metric analysis. Finally, Section 5 is the core of the paper, containing the proof of the "only if" implication. In short, the idea is to translate the problem of finding currents in (X, d, µ) to finding "network flows" in certain graphs derived from δ-nets in X. The existence of such flows is guaranteed by the famous max flow -min cut theorem of Ford and Fulkerson [3]. Then, the main task will be to verify that there are no "small cuts" in the graph, and this can be done by using the Poincaré inequality (1.1) with p = 1.
1.1. Basic notation. Open balls in a metric space (X, d) will be denoted by B(x, r), with x ∈ X and r > 0. A measure on (X, d) will always refer to a Borel measure µ with µ(B(x, r)) < ∞ for all balls B(x, r) ⊂ X. The notation A B means that there exists a constant C ≥ 1 such that A ≤ CB: the constant C will typically depend on the "data" of the ambient space, for example the doubling constant of µ, or the constant in the Poincaré inequality (1.1) (whenever (1.1) is assumed to hold). The two-sided inequality A B A is abbreviated to A ∼ B.

ACKNOWLEDGEMENTS
T.O. is grateful to David Bate and Sean Li for a week of discussions in the summer of 2018, which greatly inspired this project. We are particularly grateful to David Bate for pointing out that the main result might work in all doubling spaces, and not just Q-regular ones.

BACKGROUND ON CURRENTS
The main result in the paper mentions currents in metric spaces, so we include here a brief introduction. We claim no originality for anything in this section. We use the definition of metric currents given by Ambrosio and Kirchheim, see Definition 3.1 in [1].
Let X be a complete metric space, and let Lip(X) and Lip b (X) be the families of Lipschitz, and bounded Lipschitz functions on X.
A k-dimensional current, or just a k-current, is then a (k + 1)-multilinear functional T ∈ M F k (X) with finite mass, satisfying a few additional requirements which we will not need explicitly, see Definition 3.1 in [1]. If T is a k-current, so that T is a finite Borel measure, then bounded Lipschitz functions are dense in L 1 (X, T ), and in particular the space of bounded Borel functions B(X) equipped with the L 1 ( T )-norm. This fact, and (3.1), together imply that T has a canonical extension to B(X) × [Lip(X)] k , which we also denote by T .
We review a few basic concepts related to currents.  A k-current T is called normal, if ∂T is a (k − 1)-current, in particular, ∂T has finite mass.
Definition 3.4 (Push-forward). Let X, Y be complete metric spaces, and let ϕ : The following is a special instance of Definition 2.5 in [1].

Definition 3.5 (Restriction)
. Let X be a complete metric space, T ∈ M F 1 (X), and g ∈ Lip b (X). Then we define an element T ⌊ g ∈ M F 1 (X) by setting If T is a 1-current, then Definition 3.5 can be extended to g ∈ B(X) using the canonical extension of T to B(X) × Lip(X), see [1, p.11]. Moreover, in that case, T ⌊ g is again a current, see [1, p.16 and p.19]. If E is a Borel subset of X and g = χ E , we write T ⌊ E for the restriction T ⌊ g . We have for all (f, π 1 ) ∈ D 1 (X). Since χ E is merely Borel but not Lipschitz, (3.6) does not follow directly from the definition of T given by (3.1), but it can be deduced by the density argument alluded to earlier, see [1, (2.3)]. Finally, (3.6) and the minimality of We record the following lemma, which follows from general measure theory: Assume that T is a k-current on a σ-compact metric space X. Then, for any Borel set B ⊂ X and any ǫ > 0, there exists a compact set K ⊂ B with T (B \ K) < ǫ.
Proof. By assumption T is a finite Borel measure. The claim now follows from [10, Theorem 1.10], and the Note directly below it.
We next describe a simple example, which will be useful later on.
Example 3.8. Given a non-degenerate interval [a, b] ⊂ R we may define the 1-current a, b as follows: . This is a particular case of Example 3.2 in [1], and it is noted shown by the following computation: Next, consider an isometry γ : [a, b] → X, where X is any complete metric space. Then γ ♯ a, b defines a current in X given by (spelling out Definition 3.4) It is noted in [1, (2.6)], and in the discussion directly below, that The last equation follows from the isometry assumption. Finally, because boundary and pushforward commute by [1, (2.1)], we have We end the section by recalling (a slightly simplified) version of the compactness theorem for normal currents. The original reference, and the full version of the theorem, is [1, Theorem 5.2].
and such that spt T n ⊂ K for some fixed compact set K ⊂ X, for all n ∈ N. Then there exists a subsequence (T nm ) km∈N , and a normal k-current T supported on K, such that

GENERALISED PENCILS OF CURVES IMPLY THE 1-POINCARÉ INEQUALITY
In this section we prove that if X as in Theorem 1.4 admits generalised pencils of curves, then it supports a weak 1-Poincaré inequality. It is well known, see for instance [7,Theorem 8.1.7], that doubling metric measure spaces which support a Poincaré inequality can be characterised in terms of the validity of pointwise inequalities between functions and their upper gradients. We will use the existence of GPCs to derive such an inequality between an arbitrary Lipschitz function u : X → R and its (upper) pointwise Lipschitz constant The desired inequality will be based on the following lemma.
Lemma 4.1. Let X be a complete σ-compact metric space and let u : X → R be a Lipschitz function. Then, for any 1-current T , we have Proof. The idea of the proof is the following: By definition of ∂T and since T is a finite Borel measure on X satisfying (3.1), we know that where Lip(u) is the Lipschitz constant of u, that is, the smallest constant L ∈ [0, ∞) for which |u(x)−u(y)| ≤ Ld(x, y) holds for all x, y ∈ X. The desired inequality in Lemma 4.1 is similar, but Lip(u) is replaced by the pointwise Lipschitz constant Lip(u, ·). To achieve this, we will essentially decompose X into pieces where Lip(u, ·) is almost constant.
We now turn to the details. Write We perform countable decompositions of the sets E and Z. Consider and that for every j ∈ Z fixed, the sequences (E 1/i,j ) i∈N and (Z 1/i,j ) i∈N are nested. Then, we can write Notice that the restriction of u to the set A similar argument applies if x ∈ X, and y, z ∈ B(x, 1/(2i)) ∩ Z 1/i,j =: Z x 1/(2i),j . Then the conclusion is that the restriction of u to the set Z x 1/(2i),j is 2 −j -Lipschitz, but one does not care about (and cannot have) the lower bound (4.4).
Since (X, d) is σ-compact, it is separable and hence we can pick a countable dense subset {x n } n∈N ⊆ X. Then for arbitrarily small δ ∈ [0, 1/2), the sets constitute a countable cover of X by Borel sets. Using this cover, we can easily construct a countable disjoint cover of X by Borel sets {E i } i∈N and {Z i } i∈N such that the function u can be decomposed as where δ > 0 can be taken arbitrarily small. Here we have used that the properties (4.3) and (4.4) (and their counterparts for Z 1/(2i),j ) are preserved under taking subsets. Moreover, Lemma 3.7 allows us to remove for every i ∈ N a Borel set N i from E i (or similarly Z i ) such that Next, we use the McShane extension theorem to find Lipschitz functions and we can write We now estimate the terms involving χ E i . By definition of the restriction operation, see Definition 3.5 and the comment below it, we can write Recall that u| E i = u E i . This is useful information since, according to [1, (3.6)], the values of a 1-current agree on (f, π 1 ) and (f ′ , π ′ 1 ) whenever f = f ′ and π 1 = π ′ 1 on the support of T . Using that spt (T ⌊ E i ) ⊆ E i , we apply this fact to the current T ⌊ E i and the pairs (f, π 1 ) = (1, u) and (f ′ , π ′ 1 ) = (1, u E i ), i ∈ N. This shows that Finally, by (3.6) and the property (4.6) of u E i , it holds for every i ∈ N that Combining (4.8), (4.9), and (4.10), and using the pairwise disjointedness of the sets Letting ǫ → 0 and δ → 0 in (4.7) completes the proof of Lemma 4.1.
We next apply Lemma 4.1 to deduce the validity of a weak 1-Poincaré inequality from the existence of GPCs.
Proof of "if" implication in Theorem 1.4. By a result of Keith [8], see also Theorem 8.4.2 in [7], it suffices to verify the Poincaré inequality for a priori Lipschitz continuous functions u and for the pointwise Lipschitz constant ρ = Lip(u, ·) instead of arbitrary upper gradients. So, let u : X → R with Lip(u) < ∞.
Recalling that we will first check that for distinct points s, t ∈ X. Start by fixing such points s, t, let T be a GPC joining s to t, and recall that spt T ⊂ B(s, C 0 d(s, t)). Then, . This proves (4.11), which is (almost) a well-known sufficient condition for the weak 1-Poincaré inequality. The only technicality here is that we only know (4.11) for the particular upper gradient Lip(u, ·). To complete the proof, we will now briefly argue that that this suffices to imply the weak 1-Poincaré inequality in full generality. Indeed, [6, Theorem 9.5] lists several conditions that imply weak Poincaré inequalities in doubling spaces (see also the references in [6]). Our estimate (4.11) shows that condition (2) in [6, Theorem 9.5] holds for p = 1, µ, u Lipschitz, the particular upper gradient ρ = Lip(u, ·), C 2 = C 3 = C 0 . It then follows from the proof in [6] that also condition (3) in the theorem holds for the same pair (u, ρ) (by this, we mean that to obtain condition (3) for u and ρ, one only needs to have condition (2) for u and ρ, and no other upper gradients). We rephrase condition (3) in a slightly peculiar manner for future application: there exists a constant C ≥ 1 (again depending on C 0 ) such that if 2B := B(z, 2r) is any ball in X, and x, y ∈ 2B, then then (4.12) Here M R is the restricted maximal function ρ(y) dµ(y), R > 0.
Next we need to verify that inequality (4.12) implies that the very same pair (u, ρ) satisfies the weak 1-Poincaré inequality. To this end, we apply Theorem 8.1.18 in [7] (originally due to Hajłasz [4]) with h = M 4Cr ρ and Q = log 2 C µ for the doubling constant C µ of µ, to deduce that Finally, following verbatim the argument on p. 224 of [7], we conclude that The assumptions in Theorem 5.1 are superficially stronger than in the remaining implication of Theorem 1.4, so we start by briefly discussing how Theorem 1.4 reduces to the special case in Theorem 5.1.
We can thus apply Theorem 5.1 to (X, g, µ) in order to find a GPC between any pair of distinct points s, t ∈ X. Then T is also a GPC joining s to t in (X, d, µ), since the conditions (P1)-(P3) in Definition 1.3 are obviously invariant under bi-Lipschitz changes of metric. The least obvious is (P3), where one needs to recall that µ is a doubling measure, whence Θ d (s, d(s, y)) ∼ Θ g (s, g(s, y)) and Θ d (t, d(t, y)) ∼ Θ g (t, g(t, y)) for all y ∈ X.
As in the assumptions of Theorem 5.1, we now suppose that (X, d, µ) is a complete geodesic doubling metric measure space supporting the Poincaré inequality (1.1) with p = 1 and λ = 1.

Proof of Theorem 5.1.
Fix two points s, t ∈ X, and write B 0 := B(s, C 0 d(s, t)) for the ball inside (the closure of) which we should find the current T , as in (ii); the constant C 0 ≥ 1 will be specified later, and its size only depends on the data of (X, d, µ), such as the doubling constant of µ, and the constant in the Poincaré inequality. We find the current T by initially constructing a sequence of approximating currents, each of them a sum of finitely many currents of the form discussed in Example 3.8. We start by defining a sequence of covers of X by balls. For n ∈ N , write r n := 2 −n , and let X n ⊂ B 0 be an r n -net, that is, some maximal family of points X n ⊂ B 0 satisfying d(x, x ′ ) ≥ r n for all distinct x, x ′ ∈ X n . We assume that r n is far smaller than d(s, t), and we require that We note that the collection of open balls B n := {B(x, 2r n ) : x ∈ X n } is now a cover of B 0 . In fact, already the balls B(x, r n ) would be a cover of B 0 : by the maximality of X n , for every y ∈ B 0 there exists x ∈ X n such that y ∈ B(x, r n ).

(5.3)
Moreover, every ball B ∈ B n only has boundedly many "neighbours": This follows by using the doubling property of µ and the consequential relative lower volume decay (see [7,Lemma 8.1.13]), and noting that the balls B(x, r n /2), x ∈ X n , are disjoint.
We will now construct a current T n supported in Ω n = B(s, C 0 d(s, t) + Cr n ) for suitable constants C 0 , C ≥ 1 (depending on the constants in the 1-Poincaré inequality). The current T n will be constructed using the max flow min cut theorem from graph theory, and a subsequence of the currents T n will eventually be shown, using the compactness theorem for normal currents, to converge to the desired current T supported onB 0 .

Graphs and flows.
To apply the max flow min cut theorem, we need to define a graph G n = (V n , E n ) associated to our problem. We set V n := X n , and Note that (x, x ′ ) ∈ E n if and only if (x ′ , x) ∈ E n , and that the maximum degree of any vertex is uniformly bounded by (5.4). We also define a capacity function c n : E n → Q + satisfying x ∈ X, r > 0.
We do not specify the values of c n (x, x ′ ) more precisely: we will only use that c n (x, y) is a rational number within a constant multiple of the right hand side of (5.5). Note that c n (x, x ′ ) ∼ c n (x ′ , x) for all (x, x ′ ) ∈ E n ; in fact, we may as well define c n (x, x ′ ) = c n (x ′ , x). A flow in G n is a function f : E n → R satisfying the following three conditions: Here, and in the sequel, we write where E n (U , W) = {(x, x ′ ) ∈ E n : x ∈ U and x ′ ∈ W}, and analogously The norm of a flow f : E n → R is defined to be the quantity A cut is any pair (S, S c ), where S ⊂ V n is a set with s ∈ S and t ∈ S c . As usual, S c denotes the complement of S (in V n ). The "total flow" of f over any cut (S, S c ) equals f : where the min runs over all cuts (S, S c ). In other words, the norm of any flow is bounded from above by the capacity of any cut in the graph. A well-known theorem in graph theory due to Ford and Fulkerson [3] states that if c n is integer-valued, then (5.7) is sharp: there exists a flow f with f = min c n (S, S c ). We learned the theorem from Diestel's graph theory book, see [2, Theorem 6.2.2]. Our capacity c n is not integer valued, but since c n (x, x ′ ) ∈ Q + , and the cardinality of E n is finite, we may assume that c n (x, x ′ ) ∈ N by initially multiplying all quantities by a suitable integer.
The reader should view flows in G n as discrete models for the current T n : we will make the connection rigorous in Section 5.4. For now, we wish to find a uniform lower bound for the numbers c n (S, S c ), where (S, S c ) is an arbitrary cut. We claim that c n (S, S c ) 1. (5.8) To this end, fix a cut (S, S c ), and recall that s ∈ S and t ∈ S c by definition. Also by definition, .
To be precise, (5.9) only holds if none of the edges in E n (S, S c ) start or end in {s, t}. We may assume this, since, for example, if (s, x ′ ) ∈ E n (S, S c ), then c n (S, S c ) ≥ c n (s, x ′ ) 1, and (5.8) follows. In fact, the same argument holds a little more generally: if (x, x ′ ) ∈ E n (S, S c ) satisfies dist(x, {s, t}) r n or dist(x ′ , {s, t}) r n , then again c n (S, S c ) 1, using the assumption that µ is doubling. So, without loss of generality, we assume that min{dist(x, {s, t}), dist(x ′ , {s, t})} ≥ Cr n , (x, x ′ ) ∈ E n (S, S c ). (5.10) where C ≥ 1 is a suitable large constant to be specified later. If C ≥ 20, say, then (5.10) has the following consequence: for all y ∈ B(x, 5r n ) ∪ B(x ′ , 5r n ), and all pairs (x, x ′ ) ∈ E n (S, S c ), since d(x, x ′ ) ≤ 4r n for such pairs. Note that (5.11), combined with the doubling of µ, also allows us to replace the denominators in (5.10) by some comparable quantities, as indicated by (5.11), for example (5.12) Evidently, the proof of (5.8) should somehow use our only assumption: the Poincaré inequality (1.1) with p = 1. To this end, we define a Lipschitz function u = u n : B 0 → R associated to the cut (S, S c ), using a Lipschitz partition of unity on B 0 , subordinate to the cover B n . For x ∈ V n , let Then 13) and ψ x and is (C/r n )-Lipschitz on B 0 : this is easy to check, noting that Evidently, u takes values in [0, 1], and is L n -Lipschitz on B 0 for some L n ∼ 1/r n . For y ∈ X and any subset U ⊂ V n , write U (y) := {x ∈ U : y ∈ B(x, 2r n )} = {x ∈ U : φ x (y) > 0} Clearly S(y), S c (y) ⊂ V n (y) and V n (y) = S(y) ∪ S c (y) for all y ∈ X.
Proof. If u(y) = 1, then . Hence which forces S(y) = V n (y). The converse implication is clear. If u(y) = 0, then ψ x (y) = 0 for all x ∈ S, so S(y) = ∅. Consequently, V n (y) ⊂ S c , as claimed. The converse implication is again clear. Proof. This is, in fact, a corollary of Lemma 5.14 and (5.10). Start with s: if u(s) < 1, then φ x (s) > 0 for some x ∈ S c , hence s ∈ B(x, 2r n ) by (5.13). On the other hand, s ∈ S (by the very definition of a cut), so the fact that s ∈ spt ψ x ⊆ B(x, 2r n ) implies (s, x) ∈ E n (S, S c ). This contradicts (5.10) as soon as C ≥ 2, and hence we deduce that u(s) = 1.
The treatment of t is essentially symmetric: if u(t) > 0, then φ x (t) > 0 for some x ∈ S. But since t ∈ S c , this implies that (x, t) ∈ E n (S, S c ), again violating (5.10).
Next, still using Lemma 5.14, we investigate where Lip(u, y) = 0. where Proof. Pick y ∈ B 0 with Lip(u, y) = 0. We claim that this has the following consequence: Assume to the contrary that there is some x ∈ V n (y) with N (x) ⊂ S or N (x) ⊂ S c : we start with the case N (x) ⊂ S. Pick z ∈ B 0 ∩ B(x, 2r n ) arbitrarily, and consider any x ′ ∈ V n (z). Then z ∈ B(x ′ , 2r n ) by definition, so This shows that V n (z) ⊂ S, hence u(z) = 1 by Lemma 5.14. But z ∈ B 0 ∩ B(x, 2r n ) was arbitrary, so we have inferred that u ≡ 1 on the neighbourhood B 0 ∩ B(x, 2r n ) of y. In particular Lip(u, y) = 0, a contradiction.
Next, consider the case N (x) ⊂ S c . As before, pick z ∈ B 0 ∩ B(x, 2r n ) arbitrarily, and deduce as above that x ′ ∈ S c for all x ′ ∈ V n (z). This implies by Lemma 5.14 that u(z) = 0, and hence u ≡ 0 on B 0 ∩ B(x, 2r n ). This contradicts Lip(u, y) = 0. Now that we have proven the claim in italics, we finish the proof of the lemma. Fix y ∈ B 0 with Lip(u, y) = 0, pick any x ∈ V n (y), and assume first that x ∈ S. Then there exists x ′ ∈ S c with (x, x ′ ) ∈ E n . Hence x ∈ Bd(S) by definition, and y ∈ B(x, 2r n ) ⊂ B(x, 5r n ), as claimed. Next, if x ∈ S c , then we have shown that there exists x ′ ∈ S with (x ′ , x) ∈ E n . This means that x ′ ∈ Bd(S). Since B(x, 2r n ) ∩ B(x ′ , 2r n ) = ∅, we infer that y ∈ B(x ′ , 5r n ), and the proof is complete.
We extend u to an L n -Lipschitz map X → R without change in the notation. Then, we note that (u, Lip(u, ·)) is a function -upper gradient pair on X, and we apply Theorem 9.5 in [6], more precisely the implication "(4) =⇒ (2)", which requires the space (X, d) to be geodesic. This implication gives the following estimate: d(s, y)) + Lip(u, y) Θ(t, d(t, y)) dµ(y), assuming that s, t are "deep enough inside" the ball B 0 . This can be arranged by choosing C 0 ≥ 1 in the definition of B 0 large enough. Then, if (by slight abuse of notation) we denote by Bd(S) the set on the right hand side of (5.17), we obtain further d(s, y)) + 1 Θ(t, d(t, y)) dµ(y).
By multiplying the c n by a constant (rational) factor, we may now arrange c n (S, S c ) ≥ 1 for all cuts (S, S c ) with s ∈ S and t ∈ S c . Then, we are in a position to apply the max flow min cut theorem: there exists a flow f n : E n → R such that f n ≥ 1. Moreover, recalling (5.7), the norm of the flow f n is bounded from above by the capacity of the cut ({s}, V n \ {s}). Since there are only boundedly many edges in E n of the form (s, x), x ∈ V n , and the capacity of each one of them is c n (s, x) ∼ 1, we get f n ∼ 1. where I e is the current discussed in Example 3.8. First, we compute the boundary of T n , based on the facts that the boundary operation is linear, and we already know (recall (3.10)) the boundary of each term γ e♯ I e : ∂T n = e∈En f n (e)∂(γ e♯ I e ) = (x,y)∈En (5.20) To simplify the expression further, we use the flow property (F2) of f n , which says that Consequently, all that remains in (5.20) are the terms containing s and t: In the last equation, we again used f n (t, y) = −f n (y, t), and the little proposition stated in (5.6) that f n = f (S, S c ) for any cut (S, S c ), in particular for (S, S c ) = (V n \ {t}, {t}).
Recalling (5.19), this yields that which in particular verifies that T n is a normal current. Next, we estimate the measures T n and find a uniform upper bound for T n (X). We recall from (3.9) that γ ♯ I e = H 1 ⌊ |γe| . It follows that T n ≤ e∈En |f n (e)|H 1 ⌊ γe(Ie) ≤ e∈En c n (e)H 1 ⌊ γe(Ie) . (5.23) Recalling that |f n (e)| ≤ c n (e) (using the flow property (F3) and the fact that c n (x, x ′ ) = c n (x ′ , x) for all (x, x ′ ) ∈ E n ), we may now easily estimate T n (B) from above for all balls B = B(x, r) ⊂ X with r ≥ r n . We claim that T n (B) ˆ1 0B 1 Θ(s, d(s, y)) + 1 Θ(t, d(t, y)) dµ(y), B = B(x, r) ⊂ X, r ≥ r n . (5.24) We start by disposing of a little technicality. Note that there are only boundedly many edges in E n of the form (x, y) where min{d(x, s), d(y, s)} ≤ 5r n . We denote these edges by E n (s), and we use the trivial estimate c n (e) 1 for all edges e ∈ E n (s). If B happens to intersect γ e (I e ) for one of the edges e ∈ E n (s), we estimate as follows: But if B intersects γ e (I e ) for an edge e ∈ E n (s), then 10B contains B(s, r n ), and the second term above is bounded from above by the right hand side of (5.24): e∈En(s) r n ˆB (s,rn) dµ(y) Θ(s, d(s, y)) ≤ˆ1 0B 1 Θ(s, d(s, y)) + 1 Θ(y, d(t, y)) dµ(y), using in the first inequality that d(s, y) ∼ r n for y ∈ B(s, r n ) \ B(s, r n /2) =: A(s, r n ), and µ(A(s, r n )) ∼ µ(B(s, r n )) by the doubling hypothesis (this also requires A(s, r n ) = ∅, which easily follows from the path connectedness of (X, d); see also [7, (8.1.17)]). We may dispose similarly of the situation where B meets γ e (I e ) for some e ∈ E n (t) (defined in the same way as E n (s)). In other words, it remains to estimate the sum e∈En\(En(s)∪En(t)) c n (e)H 1 (γ e (I e ) ∩ B) (x,y)∈En(B) µ(B(x, r n )) Θ(s, d(s, x)) + µ(B(x, r n )) Θ(t, d(t, x)) . (5.25) where E n (B) := {e ∈ E n \ [E n (s) ∪ E n (t)] : γ e (I e ) ∩ B = ∅}. We used in (5.25) that d(t, y) ∼ d(t, x) and µ(B(y, r n )) ∼ µ(B(x, r n )), (x, y) ∈ E n (B).
To proceed, we recall again that every x ∈ V n only has boundedly many neighbours in G n , and all the vertices x ∈ V n with at least one edge (x, y) ∈ E n (B) must lie at distance ≤ r n from B, and at distance ≥ 5r n from {s, t}. We denote the collection of such vertices by V n (B). The observations above, and the bounded overlap of the balls B(x, r n ), x ∈ V n , allow us to continue (5.25) as follows: µ(B(x, r n )) Θ(s, d(s, x)) + µ(B(x, r n )) Θ(t, d(t, x)) ˆ1 0B 1 Θ(s, d(s, y)) + 1 Θ(t, d(t, y)) dµ(y).
Here we used, once again, that d(s, y) ∼ d(s, x) and d(t, y) ∼ d(t, x) for all y ∈ B(x, r n ), whenever dist(x, {s, t}) ≥ 5r n . This concludes the proof of (5.24). Finally, since the sets γ e (I e ), e ∈ E n , are geodesics connecting vertices in V n , we infer that spt T n = spt T n ⊂ B(s, C 0 d(s, t) + 5r n ) ⊂ 2B 0 .