# From curves to currents

Dídac Martínez-Granado  Department of Mathematics
University of California, Davis
One Shields Ave, Davis, California 95616
USA<EMAIL_ADDRESS>and  Dylan P. Thurston  Department of
Mathematics
Indiana University, Bloomington, Indiana 47405
USA<EMAIL_ADDRESS>
###### Abstract.

Many natural real-valued functions of closed curves are known to extend
continuously to the larger space of geodesic currents. For instance, the
extension of length with respect to a fixed hyperbolic metric was a motivating
example for the development of geodesic currents. We give a simple criterion
on a curve function that guarantees a continuous extension to geodesic
currents. The main condition of our criterion is the smoothing property, which
has played a role in the study of systoles of translation lengths for Anosov
representations. It is easy to see that our criterion is satisfied for almost
all the known examples of continuous functions on geodesic currents, such as
non-positively curved lengths or stable lengths for surface groups, while also
applying to new examples like extremal length. We use this extension to obtain
a new curve counting result for extremal length.

###### Contents

  1. 1 Introduction
  2. 2 Background on curves and currents
  3. 3 Convexity and continuity
  4. 4 Examples
  5. 5 Counting problems
  6. 6 Proof outline
  7. 7 Defining the extension
  8. 8 Constructing global cross-sections
  9. 9 Join lemma
  10. 10 Continuity of the extension
  11. 11 The extension extends
  12. 12 Hyperbolic geometry estimates
  13. 13 Stable functions

##  1\. Introduction

Geodesic currents on surfaces are measures that realize a suitable closure of
the space of weighted (multi-)curves on a surface. They were first introduced
by Bonahon in his seminal paper [Bon86]. Many metric structures can be
embedded in the space of currents, such as hyperbolic metrics [Bon88, Theorem
12] or half-translation structures [DLR10, Theorem 4]. Thus, geodesic currents
allow one to treat curves and metric structures on surfaces as the same type
of object. Via this unifying framework, counting curves of a given topological
type and counting lattice points in the space of deformations of geometric
structures become the same problem [RS19, Main Theorem]. Geodesic currents
also play a key step in the proof of rigidity of the marked length spectrum
for metrics, via an argument by Otal [Ota90, Théorème 2]. Finally, they
provide a boundary of the Teichmüller space, in both the compact [Bon88,
Proposition 17] and non-compact [BŠ21, Theorem 2] cases.

In this paper we consider the problem of extending continuously a function
defined on the space of weighted multi-curves to its closure, the space of
geodesic currents.

Previous work of Bonahon extended the notion of geometric intersection number
as a continuous function of two geodesic currents [Bon86, Proposition 4.5].
This allowed him to extend hyperbolic length to geodesic currents by following
the principle of realizing it as an intersection number with a distinguished
geodesic current [Bon88, Proposition 14].

The same principle using intersection numbers has been used by many authors to
extend length for many other metrics: Otal for negatively curved Riemannian
metrics [Ota90, Proposition 3], Croke-Fathi-Feldman for non-positively curved
Riemannian metrics [CFF92, Theorem A], Hersonsky-Paulin for negatively curved
metrics with conical singularities [HP97, Theorem A], Bankovic-Leininger for
non-positively curved Euclidean cone metrics [BL18], Duchin-Leininger-Rafi
more explicitly for singular Euclidean structures associated to quadratic
differentials [DLR10, Lemma 9], and Erlandsson for word-length with respect to
simple generating sets of the fundamental group [Erl19, Theorem 1.2].

Another line of results on extending functions to geodesic currents was also
started by Bonahon, who showed how to extend stable lengths to geodesic
currents, not just for surface groups but for general hyperbolic groups
[Bon91, Proposition 10]. This result was recently improved by Erlandsson-
Parlier-Souto [EPS20, Theorem 1.5], who used the return map of the geodesic
flow to remove technical assumptions. These constructions apply, for instance,
to arbitrary Riemannian metrics and the stable version of word-lengths for
arbitrary generating sets.

The problem of extending functions to geodesic currents is interesting in
itself, since, by a result of Rafi and Souto reviewed in Section 5, it
provides a way to compute asymptotics of the number of curves of a fixed type
with a bounded “length”, for a notion of “length” that extends to currents
[RS19]. Their result builds on work by Mirzakhani [Mir16, Theorem 7.1] and
Erlandsson-Souto [ES16, Proposition 4.1]. Recently, Erlandsson and Souto have
given a new argument to compute these asymptotics [ES20, Theorem 8.1].

Our main theorem gives a simple criterion on functions defined on multi-curves
that guarantees they extend to geodesic currents. Our result subsumes most of
the previous extension results mentioned above, and provides new extensions
for other notions of “length”, such as extremal length, thus yielding counting
asymptotics for them.

Our proof does not use Bonahon’s principle on intersection numbers. Although
we drew some inspiration from the dynamics of Erlandsson-Parlier-Souto
[EPS20], our techniques are distinct.

###  1.1. Main results

Notation | Meaning
---|---
$S$ | topological surface
$\Sigma$ | Riemannian surface
$UT\Sigma$ | unit tangent bundle over $\Sigma$
$\phi_{t}$ | geodesic flow on $UT\Sigma$
$\tau$ | cross-section to the geodesic flow
$\psi$ | bump function on a cross-section
$\mathcal{S}(S)$ | unoriented simple multi-curves
$\mathcal{C}(S)$ | unoriented multi-curves
$\mathcal{S}^{+}(S)$ | oriented simple multi-curves
$\mathcal{C}^{+}(S)$ | oriented multi-curves
$G(S)$ | unoriented geodesics on $\tilde{S}$
$G^{+}(S)$ | oriented geodesics on $\tilde{S}$
$\mathbb{R}\mathcal{S}(S)$ | weighted unoriented simple multi-curves
$\mathbb{R}\mathcal{C}(S)$ | weighted unoriented multi-curves
$\mathbb{R}\mathcal{S}^{+}(S)$ | weighted oriented simple multi-curves
$\mathbb{R}\mathcal{C}^{+}(S)$ | weighted oriented multi-curves
$\mathcal{GC}(S)$ | unoriented geodesic currents
$\mathcal{GC}^{+}(S)$ | oriented geodesic currents
$\gamma$ | concrete multi-curve on $S$
$C$ | multi-curve on $S$
Table 1. Notation for the objects related to surfaces, curves, and geodesic
currents.

We start by summarizing our main results. Complete definitions of the terms
are deferred to Section 2.

######  Definition 1.1.

For $S$ a compact topological surface without boundary, let
$f\colon\mathcal{C}^{+}(S)\to\mathbb{R}$ be a function defined on the space of
oriented multi-curves, not-necessarily-simple oriented curves; see Definition
2.1, and see Table 1 for a summary of notation. We will also refer to $f$ as a
_curve functional_ for short. (_Functional_ means that it takes values in
scalars; it is not assumed to be linear.) We will also refer to _unoriented_
or _weighted_ curve functionals for real-valued functions defined on the
appropriate type of multi-curves. We define several properties that $f$ might
satisfy.

  * •

Quasi-smoothing: There is a constant $R\geq 0$ with the following property.
Let $C$ be an oriented curve on $S$, and let $x$ be an essential crossing of
$C$. Let $C^{\prime}$ be the oriented smoothing of $C$ at $x$. Then $f(C)\geq
f(C^{\prime})-R$. Schematically, we have

(1.2) $f\left(\vbox{\hbox{$\includegraphics{curves-3}$}}\right)\geq
f\left(\vbox{\hbox{$\includegraphics{curves-4}$}}\right)-R$

See Definition 2.6 for “essential crossing”. Loosely, it is a crossing that
cannot be removed by homotopy. See Definition 2.7 for “oriented smoothing”.

  * •

Smoothing: We take $R=0$ in the above definition of quasi-smoothing:

(1.3) $f\left(\vbox{\hbox{$\includegraphics{curves-3}$}}\right)\geq
f\left(\vbox{\hbox{$\includegraphics{curves-4}$}}\right)$

  * •

Convex union: Let $C_{1}$ and $C_{2}$ be two oriented curves on $S$. Then

(1.4) $f(C_{1}\cup C_{2})\leq f(C_{1})+f(C_{2}).$

  * •

Additive union: The inequality in convex union becomes an equality:

(1.5) $f(C_{1}\cup C_{2})=f(C_{1})+f(C_{2}).$

Many natural curve functionals satisfy the additive union property; for
instance, length with respect to an arbitrary length metric on $S$ satisfies
it by definition. The square root of extremal length is an example of a curve
functional satisfying convex union, but not additive union (Section 4.8). The
name “convex union” comes from the fact that, if we extend to weighted curves
and additionally assume homogeneity (Definition 1.7), then for a fixed
oriented multi-curve with varying weights, $f$ is a convex as a function of
the weights (Proposition 3.4).

There are many curve functionals satisfying the smoothing property, such as
hyperbolic length, extremal length, intersection number with a fixed curve, or
length from a length metric on $S$. For an example of a natural curve
functional that satisfies quasi-smoothing but not smoothing, we have the word-
length of with respect to an arbitrary generating set of $\pi_{1}(S)$ (Example
4.10). The (quasi-)smoothing property is usually easy to check.

The smoothing property plays an important role in the study of translation
lengths associated to Anosov representations, as we discuss in Section 4.7,
following [MZ19] and [BIPP19]. These works use the smoothing property to
reduce the study of length systoles to the case of simple closed curves.
Although these papers point out the parallelism between the smoothing property
of Anosov translation lengths and that of hyperbolic length ([BIPP19, Section
4] or negatively curved lengths [MZ19, Corollary 1.3], the results in our
paper reveal the much more universal nature of the smoothing property. Indeed,
we show many other natural notions of lengths which are not associated to
negatively curved structures satisfy the smoothing condition, such as lengths
on Riemannian metrics with no curvature assumption, lengths coming from more
general length space structures, extremal lengths, or word-lengths with
respect to certain generating sets; see Section 4.

######  Definition 1.6.

For $C$ an oriented multi-curve, $nC$ is the oriented multi-curve that
consists of $n$ parallel copies of $C$ (so with $n$ times as many components
or, in the context of weighted oriented multi-curves, with weights multiplied
by $n$), and $C^{n}$ is the oriented multi-curve with as many components as
$C$, in which each component of $C$ is covered by an $n$-fold cover. That is,
if $g\in\pi_{1}(S,x)$ represents $C$, $g^{n}$ represents $C^{n}$.

######  Definition 1.7.

Let $f$ be a curve functional and let $n>0$ be an integer. We define some
properties $f$ might satisfy:

  * •

Homogeneity: For an arbitrary oriented multi-curve $C$,

(1.8) $f(nC)=nf(C).$

  * •

(Weak) stability: For an arbitrary oriented multi-curve $C$,

(1.9) $f(C^{n})=f(nC)$

  * •

Strong stability: For arbitrary oriented multi-curves $C,D$,

(1.10) $f(D\cup C^{n})=f(D\cup nC)$

Additive union implies homogeneity, and if $f$ satisfies convex union,
$f(nC)\leq nf(C)$. If $f$ satisfies quasi-smoothing, then $f(nC)-nR\leq
f(C^{n})$, since the self-crossings in $C^{n}$ are essential crossings by
definition (see Definition 2.6).

We furthermore note that curve functionals are not necessarily positive.

With this background, we can state our main theorems on extensions of curve
functionals to the space $\mathcal{GC}^{+}(S)$ of oriented geodesic currents.

######  Theorem A.

Let $f$ be a curve functional satisfying the quasi-smoothing, convex union,
stability, and homogeneity properties. Then there is a unique continuous
homogeneous function $\bar{f}\colon\mathcal{GC}^{+}(S)\to\mathbb{R}_{\geq 0}$
that extends $f$.

In the case of unoriented curves, there are two possible smoothings of an
essential crossing, not distinguished from each other. Then we have the
following version of the theorem, deduced from Theorem A in Section 2.5.

######  Corollary 1.11.

Let $f$ be an unoriented curve functional satisfying quasi-smoothing for both
possible smoothings of a crossing, in addition to the convex union, stability,
and homogeneity properties. Then there is a unique continuous homogeneous
function $\bar{f}\colon\mathcal{GC}(S)\to\mathbb{R}$ that extends $f$.

Theorem A should be thought of as an analogue of the classical theorem that a
convex function defined on the rational points in a finite-dimensional vector
space automatically extends continuously to a convex function defined on the
whole vector space (Proposition 3.1(iv)). As in the classical case, the
functions on geodesic currents arising from this construction are restricted,
as the next example shows. (This example is almost the only function we are
aware of where our techniques do not suffice to prove continuity of the
extension. See Section 4.1)

######  Example 1.12.

Consider the curve curves given by the square-root of self-intersection
number, i.e., $f(C)\coloneqq\sqrt{i(C,C)}$. Since intersection number is a
continuous two-variable function [Bon86, Proposition 4.5], it follows that $f$
extends continuously to geodesic currents. However, $f$ does not satisfy
convex union. For instance, take $C_{1}$ and $C_{2}$ be two simple closed
curves intersecting once. For any multi-curve, $i(C,C)$ is twice the self-
intersection number of $C$. Thus $f(C_{1}\cup C_{2})=\sqrt{2}$, but
$f(C_{1})+f(C_{2})=0$, contradicting convex union. On the other hand, $f$
clearly satisfies smoothing.

On the other hand, the stability and homogeneous properties are necessary
conditions for an extension to exist for elementary reasons, as the multi-
curves $nC$ and $C^{n}$ should represent the same currents (Example 4.10).
However, a curve functional that satisfies quasi-smoothing and convex union
can be modified to get a curve functional satisfying all the hypotheses of
Theorem A.

######  Theorem B.

Let $f$ be a curve functional satisfying quasi-smoothing and convex union.
Then the stabilized curve functional

$\|f\|(C)\coloneqq\lim_{n\to\infty}\frac{f(C^{n})}{n}.$

satisfies quasi-smoothing, convex union, strong stability, and homogeneity,
and thus extends to a continuous function on $\mathcal{GC}^{+}(S)$.

Theorem B is proved in Section 13, although the implication that weak
stability implies strong stability is used in the proof of Theorem A.

If the convex union property of $f$ is strengthened to additive union and the
quasi-smoothing property is strengthened to smoothing, then in fact this
extension to geodesic currents comes from intersection with a fixed current
(as in the proofs of extension that used Bonahon’s principle). This will
appear in a forthcoming paper. For this stronger result, the strict smoothing
property is necessary, since intersection number cannot increase after
smoothing an essential crossing.

###  1.2. Acknowledgments

We thank Francisco Arana, Martin Bridgeman, Maxime Fortier Bourque, Kasra
Rafi, and Tengren Zhang for helpful conversations, and the anonymous referee
for careful reading and suggestions. The first author was supported by the
Mathematics Department Indiana University Bloomington, via the Hazel King
Thompson fellowship and the Indiana University Graduate School under the
Dissertation Research Fellowship. The second author was supported by the
National Science Foundation under Grant Numbers DMS-1507244 and DMS-2110143.

##  2\. Background on curves and currents

Throughout this paper, $S$ is a fixed oriented compact 2-manifold without
boundary. (For a discussion of the more general surface case, see Remark
2.26.) If we fix an arbitrary (hyperbolic) metric on $S$, we will denote it by
$\Sigma$. The various types of curves and associated objects we consider are
summarized in Table 1.

###  2.1. Curves

######  Definition 2.1 (multi-curve).

A _concrete multi-curve_ $\gamma$ on a surface $S$ is a smooth 1-manifold
without boundary $X(\gamma)$ together with a map (also called $\gamma$) from
$X(\gamma)$ into $S$. $X(\gamma)$ is not necessarily connected. We say that
$\gamma$ is _trivial_ if it is homotopic to a point. Two concrete multi-curves
$\gamma$ and $\gamma^{\prime}$ are _equivalent_ if they are related by a
sequence of the following moves:

  * •

_homotopy_ within the space of all maps from $X(\gamma)$ to $S$;

  * •

_reparametrization_ of the 1-manifold; and

  * •

_dropping_ trivial components.

The equivalence class of $\gamma$ is denoted by $[\gamma]$, and we will call
it a _multi-curve_. If $X(\gamma)$ is connected, we will call $[\gamma]$ a
_curve_ ; a curve is equivalent to a conjugacy class in $\pi_{1}(S)$. When we
just want to refer to the equivalence class of a (multi-)curve, without
distinguishing a representative, we will use capital letters such as $C$. A
concrete multi-curve $\gamma$ is _simple_ if $\gamma$ is injective, and a
multi-curve is simple if it has a concrete representative that is simple. We
write $\mathcal{S}(S)$ for the space of simple multi-curves on $S$ and
$\mathcal{C}(S)$ for the space of all multi-curves.

We also consider _oriented multi-curves_ , which we will still denote by
$\gamma$, in which $X(\gamma)$ is oriented. We add the further condition in
the equivalence relation that the reparametrizations must be orientation-
preserving. In this paper, unless stated otherwise, we will be working with
oriented multi-curves. The spaces of oriented simple and general multi-curves
are denoted $\mathcal{S}^{+}(S)$ and $\mathcal{C}^{+}(S)$, respectively.

######  Definition 2.2 (weighted multi-curve).

A _weighted multi-curve_ $C=\bigcup_{i}a_{i}C_{i}$ is a multi-curve in which
each connected component is given a non-negative real coefficient $a_{i}$. If
coefficients are not specified, they are $1$. We add further moves to the
equivalence relation:

  * •

_merging_ two parallel components and adding their weights; and

  * •

_nullifying_ , deleting a component with weight $0$.

For instance, $C\cup C$ is equivalent to $2C$. The space of weighted multi-
curves up to equivalence is denoted by appending an $\mathbb{R}$ in front of
their non-weighted names, so $\mathbb{R}\mathcal{S}(S)$ is the space of
weighted simple multi-curves and $\mathbb{R}\mathcal{C}(S)$ is the space of
weighted general multi-curves. This is a slight abuse of notation since the
weights are required to be non-negative.

######  Remark 2.3.

Since the weighted curve functionals we are considering are not necessarily
positive, they may increase after dropping a component (see Definition 2.2).

###  2.2. Crossings

Loosely speaking, an essential crossing is a crossing of a multi-curve that
can’t be homotoped away. We make this definition precise as follows. We cover
cases where $\gamma$ does not have transverse crossings for convenience of
some of the examples.

######  Definition 2.4 (linked points on a circle).

We say that two sets of two points $\\{a,b\\}$ and $\\{c,d\\}$ in $S^{1}$ are
_linked_ if the four points are distinct and both connected components of
$S^{1}-\\{a,b\\}$ have an element of $\\{c,d\\}$.

######  Definition 2.5 (lift of a concrete curve).

Given a concrete multi-curve $\gamma$ on $S$ and a choice $p\in X(\gamma)$,
set $x=\gamma(p)\in S$. Pick a lift $\widetilde{x}\in\widetilde{S}$ of $x$.
The unique lifting property gives a unique lift
$\widetilde{\gamma_{p}}:\widetilde{X}(\gamma;p)\to\widetilde{S}$ of $\gamma$
with $\widetilde{\gamma_{p}}(\widetilde{p})=\widetilde{x}$, where
$X(\gamma;p)$ is the component of $X(\gamma)$ containing $p$ and
$\widetilde{X}(\gamma;p)$ is its universal cover with basepoint
$\widetilde{p}$.

######  Definition 2.6 (essential crossing).

Let $\gamma$ be a concrete multi-curve on $S$, and suppose we have points
$p,q\in X(\gamma)$ so that $x\coloneqq\gamma(p)=\gamma(q)\in S$. Pick a lift
$\widetilde{x}\in\widetilde{S}$ of $x$, and let $\widetilde{\gamma_{p}}$ and
$\widetilde{\gamma_{q}}$ be the corresponding lifts of components of $\gamma$
following Definition 2.5. Then the pair $(p,q)$ form an _essential crossing_
if the following two conditions hold:

  1. (1)

both components of $X(\gamma)$ containing $p$ and $q$ are not null-homotopic,
so that $\widetilde{\gamma_{p}}$ and $\widetilde{\gamma_{q}}$ are quasi-
geodesic components of $\widetilde{\gamma}$; and

  2. (2)

either

    1. (a)

the endpoints $\\{a,b\\}$ of $\widetilde{\gamma_{p}}$ and the endpoints
$\\{c,d\\}$ of $\widetilde{\gamma_{q}}$ are linked in $S^{1}_{\infty}$; or

    2. (b)

$p$ and $q$ lie on the same component of $X(\gamma)$, $[\gamma]=[\delta^{n}]$
for some $n>1$ and some primitive $\delta\in\pi_{1}(S,x)$, the loop from $p$
to $q$ in $X(\gamma)$ maps to $[\delta^{m}]$ for some $0<m<n$, and so the loop
from $q$ to $p$ maps to $[\delta^{n-m}]$.

In case (2)b, the endpoints of $\widetilde{\gamma_{p}}$ and
$\widetilde{\gamma_{q}}$ are the same.

This definition might be somewhat looser than expected. For instance, in the
chain of three crossings

the circled middle crossing is essential iff the other two are, since “linking
at infinity” doesn’t see the direction of crossing. This does not matter for
our purposes.

######  Definition 2.7 (smoothings).

Let $(p,q)\in X(\gamma)$ be an essential crossing of $\gamma$ on $S$. To make
a _smoothing_ $\gamma^{\prime}$ of $(p,q)$, cut $X(\gamma)$ at $p$ and $q$ and
reglue the resulting four endpoints in one of the two other possible ways,
getting a new $1$-manifold $X(\gamma^{\prime})$. The map $\gamma^{\prime}$
agrees with $\gamma$; this is well-defined since $\gamma(p)=\gamma(q)$. In
pictures we will homotop $\gamma^{\prime}$ slightly to round out the resulting
corners. If $\gamma$ is oriented, then the _oriented smoothing_ is the
smoothing that respects the orientation on $X(\gamma)$:

$\vbox{\hbox{$\includegraphics{curves-3}$}}\mathrel{\searrow}\vbox{\hbox{$\includegraphics{curves-4}$}}$

If we obtain a concrete curve $\gamma^{\prime}$ from $\gamma$ by a sequence of
$k$ smoothings of essential crossings, we will write
$\gamma\mathrel{\searrow}_{k}\gamma^{\prime}$. (We check whether the crossings
are essential at each stage of this process; this is more restrictive than
checking at the beginning.)

######  Lemma 2.8.

Essential crossings are unavoidable in a homotopy class, in the sense that if
$\gamma$ and $\gamma^{\prime}$ are homotopic concrete multi-curves and
$(p,q)\in X(\gamma)$ is an essential crossing of $\gamma$, then there is an
essential crossing $(p^{\prime},q^{\prime})\in X(\gamma^{\prime})$ so that the
smoothings of $(p,q)$ and of $(p^{\prime},q^{\prime})$ are homotopic.

###### Proof.

For both types of essential crossings, there is a representative
$\gamma^{\prime}$ with minimal crossing number for which the result is clear:

  1. (1)

For crossings of the first type, take the geodesic representative on $S$,
perturbed slightly to make it transverse.

  2. (2)

For crossings of the second type $[\gamma]=[\delta^{n}]$, take the geodesic
representative and perturb it slightly in a standard way in a neighborhood of
$\delta$.

In general, take the given representative $\gamma$ and perturb it slightly to
make it transverse (without introducing new types of crossings). If $\gamma$
is connected, then by a result of Hass and Scott [HS94, Theorems 1.8 and 2.1]
(see also de Graaf and Schrijver [dGS97]), $\gamma$ can be turned into any
desired minimal form $\gamma^{\prime}$ using only Reidemeister I, II, and III
moves, with the Reidemeister I and II moves being used only in the forward
(simplifying) direction. Since we know that $\gamma^{\prime}$ has a crossing
of the desired type, we can trace the crossings backwards through these moves:
a Reidemeister III move does not change the homotopy types of curves
achievable by a single smoothing, and we can ignore the additional crossings
created by backwards Reidemeister I and II moves.

For multi-curves, the papers above also prove that any diagram can be
connected to a minimal diagram by a series of forward Reidemeister moves, and
that the only obstruction to connecting two minimal diagrams for a multi-curve
is swapping the location two components $\gamma_{1},\gamma_{2}$ that are
homotopically powers of the same primitive curve $\delta$ [HS94, pp. 31–32].
But any of these minimal representatives in a neighborhood of $\delta$ contain
all essential crossings between $\gamma_{1}$ and $\gamma_{2}$ (necessarily
related to an essential self-crossing of $\delta$). ∎

######  Remark 2.9.

We can also see directly that essential crossings of the first type exist by
considering the lift to the universal cover. Lemma 2.8 is false on non-
orientable surfaces (consider the double cover of the core curve of a Möbius
strip).

We have similar notions for weighted curves.

######  Definition 2.10.

Let $C\in\mathbb{R}\mathcal{C}^{+}(S)$ be a weighted oriented curve, and let
$\gamma$ be a concrete representative of the underlying unweighted curve. Let
$(p,q)\in X(\gamma)$ be an essential crossing of $\gamma$ on $S$, and let
$\gamma^{\prime}$ be the smoothing as defined above. If the corresponding
components of $C$ have equal weight $w>0$, then we can make a weighted curve
$C^{\prime}$ by giving every component in $[\gamma^{\prime}]$ not involved in
the smoothing the same weight it had in $C$, and giving the one or two new
components weight $w$. In this case we say that $C^{\prime}$ is obtained from
$C$ by a _smoothing of weight $w$_, and write
$C\mathrel{\searrow}_{w}C^{\prime}$.

Using this, we define conditions on a weighted curve functional, extending
Definition 1.1.

  * •

Weighted quasi-smoothing: There is a constant $R>0$ so that,
$C\mathrel{\searrow}_{w}C^{\prime}$ are weighted curves related by a smoothing
of weight $w$,

$f(C)\geq f(C^{\prime})-wR.$

  * •

Weighted smoothing: Take $R=0$ in the definition of weighted quasi-smoothing.

See Proposition 3.6 for justification for these definitions.

###  2.3. Space of geodesics

######  Definition 2.11 (Boundary at infinity).

Endow $S$ with a complete hyperbolic metric $g$; we denote the pair $(S,g)$ by
$\Sigma$. Then we can consider the metric universal covering
$p\colon\widetilde{\Sigma}\to\Sigma$, with $\widetilde{\Sigma}$ isometric to
the hyperbolic plane. Two quasi-geodesic rays
$c,c^{\prime}\colon[0,\infty)\to\widetilde{\Sigma}$ are said to be
_asymptotic_ if there exists a constant $K$ for which
$d(c(t),c^{\prime}(t))\leq K$ for all $t\geq 0$. We define
$\partial_{\infty}\Sigma$, the _boundary at infinity_ of $S$, to be the set of
equivalence classes of asymptotic quasi-geodesic rays. This boundary at
infinity is independent of the hyperbolic structure on $S$ up to canonical
homeomorphism.

######  Definition 2.12 (Space of oriented geodesics).

Let $G^{+}(\Sigma)$ denote the space of _oriented geodesics_ in
$\widetilde{\Sigma}$, i.e.,

$G^{+}(\Sigma)\coloneqq\partial_{\infty}\Sigma\times\partial_{\infty}\Sigma-\Delta.$

Since this is independent of the hyperbolic structure, we will also write
$G^{+}(S)$.

###  2.4. Geodesic currents

######  Definition 2.13 (Geodesic current definition 1).

We define $\mathcal{GC}^{+}(S)$, the space of oriented geodesic currents on
$S$, to be the space of $\pi_{1}(S)$-invariant (positive) Radon measures on
$G^{+}(S)$.

Since the action of $\pi_{1}(S)$ on $G^{+}(S)$ is not discrete, this
definition is hard to visualize. We give alternate definitions that play a key
role in our proofs. For a hyperbolic surface $\Sigma$, let $UT\Sigma$ be the
unit tangent bundle and let $\phi_{t}$ be the geodesic flow on it.

######  Definition 2.14 (Geodesic current definition 2).

We can also define oriented geodesic currents to be the space of (positive)
finite Radon measures $\mu$ on $UT\Sigma$ which are invariant under the
geodesic flow, in the sense that $(\phi_{t})_{*}(\mu)=\mu$ for all
$t\in\mathbb{R}$.

We can also look at induced measures on cross-sections.

######  Definition 2.15 (Geodesic current definition 3).

A geodesic current is a _transverse invariant measure_ : a family of measures
$\\{\mu_{\tau}\\}_{\tau}$, where $\tau\subset UT\Sigma$ is a submanifold-with-
boundary of the unit tangent bundle of real codimension 1 transverse to the
geodesic foliation $\mathcal{F}$, with the following invariance property: if
$x_{1}\in\tau_{1},x_{2}\in\tau_{2}$ are two points on transversal submanifolds
on the same leaf of $\mathcal{F}$, and $\phi\colon U_{1}\to U_{2}$ a holonomy
diffeomorphism between neighborhoods of $x_{1}$ and $x_{2}$ respectively, then
$\phi_{*}\mu_{\tau_{1}}=\mu_{\tau_{2}}$.

The equivalence of the three definitions was known to Bonahon [Bon86, Chapter
4]. Details can be found in [AL17, Section 3.4]. Briefly, given a measure
$\mu$ on $UT\Sigma$ as in Definition 2.14 and a cross-section $\tau$, there is
an induced _flux_ $\mu_{\tau}$ on $\tau$, as explained in Definition 7.15;
this gives a geodesic current in the sense of Definition 2.15. Lifting to the
universal cover then gives a geodesic current in the sense of Definition 2.13.
We can also relate Definitions 2.13 and 2.14 directly by connecting both to
measures on $UT\mathbb{H}^{2}$ that are invariant under both $\pi_{1}(\Sigma)$
and the geodesic flow [BO07, Proposition 8.1].

Because Definitions 2.13 and 2.15 are invariant under the mapping class group,
we will also write $G^{+}(S)$ and $\mathcal{GC}^{+}(S)$ in the sequel. We will
also write $\pi_{1}(S)$. On the other hand, we will emphasize the dependence
of $UT\Sigma$ on the hyperbolic structure.

######  Remark 2.16.

For $\Sigma,\Sigma^{\prime}$ hyperbolic surfaces, any homeomorphism
$\psi\colon\Sigma\to\Sigma^{\prime}$, there is a homeomorphism
$\hat{\psi}\colon UT\Sigma\to UT\Sigma^{\prime}$ that is an orbit equivalence,
and it is tempting to use this to define an induced map between geodesic
currents in the sense of Definition 2.14. But this does not quite work:
$\hat{\psi}_{*}$ does not take geodesic currents to geodesic currents. Orbit
equivalence means that $\psi(\phi_{t}(x))=\phi_{f(t)}(\psi(x))$ for some
monotonic function $f\colon\mathbb{R}\to\mathbb{R}$, but this is not enough to
guarantee that $(\phi_{t})_{*}(\hat{\psi}_{*}\mu)=\hat{\psi}_{*}\mu$, and
indeed this is usually false. See [Wil14, Theorem 3.6].

###  2.5. Oriented vs unoriented currents

We will be mostly working in the setting of oriented geodesic currents, but
most of our natural examples (like measured laminations) use unoriented
currents.

######  Definition 2.17 (Unoriented geodesic currents).

To define the subspace $\mathcal{GC}(S)\subset\mathcal{GC}^{+}(S)$ of
_unoriented geodesics currents_ , let $\sigma:G^{+}(S)\to G^{+}(S)$ be the
_flip map_ that switches the two factors in the definition of $G^{+}(S)$,
reversing the orientation of the geodesic. This induces a map
$\sigma_{*}\colon\mathcal{GC}^{+}(S)\to\mathcal{GC}^{+}(S)$. Set

$\mathcal{GC}(\Sigma)\coloneqq\\{\mu\in\mathcal{GC}^{+}(\Sigma)\mid\sigma_{*}(\mu)=\mu\\}.$

There is a map $\Pi\colon\mathcal{GC}^{+}(S)\to\mathcal{GC}^{+}(S)$ given by
$\Pi(\mu)\coloneqq\frac{1}{2}(\mu+\sigma_{*}(\mu))$ with image the subset of
unoriented currents.

In the proof of the main result, we shall work with oriented currents
$\mathcal{GC}^{+}(S)$; oriented currents are more general and just as easy to
work with for our proof.

The maps $\sigma$ and $\Pi$ have obvious analogues for curves.

###### Proof of Corollary 1.11, assuming Theorem A.

For a curve functional as in the statement, let
$g\colon\mathbb{R}\mathcal{C}^{+}(S)\to\mathbb{R}$ be $f\circ\Pi$. Then $g$
satisfies quasi-smoothing, with the same constant as $f$, and thus by Theorem
A extends uniquely to a continuous function
$\bar{g}\colon\mathcal{GC}^{+}(S)\to\mathbb{R}$. The desired extension
$\bar{f}$ is the restriction of $\bar{g}$ to the subspace of unoriented
currents. ∎

###  2.6. Curves as currents

For an oriented multi-curve $C$ on a hyperbolic surface $\Sigma$, we can
construct a geodesic current as follows.

For Definition 2.13, consider all lifts of all non-trivial components of $C$
to $\widetilde{\Sigma}$. Each lift gives a quasi-geodesic in
$\widetilde{\Sigma}$, and thus a unique fellow-traveling geodesic in
$G^{+}(S)$; we thus get an infinite countable subset of $G^{+}(S)$, which is
easily seen to be discrete and $\pi_{1}(S)$-invariant. Define the geodesic
current to be the $\delta$-function of this subset.

For Definition 2.14, take the geodesic representative $\gamma$ of $C$, and
consider the canonical lift $\widetilde{\gamma}$ of $\gamma$ to $UT\Sigma$;
this is an orbit of $\phi_{t}$. Let $\mu_{C}$ be the length-normalized
$\delta$-function on this orbit. That is, for an open set $U$ we set
$\mu_{C}(U)$ to be the total length of $\widetilde{\gamma}\cap U$ with respect
to the natural Riemannian metric on $UT\Sigma$.

For Definition 2.15 on a cross-section $\tau$, again take
$\widetilde{\gamma}\subset UT\Sigma$, and let $\mu_{\tau}$ be the
$\delta$-function on the discrete set of points $\widetilde{\gamma}\cap\tau$.
(This is compatible with the length normalization in the previous paragraph.)

From any of these points of view the inclusion naturally extends to weighted
multi-curves.

Weighted closed (multi-)curves are dense in the space of geodesic currents
[Bon86, Proposition 4.4].

Geometric intersection number extends continuously to geodesic currents
[Bon86, Proposition 4.5]. The space of measured laminations (defined in [PH92,
Section 1.7]) can be characterized [Bon88, Proposition 17] as a subset of
(unoriented) geodesic currents:

$\mathcal{ML}(S)\coloneqq\\{\alpha\in\mathcal{GC}(S)\mid\sigma(\alpha)=\alpha,i(\alpha,\alpha)=0\\}.$

The following square of inclusions is useful to keep in mind:

$\mathbb{R}\mathcal{S}(S)$$\mathcal{ML}(S)$$\mathbb{R}\mathcal{C}(S)$$\mathcal{GC}(S)$.

Here the horizontal inclusions have dense image: Douady and Hubbard showed
that weighted simple multi-curves are dense in $\mathcal{ML}$ [DH75, Theorem].
Soon after, Masur showed that weighted simple curves are also dense [Mas79,
Theorem 1].

###  2.7. Topology on currents and measures

Let $\mathcal{M}(X)$ denote the space of positive Borel measures on a
topological space $X$. $\mathcal{M}_{1}(X)$ will denote the space of Borel
probability measures on $X$. The topology on $\mathcal{M}(X)$ is the _weak ∗
topology_, i.e., the smallest topology so that, for all continuous, compactly-
supported functions $f$ on $G^{+}(S)$, the functional

$\mu\mapsto\int_{G^{+}(S)}fd\mu$

is continuous.

The topology on $\mathcal{GC}^{+}(S)$ (in Definition 2.13) is the weak∗
topology as a subspace of measures on $G^{+}(S)$, We could also look at the
weak∗ topology on currents as a subspace of measures on $UT\Sigma$ (Definition
2.14); these two points of view give the same topology [BO07, Proposition
8.1]. On the other hand, if we take $\tau$ to be a closed cross-section
(including the boundary), the map $\mu\mapsto\mu_{\tau}$ relating Definitions
2.14 and 2.15 is _not_ usually continuous with respect to the weak∗
topologies, so it is delicate to use the weak∗ topology on
$\mathcal{M}(\tau)$; see Lemma 10.4 and Remark 10.6.

There are in fact two topologies on spaces of measures that are sometimes
called the weak∗ topology; the one above is also called the _wide topology_
[Min10, textitinter alia]. There is also the _narrow topology_ on measures
$\mathcal{M}(X)$ on a space $X$, defined as the smallest topology so that, for
all continuous bounded $f$ on $X$, the functional $\mu\mapsto\int_{X}fd\mu$ is
continuous. (That is, replace _compactly supported_ with _bounded_ in the
functions considered.)

######  Remark 2.18.

Some authors call the weak∗ topology the vague topology, and use the term weak
topology for the narrow one (for example, Bauer’s textbook [Bau01]). However,
this conflicts with the notion of weak topology used for Banach spaces, and we
prefer the wide/narrow usage.

In general, the weak∗ or wide topology is weaker than the narrow topology, but
in some particular cases they are equivalent.

A topological space $X$ is called _Polish_ if its topology has a countable
base and can be defined by a complete metric.

######  Theorem 2.19 ([Bau01, Theorem 31.5]).

Let $X$ be a locally compact topological space. Then $X$ is Polish if and only
if $\mathcal{M}(X)$ is Polish with respect to the weak∗-topology.

Thus, $\mathcal{GC}^{+}(S)$ is second countable completely metrizable and
second countable, so in particular sequential continuity is the same as
continuity. Although we will be dealing with Radon measures, for Polish spaces
it is equivalent to consider the a priori more general class of Borel
measures.

######  Theorem 2.20 ([Bau01, Theorem 26.3]).

On a Polish space, a locally finite Borel measure is a $\sigma$-finite Radon
measure.

The narrow and wide topology agree in certain sequences on locally compact
spaces.

######  Theorem 2.21 ([Bau01, Theorem 30.8]).

Let $X$ be a locally compact topological space, and $\mu_{n}$ a sequence of
Radon measures of uniformly bounded mass converging to a Radon measure $\mu$
in the wide topology. Then $\mu_{n}$ converges to $\mu$ in the narrow topology
if and only if $\lim_{n}\mu_{n}(X)\to\mu(X)$.

######  Proposition 2.22 ([Bau01, Corollary 30.9]).

Let $X$ be a locally compact topological space, and $\mu$, $\mu_{n}$, Borel
probability measures. Then $\mu_{n}\to\mu$ in the wide topology if and only if
$\mu_{n}\to\mu$ in the narrow topology.

In particular, when $X$ is Polish, the two topologies agree for the space
$\mathcal{M}_{1}(X)$ of Borel probability measures.

######  Proposition 2.23.

If $X$ is a locally compact Polish space, the weak∗ and narrow topologies
agree on $\mathcal{M}_{1}(X)$.

######  Convention 2.24.

For any topological space $X$, we will always use the weak∗ topology on
$\mathcal{M}(X)$. We will also work with the dense subspace
$\mathbb{R}X\subset\mathcal{M}(X)$ of finitely-supported measures on $X$ (also
called weighted linear combinations of $X$), with its inherited subspace
topology. (The weights are positive, but we usually omit that from the
notation.)

######  Remark 2.25.

If we limit to sums with at most $k$ terms in the linear combination (or
points in the support of the measure), we get a further subspace temporarily
denoted $\mathbb{R}_{(k)}X\subset\mathbb{R}X\subset\mathcal{M}(X)$. We can
view $\mathbb{R}_{(k)}X$ as a quotient of $(\mathbb{R}_{\geq 0}\times X)^{k}$,
quotienting by the action of the symmetric group and other evident
equivalences; as such, it inherits an obvious topology, which agrees with the
subspace topology.

######  Remark 2.26.

Geodesic currents can also be defined more generally for finite type
hyperbolic surfaces. Depending on if we consider ends as cusps or funnels, we
get two different spaces, that we will call $\mathcal{GC}_{\mathrm{cusp}}(S)$
and $\mathcal{GC}_{\mathrm{open}}(S)$, respectively. In the first case, we
define the space of geodesic currents analogously to Definition 2.13 for
closed surfaces, i.e., as invariant measures supported on the space of
geodesics of the universal cover, noting that now the space of geodesics
contains arcs going from cusp to cusp. In the second case, we consider
geodesic currents supported on geodesics projecting to the convex core of the
surface. Extending continuously curve functionals to these spaces is more
delicate.

In the case of $\mathcal{GC}_{\mathrm{cusp}}(S)$, let $S$ be a surface with
two open ends and let $\Sigma$ be a complete hyperbolic metric of finite area,
with respect to which the ends of $S$ are cusps. Let $a$ be an arc going from
cusp to cusp. Let $C_{n}$ be the closed curve going along for some time $a$,
winding $n$ times around one cusp, going along $a$ again and winding $n$ times
around the other cusp. Observe that although $C_{n}\to a$ in the weak∗
topology, and $i(a,a)=0$, we have $i(a,C_{n})=2n$, so intersection number is
not a continuous function on geodesic currents.

The case of $\mathcal{GC}_{\mathrm{open}}(S)$ is different, since intersection
number is continuous. Indeed, let $\overline{\Sigma}$ denote the convex core
of the complete hyperbolic surface of infinite area, which is a compact
surface with geodesic boundary. We can consider the intersection number on the
double $D(\overline{\Sigma})$ of $\overline{\Sigma}$, which is a closed
surface $\overline{\Sigma}$ embeds into. This intersection number on
$D(\overline{\Sigma})$ is continuous [Bon86, Proposition 4.5]. Restricting
this intersection number to $\overline{\Sigma}$, we obtain continuity of
intersection number on $\mathcal{GC}_{\mathrm{open}}(S)$. However, the
conditions of Theorem A alone are not enough to guarantee a continuous
extension $f\colon\mathcal{GC}_{\mathrm{open}}(S)\to\mathbb{R}$. Indeed, let
$\ell$ be the restriction to $\overline{\Sigma}$ of the hyperbolic length on
$D(\overline{\Sigma})$, and consider the modified curve functional
$\ell^{\prime}$ obtained by setting

$\ell^{\prime}(C)\coloneqq\begin{cases}0&\text{$C$ is a boundary curve}\\\
\ell(C)&\text{otherwise.}\end{cases}$

We note that $\ell^{\prime}$ satisfies additivity, stability and homogeneity
properties because $\ell$ does. Also, it satisfies smoothing because $\ell$
does and non-boundary curves don’t intersect boundary curves. However,
$\ell^{\prime}$ does not extend to a continuous function on
$\mathcal{GC}_{\mathrm{open}}(S)$: let $\gamma,\beta\in\pi_{1}(S,p)$ be
elements based at a point $p\in S$, and denote $C=[\beta]$, $D=[\gamma]$.
Assume that $C$ is a boundary curve and $D$ is not. For each $n$, define a
non-simple, non-boundary parallel curve by $C_{n}\coloneqq[\gamma\beta^{n}]$.
Observe that the sequence $\frac{1}{n}C_{n}$ converges to $C$ in the weak∗
topology, but

$\displaystyle\ell^{\prime}(C_{n})$ $\displaystyle\asymp n,$ whereas
$\displaystyle\ell^{\prime}(C)$ $\displaystyle=0,$

so $\ell^{\prime}$ can’t be a continuous function on
$\mathcal{GC}_{\mathrm{open}}(S)$. So additional conditions on $f$ are needed
to guarantee a continuous extension to $\mathcal{GC}_{\mathrm{open}}(S)$.

##  3\. Convexity and continuity

###  3.1. Convexity on the reals

The curve functionals $f$ we study have some convexity property as a function
of the weights, because of the convex union and homogeneity properties. We
first review some background on convex functions and their continuity
properties.

A function $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is called _$\mathbb{R}$
-convex_ (resp.  _$\mathbb{Q}$ -convex_) if

$f(ax+(1-a)y)\leq af(x)+(1-a)f(y)$

for all $x,y\in\mathbb{R}^{n}$ and $a\in[0,1]$ (resp.
$a\in[0,1]\cap\mathbb{Q}$). A function $f\colon\mathbb{Q}^{n}\to\mathbb{R}$
might also be $\mathbb{Q}$-convex, with the same definition. We furthermore
say that $f$ is _midpoint-convex_ if

$f\bigl{(}{\textstyle\frac{1}{2}}x+{\textstyle\frac{1}{2}}y\bigr{)}\leq{\textstyle\frac{1}{2}}f(x)+{\textstyle\frac{1}{2}}f(y).$

######  Proposition 3.1.

The following are true:

  1. (1)

A midpoint-convex function $f\colon\mathbb{Q}^{n}\to\mathbb{R}$ is
$\mathbb{Q}$-convex.

  2. (2)

An $\mathbb{R}$-convex function $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is
continuous.

  3. (3)

A $\mathbb{Q}$-convex function $f\colon\mathbb{Q}^{n}\to\mathbb{R}$ is
continuous.

  4. (4)

Every $\mathbb{Q}$-convex function $f\colon\mathbb{Q}^{n}\to\mathbb{R}$ has a
unique continuous extension to an $\mathbb{R}$-convex function
$\bar{f}\colon\mathbb{R}^{n}\to\mathbb{R}$.

###### Proof.

  1. (1)

This proof is due to Ivan Meir [Mei19], following [HLP88, Page 17]. We first
prove that midpoint inequality extends to arbitrary means:

$g((x_{1}+\dots+x_{m})/m)\leq(g(x_{1})+\dots+g(x_{m}))/m$

for any $m\in\mathbb{Z}_{\geq 1}$. We can prove this first for $m=2^{k}$ by
using midpoint convexity repeatedly. For general $m\leq 2^{i}$, we take
$x_{1},\dots,x_{m}$ plus $2^{i}-m$ copies of
$x^{\prime}=(x_{1}+\dots+x_{m})/m$, yielding

$g(x^{\prime})=g\left(\frac{(2^{i}-m)x^{\prime}+x_{1}+\cdots+x_{m}}{2^{i}}\right)\leq\frac{(2^{i}-m)g(x^{\prime})+g(x_{1})+\dots+g(x_{m})}{2^{i}},$

which implies
$g(x^{\prime})=g((x_{1}+\dots+x_{m})/m)\leq(g(x_{1})+\dots+g(x_{m}))/m$.

To prove $\mathbb{Q}$-convexity, taking $a$ copies of $x$ and $b$ copies of
$y$ we obtain

$g\left(\frac{ax+by}{a+b}\right)\leq\frac{ag(x)+bg(y)}{a+b}=\left(\frac{a}{a+b}\right)g(x)+\left(\frac{b}{a+b}\right)g(y)$

for $a,b\in\mathbb{Z}_{\geq 0}$ not both zero.

  2. (2)

See [Kuc09, Theorem 7.1.1].

  3. (3)

The proof of [Kuc09, Theorem 7.1.1] can be adapted for functions on
$\mathbb{Q}^{n}$. The proof relies on Bernstein-Doetsch Theorem, which works
in high generality for topological vector spaces (see [KK89, Theorem B]), and
the fact that any point $x\in\mathbb{Q}^{n}$ is the interior of some full-
dimensional $\mathbb{Q}$-simplex, on which $f$ is bounded.

  4. (4)

Define the extension by

$\bar{f}(x)\coloneqq\liminf_{\begin{subarray}{c}y\to x\\\
y\in\mathbb{Q}^{n}\end{subarray}}f(y).$

By continuity of $f$ on $\mathbb{Q}^{n}$, $\bar{f}$ is an extension of $f$. To
study $\bar{f}(ax+(1-a)y)$, let $x_{i},y_{i}$ be sequences in $\mathbb{Q}^{n}$
with $\lim x_{i}=x$, $\lim y_{i}=y$, $\liminf f(x_{i})=f(x)$, and $\liminf
f(y_{i})=y$. Let $a_{i}\in[0,1]\cap\mathbb{Q}$ be a sequence with $\lim
a_{i}=a$. Then

$\displaystyle\bar{f}(ax+(1-a)y)$ $\displaystyle\leq\liminf
f\bigl{(}a_{i}x_{i}+(1-a_{i})y_{i}\bigr{)}$
$\displaystyle\leq\liminf\bigl{(}a_{i}f(x_{i})+(1-a_{i})f(y_{i})\bigr{)}$
$\displaystyle=a\bar{f}(x)+(1-a)\bar{f}(y).$

Thus $\bar{f}$ is convex and therefore continuous.∎

It is not true that all $\mathbb{Q}$-convex functions
$f\colon\mathbb{R}^{n}\to\mathbb{R}$ must be continuous, but all
counterexamples are highly pathological. In particular, any measurable
$\mathbb{Q}$-convex function $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is
necessarily continuous (see [Kuc09, Theorem 9.4.2]).

######  Remark 3.2.

Proposition 3.1(2) does not hold for infinite dimensional topological vector
spaces: an unbounded linear functional is convex but not continuous.

###  3.2. Convexity for curve functionals

We now apply the results above to our setting of real-valued functions on
curves.

As an immediate consequence of Proposition 3.1(4), for a curve functional $f$
satisfying convex union and homogeneity, we can extend $f$ to a weighted curve
functional that is convex and therefore continuous for a fixed set of
components. This will play a role in the proof of Theorem A, specifically in
Proposition 10.3.

First, for any curve functional satisfying homogeneity, we adopt the
convention that we extend $f$ to rationally-weighted curves
$\mathbb{Q}\mathcal{C}^{+}(S)$ in the usual way by clearing denominators: set

(3.3) $f\Bigl{(}\sum a_{i}C_{i}\Bigr{)}\coloneqq\frac{1}{d}f\Bigl{(}\sum
da_{i}C_{i}\Bigr{)}$

for some integer $d$ sufficiently large so all the $da_{i}$ are integers. By
homogeneity of $f$, the extension does not depend on $d$.

######  Proposition 3.4.

Let $C=(C_{i})_{i=1,\dots,n}$ be a finite sequence of multi-curves, and
consider combinations $\sum_{i=1}^{n}a_{i}C_{i}$. Let $f$ be a curve
functional that satisfies homogeneity and convex union. Define a function
$f_{C}\colon\mathbb{Q}^{n}\to\mathbb{R}$ by

$f_{C}(a_{1},\dots,a_{n})\coloneqq f\left(\sum_{i=1}^{n}a_{i}C_{i}\right).$

Then $f_{C}$ is $\mathbb{Q}$-convex and thus continuous.

###### Proof.

It is immediate from the definitions that $f_{C}$ is midpoint convex. The
result follows from Proposition 3.1. ∎

######  Corollary 3.5.

If a curve functional $f$ satisfies homogeneity and convex union, then there
is a unique continuous homogeneous extension of $f$ to weighted curve
functional.

######  Proposition 3.6.

If a curve functional satisfies convex union, homogeneity, and quasi-
smoothing, the extension from Corollary 3.5 satisfies weighted quasi-smoothing
with the same constant.

###### Proof.

We first observe that $f$, as a function on integrally weighted multi-curves,
satisfies weighted quasi-smoothing. If $C=[\gamma]$,
$\gamma\mathrel{\searrow}\gamma^{\prime}$ and $k$ is an integer, then
$k\gamma\mathrel{\searrow}_{k}k\gamma^{\prime}$, since $k\gamma$ are $k$
disjoint parallel copies of $\gamma$. Thus,

$f(kC)\geq f(kC^{\prime})-kR.$

By the method of clearing denominators and homogeneity, as in Equation (3.3),
we obtain rational weighted quasi-smoothing.

Finally, by continuity of $f$ as a function of the weights of a fixed multi-
curve, we get real weighted quasi-smoothing. ∎

Theorems A and B as stated start from a curve functional of various types.
Many curve functionals naturally come as functions on weighted curves (see
Section 4). On the other hand, we have seen in Proposition 3.6 that a curve
functional satisfying convex union and homogeneity and stability properties
yields weighted curve functional satisfying the same properties.

##  4\. Examples

We give several examples of curve functionals that extend to functions on
currents, mostly as a consequence of our main theorems. This includes known
results, such as hyperbolic lengths and intersection numbers, or more
generally lengths for any length metric structure, as well as new results,
such as extremal lengths with respect to a conformal structure or with respect
to a graph. In the following applications we consider unoriented curves unless
otherwise stated.

###  4.1. Intersection number

Fix a multi-curve $D$ and consider the curve functional $\mathcal{C}(S)$
defined by $f(C)=i(C,D)$, where $i(C,D)$ is the minimal number of intersection
points between representatives of $C$ and $D$ in general position. Then $f$ is
homogeneous, additive, and stable.

There is a simple geometric argument, that we will use repeatedly, to see that
$f$ satisfies smoothing. Fix a minimal representative $\delta$ for $D$. Take a
curve $C$ with an essential self-intersection $x$, and a representative
$\gamma$ with minimal intersection with $\delta$. Then $\gamma$ has a self-
intersection point $x^{\prime}$ of the homotopy type of $x$. If we consider
the curve representative $\gamma^{\prime}\in C^{\prime}$ obtained by smoothing
at $x^{\prime}$, then, since $i(C^{\prime},D)$ is an infimum, we have

$i(C^{\prime},D)\leq i(\gamma^{\prime},\delta)=i(\gamma,\delta)=i(C,D),$

as desired.

By Theorem A, intersection number with $D$ extends to a continuous function on
geodesic currents

$i(\cdot,D)\colon\mathcal{GC}(S)\to\mathbb{R}.$

We can then fix $C$ and vary $D$ to show that, for $\mu$ a geodesic current,
$i(\cdot,\mu)$ is a continuous function on $\mathcal{GC}(S)$. In [Bon86,
Proposition 4.5], Bonahon shows that the geometric intersection number
$i\colon\mathbb{R}\mathcal{C}(S)\times\mathbb{R}\mathcal{C}(S)\to\mathbb{R}_{\geq
0}$ between two weighted multi-curves extends to a continuous two-variable
function

$i\colon\mathcal{GC}(S)\times\mathcal{GC}(S)\to\mathbb{R}_{\geq 0}.$

######  Question 4.1.

Can the arguments in this paper be extended to give an alternate proof that
geometric intersection number is a continuous two-variable function?

Following Example 1.12, proving that there is a continuous extension of
$\sqrt{i(C,C)}$ to a function on $\mathcal{GC}(S)$ is equivalent to proving
continuity of $i$ as a two-variable function, by a simple polarization
argument:

$i(C,D)=\frac{i(C\cup D,C\cup D)-i(C,C)-i(D,D)}{2}.$

###  4.2. Hyperbolic length

We continue with the original motivating example for geodesic currents [Bon88,
Proposition 14]. Fix a hyperbolic metric $g$ on $S$, and denote the hyperbolic
structure by $\Sigma$. Then, for any closed curve $C$ on $S$ (not necessarily
simple), we can consider its hyperbolic length with respect to the Riemannian
metric. In terms of the holonomy representation
$\rho_{g}:\pi_{1}(S)\to\operatorname{\mathit{PSL}}_{2}(\mathbb{R})$, this is
given by

$\ell_{g}(C)=2\cosh^{-1}\left({\textstyle\frac{1}{2}}\operatorname{tr}(\rho(c))\right)$

where $c\in\pi_{1}(S)$ is a representative of $C$. We extend $\ell_{g}$ to a
weighted curve functional by additivity and homogeneity:

$\ell_{g}(t_{1}C_{1}\cup\dots\cup
t_{n}C_{n})=\sum_{i=1}^{n}t_{i}\ell_{g}(C_{i}).$

By definition, $\ell_{g}$ is additive and homogeneous. Stability follows from
properties of the trace of $2\times 2$ matrices, or geometrically from the
length. Smoothing follows by the argument for intersection number. Thus, by
Theorem A, $\ell_{g}$ extends to a continuous function on geodesic currents.

We recall that Bonahon shows that

$\ell_{g}(C)=i(\mathcal{L}_{\Sigma},C)$

where $\ell_{g}(C)$ denotes the hyperbolic length of $C$, i.e., the length of
the $g$-geodesic representative, and $\mathcal{L}_{\Sigma}$ denotes the
_Liouville current_ , a geodesic current induced by the volume form on
$UT\Sigma$. (For an equivalent formulation in terms of Definition 2.13, see
[Bon88, § 2].)

###  4.3. Length with respect to arbitrary metrics

The argument from Section 4.2 applies equally well to show that for any
Riemannian or, more generally, length metric $g$ on $S$, length $\ell_{g}$
with respect to $g$ satisfies smoothing. For completeness and later use, we
prove that these curve functionals are stable. (Freedman-Hass-Scott give a
proof in the Riemannian case [FHS82, Lemma 1.3].)

######  Lemma 4.2.

For any orientable surface $S$ and length metric $g$ on $S$, the curve
unctional $\ell_{g}$ is stable: $\ell_{g}(C^{n})=n\ell_{g}(C)$.

###### Proof.

One inequality is true in any length space: by taking the obvious $n$-fold
representative of $C^{n}$, we see that $\ell_{g}(C^{n})\leq n\ell_{g}(C)$. The
other inequality follows from the smoothing property: since
$C^{n}\mathrel{\searrow}nC$, we have $\ell_{g}(C^{n})\geq n\ell_{g}(C)$. ∎

######  Remark 4.3.

Lemma 4.2 is false if $S$ is not orientable. For instance, if $S$ is the
projective plane, $g$ is any metric, and $C$ is the non-trivial curve on
$\Sigma$, then $\ell_{g}(C)>0$ but $C^{2}$ is null-homotopic so
$\ell_{g}(C^{2})=0$. We can get a similar inequality without torsion on a
Möbius strip, by removing a small disk from this projective plane.

###  4.4. Length with respect to embedded graphs

We can generalize further beyond length metrics. Let
$\iota\colon\Gamma\hookrightarrow S$ be an embedding of a finite graph in $S$
that is filling, in the sense that the complementary regions are disks, or
equivalently $\iota_{*}$ is surjective on $\pi_{1}$. Endow $\Gamma$ with a
length metric $g$. Then any closed multi-curve $C$ on $S$ can be homotoped so
that it factors through $\Gamma$, in many different ways. Let
$\ell_{\Gamma}(C)$ be the length of the smallest multi-curve $D$ on $\Gamma$
so that $\iota(D)$ is homotopic to $C$. It is easy to see that this length is
realized and is positive. (In fact we can see $\ell_{\Gamma}$ as a limit of
lengths with respect to Riemannian metrics, by fixing an embedding of $\Gamma$
and making the metric on the complement of a regular neighborhood of $\Gamma$
be very large, following Shepard [She91].)

As before, $\ell_{\Gamma}$ is clearly additive and homogeneous, and is stable
by the argument of Lemma 4.2. To see that $\ell_{\Gamma}$ satisfies smoothing
at an essential crossing, take a minimal-length concrete representative
$\delta$ of $D$ on $\Gamma$. Since the image $\iota\circ\delta$ has a
corresponding crossing by Lemma 2.8 and $\iota$ is an embedding, there is a
corresponding crossing of $\delta$ that can be smoothed and then tightened to
get the desired inequality.

As a special case, we can consider the case when $\Gamma$ is a _rose graph_
with only one vertex $\ast$ and edges of length $1$. Since $\iota$ is filling,
the image of the edges of $\Gamma$ give generators for
$\pi_{1}(S,\iota(\ast))$. Then the length $\ell_{\Gamma}(C)$ of a curve $C$ is
the length of $C$ as a conjugacy class in $\pi_{1}(S)$ with respect to these
generators. This is a simple generating set in the sense of Erlandsson
[Erl19], who proved this continuity and constructed an explicit multi-curve
$K$ so that $\ell_{\Gamma}(C)=i(C,K)$.

###  4.5. Stable lengths

Generalizing the previous example, let $\iota\colon\Gamma\to S$ be an
_immersion_ from a finite graph to $S$ so that
$\iota_{*}\colon\pi_{1}(\Gamma)\to\pi_{1}(S)$ is surjective, and again give a
length metric on $\Gamma$. For instance, if $\Gamma$ has a single vertex and
all edges have length $1$, this is equivalent to giving an arbitrary
generating set for $\pi_{1}(S)$. We can define $\ell_{\Gamma}(C)$ as before,
as the minimum length of any multi-curve $D$ on $\Gamma$ so that
$\iota_{*}(D)=C$.

The curve functional $\ell_{\Gamma}$ is still additive, but unlike the
previous examples it is not stable (see Example 4.10 below). Thus we cannot
hope to extend $\ell_{\Gamma}$ to currents, but rather extend the stable curve
functional $\lVert\ell_{\Gamma}\rVert$ (defined in Section 13). We do have
quasi-smoothing.

######  Lemma 4.4.

For any connected, $\pi_{1}$-surjective immersion of a length graph
$\iota\colon\Gamma\to S$, the curve functional $\ell_{\Gamma}$ satisfies
quasi-smoothing.

This is a special case of a more general result. Let $V$ be a connected length
space, with a continuous, $\pi_{1}$-surjective map $\iota\colon V\to S$. Then
for $C$ a multi-curve on $S$, a _lift_ of $C$ is a multi-curve $\tilde{C}$ in
$V$ so that $\iota_{*}\tilde{C}=C$. (Both $\tilde{C}$ and $C$ are defined up
to homotopy.) Define $\ell_{\iota,V}(C)$ to be the infimum, over all lifts
$\tilde{C}$ of $C$, of the length of $\tilde{C}$ in $V$.

######  Proposition 4.5.

Let $V$ be a connected, compact length space, with a continuous,
$\pi_{1}$-surjective map $\iota\colon V\to S$. Then
$\ell_{\iota,V}\colon\mathcal{C}(S)\to\mathbb{R}_{+}$ satisfies quasi-
smoothing.

As a corollary of Proposition 4.5 and Theorems B and A, $\lVert\ell_{V}\rVert$
extends continuously to a function on $\mathcal{GC}(S)$. This continuous
extension was first proved by Bonahon [Bon91, Proposition 10] in the context
of a hyperbolic group acting discretely and cocompactly on a length space
(replacing $\pi_{1}(S)$ acting on $\widetilde{V}$), with an additional
technical assumption that the space is uniquely geodesic at infinity. Later,
Erlandsson, Parlier, and Souto [EPS20, Theorem 1.5] lifted this assumption.

We remark that given a properly discontinuous action of $\pi_{1}(S)$ on $X$ a
CW-complex, we can construct a $\pi_{1}(S)$-surjective map $\iota\colon
X/\pi_{1}(S)\to S$. In fact, we will construct a $\pi_{1}(S)$-equivariant map
$\iota\colon X\to\widetilde{S}$. First, we define $\iota\colon
X_{0}\to\tilde{S}$ by picking a value on each $\pi_{1}(S)$ orbit of the
0-skeleton arbitrarily and extending equivariantly. Similarly on the
1-skeleton $X_{1}$, for each $\pi_{1}(S)$ orbit on $X_{1}$, pick a path in
$\widetilde{S}$ between the images of the endpoints. Continue the construction
inductively. This construction works because $\pi_{1}(S)$ acts freely—since
$\pi_{1}(S)$ is torsion-free and acts properly discontinuously—and
$\widetilde{S}$ is contractible.

In this construction, proper discontinuity of the action is crucial. For
example, it was shown by Bonahon in [Bon91, Proposition 11] that if $W$ is a
finite graph which is a deformation retract of $S$, the action of $\pi_{1}(S)$
on the universal cover $\widetilde{W}$ of $W$ is cocompact but not properly
discontinuous, and translation length of conjugacy classes of $\pi_{1}(S)$
acting on $X$ does not extend continuously to $\mathcal{GC}(S)$.

###### Proof of Proposition 4.5.

Let $\widetilde{\iota}\colon\widetilde{V}\to\widetilde{S}$ be the pull-back of
$\iota$ along the universal cover $\pi_{S}$ of $S$, part of the pull-back
square

(4.6)
$\widetilde{V}$$V$$\widetilde{S}$$S$$\scriptstyle{\pi_{V}}$$\scriptstyle{\tilde{\iota}}$$\scriptstyle{\pi_{S}}$$\scriptstyle{\iota}$

where $\widetilde{V}=V\times_{S}\widetilde{S}$. Since $\iota$ is
$\pi_{1}$-surjective, $\widetilde{V}$ is a connected covering space of $V$.
For $\widetilde{x}\in\widetilde{S}$, let
$\operatorname{diam}_{\iota}(\widetilde{x})$ be the diameter of
$\widetilde{\iota}^{-1}(\widetilde{x})\subset\widetilde{V}$. (Set
$\operatorname{diam}_{\iota}(\widetilde{x})=0$ if $\widetilde{x}$ is not in
the image of $\widetilde{\iota}$.) Set

$\operatorname{diam}_{\iota}(V)\coloneqq\sup_{\widetilde{x}\in\widetilde{S}}\operatorname{diam}_{\iota}(\widetilde{x}).$

We wish to see that $\operatorname{diam}_{\iota}(V)$ is finite. First, since
$\iota$ is $\pi_{1}$-surjective, for every $\gamma\in\pi_{1}(S,x)$ there
exists $\delta_{\gamma}\in\pi_{1}(V,\tilde{x})$ so that
$\iota_{*}\delta_{\gamma}=\gamma$. This implies

$\delta_{\gamma}\cdot\tilde{\iota}^{-1}(\tilde{x})=\tilde{\iota}^{-1}(\gamma\cdot\tilde{x}).$

But $\delta_{\gamma}$ is a deck transformation, and thus acts as an isometry
on $\widetilde{V}$, so
$\operatorname{diam}(\tilde{\iota}^{-1}(\gamma\cdot\tilde{x}))=\operatorname{diam}(\tilde{\iota}^{-1}(\tilde{x}))$
and so we can define

(4.7)
$\operatorname{diam}_{\iota}(x)=\operatorname{diam}_{\iota}(\widetilde{x}).$

for $x\in S$ and any lift $\widetilde{x}$ of $x$. (Note that
$\operatorname{diam}_{\iota}(x)$ is not in general the diameter of
$\iota^{-1}(x)$; rather than looking at the length of a shortest path
connecting two points in $\iota^{-1}(x)$, we restrict to paths that map to
null-homotopic loops.)

######  Lemma 4.8.

In the above setting, the diameter $\operatorname{diam}_{\iota}(x)$ is upper
semi-continuous as a function of $x$.

###### Proof.

For each $x_{0}\in S$, consider an evenly covered neighborhood $U$ of $x_{0}$,
and fix a lift $\tilde{x}_{0}\in\widetilde{S}$. We want to show that for all
sequences $\\{x_{i}\\}\subset U$ with $x_{i}\to x$ and for all
$\varepsilon>0$, there exists $i_{0}$ so that for all $i\geq i_{0}$

$\operatorname{diam}(\tilde{\iota}^{-1}(\tilde{x_{i}}))<\operatorname{diam}(\tilde{\iota}^{-1}(\tilde{x}))+\varepsilon.$

Now, $\tilde{\iota}$ is a pullback of a proper map, so it is a closed map:

######  Lemma 4.9 ([Sta18, Theorem 005R]).

Let $X$ be a metric space, and $f\colon X\to Y$ a proper map. For any
continuous map $g\colon Z\to Y$, the pullback map $X\times_{Y}Z\to Z$ is
closed.

By definition of diameter, and since the fibers
$\tilde{\iota}^{-1}(\tilde{x})$ are compact for any $\tilde{x}_{i}$, we can
find points $p_{i},q_{i}\in\tilde{\iota}^{-1}(\tilde{x_{i}})$ so that
$\operatorname{diam}(\tilde{\iota}^{-1}(\tilde{x_{i}}))=d(p_{i},q_{i})$. Also,
by closedness of $\tilde{\iota}$, a subsequence of the $p_{i}$ and $q_{i}$
converges to points $p,q\in\tilde{\iota}^{-1}(\tilde{x})$. Furthermore, by
continuity of distance, for any $\varepsilon>0$, we have, for $i$ large
enough,

$\operatorname{diam}(\tilde{\iota}^{-1}(\tilde{x_{i}}))=d(p_{i},q_{i})<d(p,q)+\varepsilon\leq\operatorname{diam}(\tilde{\iota}^{-1}(\tilde{x}))+\varepsilon,$

finishing the proof of Lemma 4.8. ∎

As a result of Lemma 4.8, the function $\operatorname{diam}_{\iota}(x)$ is
bounded on $S$; let $R(V)$ be this global bound.

We now finish the proof of Proposition 4.5. We are given a curve $C$ with an
essential crossing $p$ and corresponding smoothing $C^{\prime}$. Pick a
concrete curve $\tilde{\gamma}$ on $V$ that comes within $\varepsilon$ of
realizing $\ell_{\iota,V}(C)$; in particular, $\iota\circ\tilde{\gamma}$
represents $C$. By Lemma 2.8, there are points $x,y\in X(\tilde{\gamma})$ so
that $\iota(\tilde{\gamma}(x))=\iota(\tilde{\gamma}(y))$ is a crossing
corresponding to $p$. We wish to find another curve $\tilde{\gamma}^{\prime}$
on $V$, with length not too much longer, so that
$\iota\circ\tilde{\gamma}^{\prime}$ represents $C^{\prime}$. We can do this by
cutting $\widetilde{\gamma}$ at $x$ and $y$, yielding endpoints $x_{1}$,
$x_{2}$ and $y_{1}$, $y_{2}$, and reconnecting $x_{1}$ to $y_{2}$ and $y_{1}$
to $x_{2}$ by paths in $V$ that project to the identity in $\pi_{1}(S)$.

But the maximal length of a path connecting any two points $x,y\in V$ with
$\iota(x)=\iota(y)$ that projects to a null-homotopic path is exactly
$\operatorname{diam}_{\iota}(\iota(x))$. We can therefore construct a desired
representative $\tilde{\gamma}^{\prime}$ with

$\ell_{V}(\tilde{\gamma}^{\prime})\leq\ell_{V}(\tilde{\gamma})+2R(V)\leq\ell_{\iota,V}(C)+\varepsilon+2R(V).$

Since $\varepsilon$ was arbitrary, we have proved the result with quasi-
smoothing constant $2R(V)$. ∎

We show now an example of a curve functional that satisfies quasi-smoothing
but not strict smoothing.

######  Example 4.10.

Consider the torus with one puncture with fundamental group generated by the
usual horizontal loop $a$ and vertical loop $b$, as in Figure 4.1. (This does
not strictly speaking fit in the context of closed surfaces considered in this
paper, but we can embed this punctured torus in a larger surface without
essential change.)

$1$
---
$x$
---
$1-x$
---
$x$
---
$a$
---
$b$
---

Figure 4.1. A punctured torus with, in blue, a train-track carrying a slice of
measured laminations on the punctured torus depending on a parameter
$x\in[0,1]$. In red, the parallel loop $a$. In green, the meridian loop $b$.

Its fundamental group is the free group $F_{2}=\langle a,b\rangle$. We will
consider word-length $f$ with respect to the generating set $(a,a^{2},b)$.
Word-length satisfies quasi-smoothing and additive union, but not stability.
(For instance, $f(a^{2})=1\neq f(2a)=2$.) The stable word-length $\|f\|$
satisfies stability and still satisfies quasi-smoothing, but it doesn’t
satisfy strict smoothing. We will show it behaves more erratically than word-
length with respect to embedded generating sets.

Consider for example the collection of weighted curves $C(x)$ carried by the
train-track in Figure 4.1, with weights depending on a rational parameter
$x\in[0,1]$. For instance, for $x=2/5$, the curve is $1/5[aabab]$, with stable
length $(1/5)\cdot 4$. If we plot the stable word-length of $C(x)$ multiplied
by the weight, we obtain the saw-tooth graph in Figure 4.2. We note the
erratic behavior as a function of $x$. In particular, it is far from convex.
If $\|f\|$ satisfied the smoothing property, then it would be a convex
function of the train-track weights, since if $w_{1}$ and $w_{2}$ are two
rational weights on a train track $T$, the weighted multi-curve $T(w_{1})\cup
T(w_{2})$ can be smoothed to $T(w_{1}+w_{2})$ (observed in [Mir04, Appendix A]
and [Thu16, Section 3.2]).

$\bigl{(}\frac{1}{3},\frac{2}{3}\bigr{)}$
---
$\bigl{(}\frac{1}{4},\frac{3}{4}\bigr{)}$
---
$\bigl{(}\frac{1}{5},\frac{3}{5}\bigr{)}$
---
$\bigl{(}\frac{2}{5},\frac{4}{5}\bigr{)}$
---
0.2
---
0.4
---
0.6
---
0.8
---
1.0
---
0.6
---
0.4
---
0.8
---
1.0
---
0.0
---
0.0
---

$\|\ell_{\Gamma}(C(x))\|$
---

$x$
---
0.2
---
Figure 4.2. We consider the curve $C(x)$ carried by a train-track depending on
a parameter $x$, and plot the stable word-length of $C(x)$ as a function of
$x$. The graph has vertices at
$\bigl{(}\frac{1}{2n+1},\frac{n+1}{2n+1}\bigr{)}$ and
$\bigl{(}\frac{1}{2n},\frac{n+1}{2n}\bigr{)}$.

###  4.6. Asymmetric lengths

The arguments in Section 4.5 apply equally well to cases where distances may
be zero or not symmetric. For instance, we can take a directed graph $\Gamma$
with a non-negative length on each edge, together with a map
$\iota\colon\Gamma\to S$ so that the corresponding cover $\widetilde{\Gamma}$
is strongly connected (every vertex can be reached from any other vertex). The
same arguments apply to show that $\ell_{\Gamma}(\vec{C})$ satisfies the
_oriented_ quasi-smoothing property and so its stabilization
$\lVert\ell_{\Gamma}\rVert$ extends to a continuous function
$\mathcal{GC}^{+}(S)\to\mathbb{R}_{\geq 0}$.

One example would be to take a generating set for $\pi_{1}(S)$ as a monoid.
This corresponds to taking $\Gamma$ to be a graph with a single vertex and one
edge for each monoid generator.

###  4.7. Generalized translation lengths from higher representations

Let $G$ be a real, connected, non-compact, semi-simple, linear Lie group. Let
$K$ denote a maximal compact subgroup of $G$, so that $X=G/K$ is the
Riemannian symmetric space of $G$. Let $[P]$ be the conjugacy class of a
parabolic subgroup $P\subset G$. Then there is a natural notion of
$[P]$-Anosov representation $\rho\colon\pi_{1}(S)\to G$; see, for example,
[Kas18, Section 4]. When $\operatorname{rank}_{\mathbb{R}}(G)=1$ there is
essentially one class $[P]$, so we can simply refer to them as Anosov
representations, and they can be defined as those injective representations
$\rho\colon\pi_{1}(S)\to G$ where $\Gamma\coloneqq\rho(\pi_{1}(S))$ preserves
and acts cocompactly on some nonempty convex subset $V$ of $X$.

Rank $1$ Anosov representations include two familiar examples.

  1. (1)

Fuchsian representations into $G=\operatorname{\mathit{PSL}}(2,\mathbb{R})$.
Here, $K=\operatorname{\mathit{SO}}(2)$ and $X=\mathbb{H}^{2}$. The convex set
$V$ in this case is the lift of the convex core of the hyperbolic surface.

  2. (2)

Quasi-Fuchsian representations of surface groups into
$G=\operatorname{\mathit{PSL}}(2,\mathbb{C})$. In this case,
$K=\operatorname{\mathit{SU}}(2)$, $X=\mathbb{H}^{3}$, and $V$ is the lift of
the convex core of the hyperbolic quasi-Fuchsian manifold.

In general, the conjugacy classes of parabolic subgroups of $G$ correspond to
subsets $\theta$ of the set of restricted simple roots $\Delta$ of $G$. For a
given $[P]$-Anosov representation and each $\alpha\in\theta$, Martone and
Zhang define [MZ19, Definition 2.21] a curve functional

$l_{\alpha}^{\rho}\colon\mathcal{C}(S)\to\mathbb{R}_{\geq 0}$

and show that for a certain subset of Anosov representations, these can be
extended to geodesic currents as intersection numbers with some fixed geodesic
current.

For the two rank 1 examples above, this length $l_{\alpha}^{\rho}(C)$
corresponds to hyperbolic length of the closed geodesic in the homotopy class
$C$ in the quotient hyperbolic manifold $\mathbb{H}^{2}/\rho(\pi_{1}(S))$ or
$\mathbb{H}^{3}/\rho(\pi_{1}(S))$.

Bonahon showed the length in the Fuchsian case extends to geodesic currents
[Bon88, Proposition 14]. Length in the quasi-Fuchsian case also extends to
geodesic currents [BT05, Lemma 4.3]. Our techniques give another proof in this
second case.

######  Proposition 4.11.

Translation length $l_{\rho}$, for
$\rho\colon\pi_{1}(S)\to\operatorname{\mathit{PSL}}(2,\mathbb{C})$ a quasi-
Fuchsian representation, extends to geodesic currents.

###### Proof.

Let $V$ to be the convex core of $\mathbb{H}^{3}/\rho(\pi_{1}(S))$. We have an
obvious retract $r\colon V\to S$ (defined up to homotopy). Now use Proposition
4.5, taking $\iota=r$. ∎

######  Question 4.12.

The _complex_ translation length of a quasi-Fuchsian representation also
extends continuously to a function on geodesic currents [BT08, Section 6]. Is
there a version of our main theorem that would prove that a complex-valued
curve functional like this extends to currents?

######  Remark 4.13.

All the proposed generalizations of the definition of convex cocompact
representations for higher rank groups turn out to yield products of
representations of rank $1$ (see [KL05, Theorem 1.3] and [Qui05, Théorème]) so
it’s not clear that the approach using Proposition 4.5 will allow one to
extend curve functionals in higher rank to geodesic currents.

For $G=\operatorname{\mathit{PSL}}(3,\mathbb{R})$, there is another cocompact
action, not on a convex subset of the symmetric space, but on a convex subset
of $G/P=\mathbb{R}P^{2}$. In this case, there is a natural metric on this
convex subset, the Hilbert metric. One can easily show using this metric that
smoothing is satisfied. In general, in higher rank one can construct similar
cocompact actions on convex domains of $G/P$ (see [GW12]), but there is not a
known canonical choice of metric. Martone and Zhang show [MZ19, Theorem 2.1]
that some types of representations known as _positively ratioed_ can be
realized as intersection numbers with a distinguished geodesic current. This
immediately implies this subclass of representations satisfy the smoothing
property. It would be interesting to subsume their extension result under our
scope. More specifically:

######  Question 4.14.

Can we prove quasi-smoothing for the translation length for a subclass of
$[P]$-Anosov representations, as in [MZ19, Definition 2.25], directly from the
definition of translation length?

###  4.8. Extremal length

We now turn to curve functionals that satisfy only convex union and not
additive union, starting with the original motivation for this work, extremal
length.

######  Definition 4.15.

Fix $\Sigma$ a Riemann surface with a metric $g$. Let $C=\bigcup t_{i}C_{i}$
be a weighted multi-curve on $\Sigma$. For
$\rho\colon\Sigma\to\mathbb{R}_{\geq 0}$ a measurable rescaling function, the
_area_ of $\rho$ is

$\operatorname{Area}(\rho g)\coloneqq\int_{x\in\Sigma}\rho(x)^{2}\mu_{g}(x),$

where $\mu_{g}$ is the Lebesgue measure of $g$. The _length_ of $C$ is

$\ell_{\rho g}(C)\coloneqq\inf_{\gamma\in
C}\sum_{i}t_{i}\int_{x\in\gamma_{i}}\rho(x)\,dx,$

where $dx$ is measured with respect to $g$ arc-length, and the infimum runs
overall all representatives $\gamma=\bigcup_{i}\gamma_{i}$ of $C$, where
$\gamma_{i}$ is a representative of the component $C_{i}$ of $C$. When $\rho$
is continuous, $\ell_{\rho}(C)$ is the length with respect to the metric $g$
rescaled by $\rho$. The square root of the _extremal length_ of $C$ is

$\sqrt{\operatorname{EL}}(C)\coloneqq\sup_{\rho}\frac{\ell_{\rho
g}(C)}{\sqrt{\operatorname{Area}(\rho g)}}.$

Observe that the supremand is unchanged under multiplying $\rho$ by a positive
constant. It is a standard result that the supremum is realized by some
generalized metric (not necessarily Riemannian) [Rod74, Theorem 12] and that,
when $C$ is a simple multi-curve, the optimum metric $\rho g$ is the cone
Euclidean metric associated to a quadratic differential [Jen57]. Very little
is known about the optimum metric when $C$ is not simple, except in special
cases [WZ94, Cal96, HZ20, NZ19].

######  Lemma 4.16.

As a function of $C$ with fixed $\Sigma$, $\sqrt{\operatorname{EL}}$ satisfies
homogeneity, stability, and smoothing.

###### Proof.

This follows since $\ell_{\rho g}$ satisfies these properties for each $\rho$;
here is the argument for smoothing.

Let $C$ be a multi-curve with an essential crossing, and let $C^{\prime}$ be
the curve obtained by smoothing at the crossing. Then, for any scaling
function $\rho$,

$\frac{\ell_{\rho g}(C^{\prime})}{\sqrt{\operatorname{Area}(\rho
g)}}\leq\frac{\ell_{\rho g}(C)}{\sqrt{\operatorname{Area}(\rho g)}}.$

Since $\sqrt{\operatorname{EL}}(C^{\prime})$ and $\sqrt{\operatorname{EL}(C)}$
are the suprema of such terms, the result follows. ∎

Convex union is different, since on one side of the inequality we have a sum
of values of $\sqrt{\operatorname{EL}}$. (Extremal length does not satisfy
additivity.)

######  Lemma 4.17.

$\sqrt{\operatorname{EL}}$ satisfies convex union.

###### Proof.

Fix a curve split as a union $C=C_{1}\cup C_{2}$, and let
$\rho\colon\Sigma\to\mathbb{R}$ be the function realizing the supremum in the
definition of extremal length for $C$. Then

$\displaystyle\sqrt{\operatorname{EL}}(C_{1}\cup C_{2})$
$\displaystyle=\frac{\ell_{\rho g}(C_{1})}{\sqrt{\operatorname{Area}(\rho
g)}}+\frac{\ell_{\rho g}(C_{2})}{\sqrt{\operatorname{Area}(\rho g)}}$
$\displaystyle\leq\sqrt{\operatorname{EL}}(C_{1})+\sqrt{\operatorname{EL}}(C_{2}),$

where the last inequality holds by the supremum in the definition of
$\operatorname{EL}$. ∎

Thus, by Theorem A, $\sqrt{\operatorname{EL}}$ (and $\operatorname{EL}$)
extend uniquely to continuous functions on geodesic currents. With this
extension, we propose the following conjecture.

######  Conjecture 4.18.

For some universal constant $C$,

$\operatorname{EL}_{\Sigma}(\mathcal{L}_{\Sigma})=C\operatorname{Area}(\Sigma).$

where $\mathcal{L}_{\Sigma}$ is the Liouville current (compare Subsection
4.2).

######  Remark 4.19.

For the extremal length _without_ the square root, we instead have
inequalities

$\operatorname{EL}(C_{1})+\operatorname{EL}(C_{2})\leq\operatorname{EL}(C_{1}\cup
C_{2})\leq 2(\operatorname{EL}(C_{1})+\operatorname{EL}(C_{2})).$

The second inequality is a simple consequence of Lemma 4.17. To see the first
inequality, take optimal rescaling functions $\rho_{i}$ for
$\operatorname{EL}(C_{i})$, normalized so that
$\ell_{\rho_{i}g}(C_{i})=\operatorname{Area}(\rho_{i}g)=\operatorname{EL}(C_{i})$.
Then using $\rho_{1}+\rho_{2}$ as the test function for
$\operatorname{EL}(C_{1}\cup C_{2})$ gives the desired inequality after
elementary manipulations.

###  4.9. Extremal length with respect to elastic graphs

There is a parallel notion of extremal length with respect to elastic graphs
[Thu19], just as there is for ordinary lengths (Section 4.4).

An _elastic graph_ $(\Gamma,\alpha)$ is a 1-dimensional CW complex $\Gamma$
(i.e., allowing multiple edges and loops) together with an assignment of
positive real numbers $\alpha(e)$ for each $e\in\operatorname{Edge}(\Gamma)$,
where the _edges_ are the 1-dimensional cells of $\Gamma$.

By a concrete _multi-curve $\gamma$ on $\Gamma$_ we mean a 1-manifold
$X(\gamma)$ and a PL map $\gamma\colon X(\gamma)\to\Gamma$. Given a scaling
function $\rho\colon\operatorname{Edge}(\Gamma)\to\mathbb{R}_{\geq 0}$, the
length metric $\rho\alpha$ on $\Gamma$ gives edge $e$ the length
$\rho(e)\alpha(e)$. We define the _length_ of $\gamma$ as

$\ell_{\rho\alpha}(\gamma)\coloneqq\sum_{e\in\operatorname{Edge}(\Gamma)}n_{\gamma}(e)\rho(e)\alpha(e),$

where $n_{\gamma}(e)$ is the weighted number of times that $\gamma$ runs over
$e$. We can likewise define the length of a multi-curve $D$ on $\Gamma$ as the
infimum over of concrete multi-curves in $D$.

The _area_ of $\Gamma$ with respect to $\rho\alpha$ is defined to be

$\operatorname{Area}_{\rho}(\Gamma,\alpha)\coloneqq\sum_{e\in\operatorname{Edge}(\Gamma)}\rho(e)^{2}\alpha(e).$

Intuitively, each edge is turned into a rectangle of width $\rho(e)$, aspect
ratio $\alpha(e)$, and thus area $\rho(e)^{2}\alpha(e)$.

As for extremal length for surfaces, we define the square root of extremal
length of a multi-curve on $\Gamma$ by

(4.20)
$\sqrt{\operatorname{EL}}(D;\Gamma,\alpha)\coloneqq\sup_{\rho:\operatorname{Edge}(\Gamma)\to\mathbb{R}_{\geq
0}}\frac{\ell_{\rho\alpha}(D)}{\sqrt{\operatorname{Area}_{\rho}(\Gamma,\alpha)}}.$

It is easy to do this optimization. We get a more interesting quantity by
incorporating a filling embedding $\iota\colon\Gamma\hookrightarrow S$ of
$\Gamma$ in a surface $S$, i.e., an embedding $\iota$ that is
$\pi_{1}$-surjective. Then, for a multi-curve $C$ on $S$ and scaling $\rho$,
the length is defined as in Section 4.4:

$\ell_{\rho\alpha;\iota}(C)\coloneqq\inf_{\begin{subarray}{c}D\textrm{ on
}\Gamma\\\ \iota_{*}(D)=C\end{subarray}}\ell_{\rho\alpha}(D).$

(The “filling” condition guarantees that there are such multi-curves $D$ with
$\iota_{*}(D)=C$.) We can then define a version of extremal length, following
Equation (4.20), with respect to $\iota$:

(4.21)
$\sqrt{\operatorname{EL}}(C;\Gamma,\alpha,\iota)\coloneqq\sup_{\rho:\operatorname{Edge}(\Gamma)\to\mathbb{R}_{\geq
0}}\frac{\ell_{\rho\alpha;\iota}(D)}{\sqrt{\operatorname{Area}_{\rho}(\Gamma,\alpha)}}.$

######  Proposition 4.22.

For $\iota\colon\Gamma\to\Sigma$ a filling embedding,
$\sqrt{\operatorname{EL}}(C;\Gamma,\alpha,\iota)$ satisfies the convex union,
stability, homogeneity, and smoothing properties.

###### Proof.

As in Lemma 4.16, smoothing, stability, and homogeneity follow because
$\ell_{\rho\alpha}(C)$ satisfies them for any $\rho$. Convex union follows as
in Lemma 4.17. ∎

We can also consider extremal length with respect to an immersion $\iota$
(rather than an embedding), defined in the same way.

######  Proposition 4.23.

For $\iota\colon\Gamma\to\Sigma$ a $\pi_{1}$-surjective immersion,
$\sqrt{\operatorname{EL}}(C;\Gamma,\alpha,\iota)$ satisfies the convex union,
homogeneity, and quasi-smoothing properties.

###### Proof.

Convex union and homogeneity still hold by the same argument. We need an extra
argument for quasi-smoothing. Instead of taking the supremum over all $\rho$,
rewrite Equation (4.21) as

$\sqrt{\operatorname{EL}}(C;\Gamma,\alpha,\iota)=\sup_{\begin{subarray}{c}\rho:\operatorname{Edge}(\Gamma)\to\mathbb{R}_{\geq
0}\\\
\operatorname{Area}_{\rho}(\Gamma,\alpha)=1\end{subarray}}\ell_{\rho\alpha}(C).$

The immersed graph $\iota(\Gamma)$ has finitely many self-intersections. For
each self-intersection $x$, take the supremum over the compact set of metrics
$\\{\rho\mid\operatorname{Area}_{\rho}(\Gamma,\alpha)=1\\}$ of the diameter
$\operatorname{diam}_{\iota}(x)$ defined in Equation (4.7). By the replacement
argument in Proposition 4.5, all $\ell_{\rho\alpha}$ for $\rho$ in this set
satisfy quasi-smoothing with a uniform quasi-smoothing constant. It follows
that $\operatorname{EL}(C;\Gamma,\alpha,\iota)$ also satisfies quasi-smoothing
with the same constant. ∎

######  Remark 4.24.

We can relate extremal length for elastic graphs and surfaces by making a
choice of a ribbon structure to $\Gamma$. Given any $\varepsilon>0$, a ribbon
elastic graph $G$ can be thickened into a conformal surface with boundary
$N_{\varepsilon}(G)$ by replacing each edge $e$ of $G$ by a rectangle of size
$\alpha(e)\times\varepsilon$ and gluing the rectangles at the vertices by
using the given ribbon structure. There are then inequalities relating
$\operatorname{EL}(C;\Gamma,\alpha)$ and
$\varepsilon\operatorname{EL}(C;N_{\varepsilon}(G))$, to within a
multiplicative factor [Thu20, Props. 4.8 and 4.9]. For graphs immersed or
embedded in a surface, the situation is less clear. By suitably choosing the
elastic weights on an embedded graph, it appears that one can approximate
extremal length well; Palmer gives one approach [Pal15]. We are not aware of
precise theorems.

###  4.10. $p$-extremal-length with respect to immersed graphs

Extremal length fits into a family of energies for graphs [Thu19, Appendix A].
For $\Gamma$ a metric graph with metric $g$, a constant $p$ with $1\leq
p\leq\infty$, and $C$ a curve on $\Gamma$, define

(4.25)
$E_{p}(C;\Gamma,g)\coloneqq\\!\sup_{\sigma:\operatorname{Edge}(\Gamma)\to\mathbb{R}_{\geq
0}}\frac{\ell(C;\sigma g)}{\lVert\sigma\rVert_{p}}$

where the $L^{p}$ norm $\lVert\sigma\rVert_{p}$ is taken with respect to the
metric $g$. As in the previous section, we can also consider a
$\pi_{1}$-surjective immersion $\iota\colon\Gamma\to S$, and consider $C$ to
be a curve on $S$ rather than on $\Gamma$.

For $p=\infty$, $E_{\infty}(C)$ is in fact just the length with respect to $g$
(as in Section 4.5). Indeed, let $\sigma$ be any scaling factor, and let
$\gamma$ be the shortest representative of $C$ on $\Gamma$ with respect to $g$
(not with respect to $\sigma g$). Then

$\ell(C;\sigma g)\leq\ell(\gamma;\sigma
g)\leq\lVert\sigma\rVert_{\infty}\ell(\gamma;g)$

from which the result easily follows.

######  Proposition 4.26.

For any $\pi_{1}$-surjective immersion $\iota\colon\Gamma\to S$, the curve
functional $E_{p}(\cdot;\allowbreak\Gamma,g,\iota)$ satisfies convex union,
homogeneity, and quasi-smoothing, and thus its stabilization extends
continuously to a function on geodesic currents. If $\iota$ is a filling
embedding, then $E_{p}$ in addition satisfies stability and smoothing.

###### Proof.

This follows as in Propositions 4.22 and 4.23. To prove quasi-smoothing in the
immersed case, we restrict to those functions $\sigma$ on
$\operatorname{Edge}(\Gamma)$ where $\lVert\sigma\rVert_{p}=1$; as in the
proof of Proposition 4.23, this set is compact. ∎

##  5\. Counting problems

One direct application of Theorem A is to obtain new counting results for
curves on surfaces of a given topological type.

A _filling current_ is a geodesic current
$\alpha\in\mathcal{G}\mathcal{C}(\Sigma)$ so that $i(\alpha,\mu)>0$ for all
$\mu\in\mathcal{G}\mathcal{C}(\Sigma)\backslash\\{0\\}$. One example is a
filling multi-curve, one whose complement in $S$ consists of disks.

Rafi and Souto proved the following.

######  Definition 5.1.

A function $f$ on currents is _positive_ if $f(\mu)>0$ for all $\mu\neq 0$.

For a fixed continuous, homogeneous, and positive function
$f\colon\mathcal{GC}\to\mathbb{R}_{+}$, $\alpha$ a current, and $L$ a positive
real number, let

$N(f,\alpha,L)\coloneqq\\#\\{\phi\in\operatorname{\mathit{MCG}}\mid
f(\phi(\alpha))\leq L\\}.$

######  Theorem 5.2 (Rafi-Souto [RS19, Main Theorem]).

For a fixed continuous, homogeneous, and positive function
$f\colon\mathcal{GC}\to\mathbb{R}_{+}$ and for a fixed filling current
$\alpha\in\mathcal{G}\mathcal{C}(\Sigma)$, the limit

$\lim_{L\to\infty}\frac{N(f,\alpha,L)}{L^{6g-6}}$

exists and is equal to

$\frac{m(f)m(\alpha)}{\mathfrak{m}_{g}}$

where $m(f)$, $m(\alpha)$, and $\mathfrak{m}_{g}$ are constants depending only
on $f$, $\alpha$, and the genus $g$ respectively:

(5.3) $\displaystyle m(f)$
$\displaystyle=\mu_{\mathrm{Thu}}(\\{\lambda\in\mathcal{ML}\mid f(\lambda)\leq
1\\})$ (5.4) $\displaystyle m(\alpha)$
$\displaystyle=\mu_{\mathrm{Thu}}(\\{\lambda\in\mathcal{ML}\mid
i(\alpha,\lambda)\leq 1\\})$ (5.5) $\displaystyle\mathfrak{m}_{g}$
$\displaystyle=\int_{\mathcal{M}_{g}}\\!m(Y)\,d\omega_{\mathrm{WP}}(Y).$

Here $\mu_{\mathrm{Thu}}$ is the Thurston volume measure on $\mathcal{ML}$
induced by the symplectic pairing on $\mathcal{ML}$.111There are at least two
natural normalizations for volume on measured foliations; the volume here is
the one induced from the symplectic structure.

Dumas communicates a proof of the following theorem (attributed to
Mirzakhani).

######  Theorem 5.6 (Dumas-Mirzakhani [Dum15, Theorem 5.10]).

The function $\Lambda\colon\mathcal{M}_{g}\to\mathbb{R}_{\geq 0}$ given by
$\Sigma\mapsto m(\operatorname{EL}_{\Sigma})$ is constant, where
$m(\operatorname{EL}_{\Sigma})$ is defined by Eq. (5.3).

Let $\sqrt{\operatorname{EL}_{\Sigma}}\colon\mathcal{GC}\to\mathbb{R}_{\geq
0}$ be the continuous extension of square-root of extremal length to currents
provided by Theorem A. In order to be able to apply Theorem 1, it remains to
check that square root of extremal length is non-zero.

######  Lemma 5.7.

For any $\Sigma\in\operatorname{Teich}(S)$, the functional
$\sqrt{\operatorname{EL}_{\Sigma}}$ is positive on $\mathcal{GC}(\Sigma)$.

###### Proof.

Let $A=\sqrt{-2\pi\chi(S)}$. By Definition 4.15 applied to the hyperbolic
metric, $\ell_{\Sigma}(C)/A\leq\sqrt{\operatorname{EL}_{\Sigma}}(C)$ for all
curves $C$ and all $\Sigma\in\operatorname{Teich}(S)$. For any
$\mu\in\mathcal{GC}(\Sigma)$, there exists a sequence
$(\lambda_{i}C_{i})_{i\in\mathbb{N}}$ of weighted curves so that
$\lambda_{i}C_{i}\to\mu$ in the weak∗ sense. It thus follows that
$\ell_{\Sigma}(\mu)/A\leq\sqrt{\operatorname{EL}_{\Sigma}}(\mu)$. Since
$\ell_{\Sigma}$ is a positive function on currents,
$\sqrt{\operatorname{EL}_{\Sigma}}$ is as well. ∎

We thus get solutions to counting problems for extremal length.

######  Corollary 5.8.

For any filling current $\alpha$ and ${\Sigma}\in\operatorname{Teich}(S)$, the
limit

$\lim_{L\to\infty}\frac{N(\sqrt{\operatorname{EL}_{\Sigma}},\alpha,L)}{L^{6g-6}}$

exists, is independent of $\Sigma$, and is equal to

(5.9) $\frac{\Lambda\cdot m(\alpha)}{\mathfrak{m}_{g}}.$

###### Proof.

Using Theorem A and the results in Subsection 4.8, we can extend
$\sqrt{EL_{X}}$ as a continuous, real-homogeneous functional on geodesic
currents. Furthermore, $\sqrt{EL_{X}}$ is positive on currents, by Proposition
5.7. Thus, by Theorem 1, the result follows. Independence of $\Sigma$ follows
from Theorem 5.6. ∎

A similar counting result is true for $\alpha$ a simple multi-curve by
Mirzakhani’s work, but one has to count slightly differently. For $f$ a curve
functional, $\alpha$ a simple curve, and $L>0$, set

$n(f,\alpha,L)\coloneqq\\#\\{\phi(\alpha)\in\operatorname{\mathit{MCG}}\mid
f(\phi(\alpha))\leq L\\}.$

In general, note that $n(f,\alpha,L)\neq N(f,\alpha,L)$; in fact, $N$ will be
infinite if $\alpha$ is not filling. Even for $\alpha$ filling, $N$ will be
bigger than $n$ if $\alpha$ has non-trivial stabilizer in the mapping class
group.

We state the corresponding result for simple multi-curves.

######  Proposition 5.10.

For any simple multi-curve $\alpha$, there is a constant $c(\alpha)$ so that,
for any $f\colon\mathcal{GC}^{+}(S)\to\mathbb{R}$ continuous, positive and
real-homogeneous function, the limit

(5.11) $\lim_{L\to\infty}\frac{n(f,\alpha,L)}{L^{6g-6}}$

exists and is equal to

(5.12) $\frac{m(f)c(\alpha)}{\mathfrak{m}_{g}}.$

###### Proof.

First note that the set $\\{\lambda\in\mathcal{ML}(\Sigma)\mid f(\lambda)\leq
1\\}$ is compact because $f$ is positive on non-zero measured laminations. Let
$A\coloneqq\\{\lambda\in\mathcal{ML}\mid f(\lambda)=1\\})$; then
$\mu_{\mathrm{Thu}}(A)=0$, as proved by Rafi and Souto [RS19, p. 879].
Finally, we apply Mirzakhani’s counting result [Mir08, Theorem 1.3] and the
Portmanteau theorem (see [Bau01, Theorem 30.12]) to conclude the limit exists.
∎

######  Corollary 5.13.

For any simple multi-curve $\alpha$ and ${\Sigma}\in\operatorname{Teich}(S)$,
the limit

$\lim_{L\to\infty}\frac{n(\sqrt{\operatorname{EL}_{\Sigma}},\alpha,L)}{L^{6g-6}}$

exists, is independent of $\Sigma$, and is equal to

(5.14) $\frac{\Lambda\cdot c(\alpha)}{\mathfrak{m}_{g}}.$

######  Remark 5.15.

The constant $c(\alpha)$ in Proposition 5.13 is not the same as the constant
$m(\alpha)$ in Theorem 1. For details on how $c(\alpha)$ is defined, see
[Mir08, Equation (1.2)]. This is related to the fact that in Proposition 5.13
we count multi-curves instead of mapping classes because the stabilizer of a
simple multi-curve under the mapping class group is infinite.

The counting problem $n(\ell_{\Sigma},\alpha,L)$ with $\alpha$ an arbitrary
essential multi-curve (possibly neither simple nor filling) and
$f=\ell_{\Sigma}$ a hyperbolic length, is established in more recent work of
Mirzakhani [Mir16, Theorem 1.1]. Relying on her work, Erlandsson-Parlier-Souto
[EPS20, Theorem 1.6] give the corresponding result where $f$ is allowed to be
intersection number with other filling currents (not just a hyperbolic
Liouville current). From Mirzakhani’s work and work of Erlandsson-Souto [ES16,
Corollary 4.4], one also can get the corresponding counting problems where
$\alpha$ is allowed to be a current (not just a multi-curve). In fact, from
Erlandsson-Souto’s work one can also allow $f$ to be any continuous, positive
and real-homogeneous function on currents, although they don’t explicitly
state this in their paper. This is done in Rafi-Souto’s work (Theorem 1 above)
which also gives the expression (5.9) for the limit of the counting problem.
Rafi-Souto also relies on Mirzakhani’s work [Mir16].

Finally, recently Erlandsson-Souto [ES20, Theorem 8.1] have given an
independent proof of the counting argument in [Mir16] illuminating the
connection between counting problems for simple and non-simple multi-curves.
Corollary 5.9 also follows from Proposition 5.13 and forthcoming work of
Erlandsson and Souto [ES19, ES20].

We remark that Erlandsson-Souto’s work shows that if one knows a counting
result for simple closed curves, then one can obtain a counting result for
non-simple closed curves (for curve functionals extending continuously to
currents). The connection between these two types of counting problems is
perhaps that, in some sense, the simple closed curves are the extremal points
of the space of currents, in the sense of convex sets [Roc70]. For instance,
the systole of positive curve functional satisfying smoothing and convex union
is always a simple curve. (Here by _systole_ we mean a weight $1$, non-trivial
multi-curve $C$ with a minimal value of $f(C)$.)

##  6\. Proof outline

In this section, we prove the core theorem of the paper, Theorem 6.1, giving a
continuous extension to geodesic currents of a functional $f$ on weighted
multi-curves satisfying convex union, stability, homogeneity and weighted
quasi-smoothing.

######  Theorem 6.1.

Let $f$ be a weighted curve functional defined on weighted oriented multi-
curves satisfying the weighted quasi-smoothing, convex union, stability, and
homogeneity properties. Then there is a unique continuous homogeneous function
$\bar{f}\colon\mathcal{GC}^{+}(S)\to\mathbb{R}_{\geq 0}$ that extends $f$.

###### Proof of Theorem 6.1.

The proof proceeds by studying the geodesic flow on the unit tangent bundle to
$S$ (with respect to an arbitrary hyperbolic metric), picking a suitable
global cross-section with boundary $\tau$, and looking at a “smeared first
return map” to $\tau$.

The proof breaks up into the following steps.

  * •

Step 1: In Section 7, we define the cross-sections we consider, though we
delay proving existence. We introduce _bump functions_ and the associated
_smeared first return map_ (Def. 7.4); indeed, there are several varieties of
return maps (Table 2). The main advantage of smeared return maps is that they
are continuous (Prop. 7.7). We use these smeared returns to define our
purported extension $f_{\tau}$ to geodesic currents as a limit (Def. 7.25),
assuming a suitable global cross-section $\tau$ exists.

  * •

Step 2: In Section 8, we find a suitable “good” $\tau$, defined in Definition
8.4, by considering certain _wedge subsets_ (Def. 8.1) of $UT\Sigma$, based at
a collection of geodesic sub-segments of a closed geodesic $\delta$ (see Prop.
8.3).

  * •

Step 3: In Section 9, Proposition 9.6, we show the limit defining $f_{\tau}$
exists, by using convex union, quasi-smoothing, stability and homogeneity
assumptions on $f$, and applying a version of Fekete’s Lemma (Lemma 9.5).

  * •

Step 4: In Section 10, Proposition 10.8, we show that $f_{\tau}$ is a
continuous function on the space of oriented geodesic currents. We do this by
showing that the iterates $f^{k}_{\tau}$ used in the definition of $f_{\tau}$
are continuous for each $k$, using the continuity of the smeared homotopy
return map from Step 1, and $f$ is a continuous function of the weights of a
fixed multi-curve (Proposition 3.5). Propositions 9.4 and 10.1 give enough
control to ensure uniform convergence of the iterates $f^{k}_{\tau}$ to
$f_{\tau}$ (see Lemma 10.2), and thus continuity of $f_{\tau}$ follows.

  * •

Step 5: In Section 11, Proposition 11.5 we show that $f_{\tau}$ extends $f$
for oriented curves, by analyzing the image of the smeared return map in that
case, and mixing-and-matching the components of the multi-curve.

The definition of $f_{\tau}$ depends on many choices: the hyperbolic metric on
$S$, a choice of global cross-section $\tau$, and in fact nested cross-
sections $\tau_{0}\subset\tau\subset\tau^{\prime}$ and a choice of bump
function $\psi$ on $\tau$ (see Definition 7.1). Different choices yield, a
priori, different extensions $f_{\tau}$. But we have proved that $f_{\tau}$ is
a continuous function on the space of geodesic currents, and moreover it
restricts to $f$ on multi-curves. Since weighted multi-curves are a dense
subset of the space of geodesic currents (see Section 2.7), the extension
doesn’t depend on these choices. This proves Theorem 6.1. ∎

Now, we prove Theorem A.

###### Proof of Theorem A.

By Proposition 3.6, $f$ extends uniquely to a weighted curve functional
satisfying convex union, homogeneity, stability, and weighted quasi-smoothing
with the same constant. Then, by Theorem 6.1, the theorem follows. ∎

##  7\. Defining the extension

We now turn to the proof of Theorem A. As mentioned above, we will fix a
hyperbolic structure $\Sigma$ on $S$ (with no relation to the curve functional
$f$) and use the geodesic flow $\phi_{t}$ on the unit tangent bundle to define
the extension to currents.

In this section we will deal with _return maps_ for this flow. After some
generalities about return maps for cross-sections with boundary, we introduce
a _smeared return map_ that is continuous. We also define a _homotopy return
map_ (and a smeared version of it) that keeps track of homotopy classes of
closures of trajectories. Then we use the smeared homotopy return map to
define the extension $f_{\tau}$ of $f$. The several variants of the return map
are summarized in Table 2.

Notation | Type | Meaning
---|---|---
$p$ |  $\tau$$\mskip 5.0mu plus 5.0mu\to\mskip 5.0mu plus 5.0mu$ | $\tau$ | Ordinary first return
$P$ |  $\tau$$\mskip 5.0mu plus 5.0mu\to\mskip 5.0mu plus 5.0mu$ | $\mathbb{R}_{1}\tau$ | Smeared first return
$q$ |  $\tau$$\mskip 5.0mu plus 5.0mu\to\mskip 5.0mu plus 5.0mu$ | $\tau\times\pi_{1}(M)$ | Complete homotopy return
$Q$ |  $\tau$$\mskip 5.0mu plus 5.0mu\to\mskip 5.0mu plus 5.0mu$ | $\mathbb{R}_{1}(\tau\times\pi_{1}(M))$ | Smeared homotopy return
$m$ |  $\tau$$\mskip 5.0mu plus 5.0mu\to\mskip 5.0mu plus 5.0mu$ | $\pi_{1}(M)$ | Return curve (projection of $q$)
$[m]$ |  $\tau$$\mskip 5.0mu plus 5.0mu\to\mskip 5.0mu plus 5.0mu$ | $\mathcal{C}^{+}(S)$ | Conjugacy class of projection of $m$ to $S$
$M$ |  $\tau$$\mskip 5.0mu plus 5.0mu\to\mskip 5.0mu plus 5.0mu$ | $\mathbb{R}_{1}\pi_{1}(M)$ | Smeared return curve (projection of $Q$)
$[M]$ |  $\tau$$\mskip 5.0mu plus 5.0mu\to\mskip 5.0mu plus 5.0mu$ | $\mathbb{R}_{1}\mathcal{C}^{+}(S)$ |
$R$ |  $\mathcal{GC}^{+}(S)$$\mskip 5.0mu plus 5.0mu\to\mskip 5.0mu plus 5.0mu$ | $\mathbb{R}\mathcal{C}^{+}(S)$ | Integral of $[M]$
Table 2. Various types of return maps. $M=UT\Sigma$ is the domain of the flow
$\phi_{t}$.

###  7.1. Smeared first return map

Let $Y$ be a smooth closed manifold with a smooth flow $\phi_{t}$. For us, a
_cross-section_ is a compact smooth codimension $1$ submanifold-with-boundary
$\tau$ that is smoothly transverse to the foliation of $Y$ given by
$\phi_{t}$. A _global cross-section_ is a cross-section $\tau$ so that its
interior $\tau^{\circ}$ intersects all forward and backward orbits: for all
$x\in Y$, there exists $s<0$ and $t>0$ with
$\phi_{s}(x),\phi_{t}(x)\in\tau^{\circ}$. Any flow on a compact manifold has
global cross-section consisting of a union of finitely many disks, although
not all flows on compact manifolds admit a global cross-section consisting of
a single connected component. (For instance, the Reeb foliation on
$\mathbb{T}^{2}$ has no connected global cross-section.) By the implicit
function theorem, for any cross-section $\tau$ there is a larger cross section
$\tau^{\prime}$ with $\partial\tau\subset\tau^{\prime}{}^{\circ}$, which we
also write $\tau\Subset\tau^{\prime}$.

Let $t_{\tau}\colon Y\to\mathbb{R}$ be the first return time defined by
$t_{\tau}(x)\coloneqq\min\\{t>0\mid\phi_{t}(x)\in\tau\\}$, and let
$p_{\tau}(x)\coloneqq\phi_{t_{\tau}(x)}(x)$. Then $p_{\tau}$ (restricted to
$\tau$) is the _first return map_ associated to the cross-section $\tau$. We
will omit the subscript on $p_{\tau}$ if it is clear from context. We also
have the first return time to the interior, denoted $t_{\tau}^{\circ}$.
(Recall we assume $t_{\tau}^{\circ}(x)$ is finite.)

If $\tau$ has no boundary, then $p$ is a homeomorphism. On the other hand, if
the cross-section has a non-invariant boundary (i.e.,
$p(\partial\tau)\neq\partial\tau$) then $t_{\tau}$ and $p$ will have
discontinuities. This necessarily happens for the geodesic flow on the unit
tangent bundle of a hyperbolic surface. See [CD16, Sec. 1] for a justification
and examples of global cross-sections with boundary for this flow; we
construct our own cross-section in Section 8. However, by the continuity of
$\phi_{t}$ with respect to initial parameters, we have the following “local
continuity” claim.

######  Lemma 7.1.

Let $\tau_{1},\tau_{2}$ be cross-sections (not necessarily global) of $\phi$.
Let $x_{1}\in\tau_{1}^{\circ}$, and suppose we are given $t>0$ so
$x_{2}=\phi_{t}(x_{1})\in\tau_{2}^{\circ}$. Then there exists a neighborhood
$U_{1}$ of $x_{1}$ in $\tau_{1}$, a neighborhood $U_{2}$ of $x_{2}$ in
$\tau_{2}$, and a continuous function $t^{1}_{2}\colon
U_{1}\to\mathbb{R}_{>0}$ so that for $x\in U_{1}$, $\phi_{t^{1}_{2}}(x)\in
U_{2}$. Furthermore, $\phi_{t^{1}_{2}}\colon U_{1}\to U_{2}$ is a
diffeomorphism. If $t$ is the first return to $\tau_{2}$, then we can choose
$U_{1}$ and $U_{2}$ so that $t^{1}_{2}(x)=t_{\tau_{2}}(x)$ is also the first
return time.

This is presumably standard (Basener gives this as “a useful technical lemma,
the proof of which is trivial” [Bas02, Lem. 1]), but we give a proof for
completeness.

###### Proof.

Pick an initial neighborhood $U_{1}^{\prime}$ of $x_{1}$ in $\tau$, and let
$\varepsilon$ be small enough so that
$V_{1}\coloneqq\phi_{(-\varepsilon,\varepsilon)}(U_{1}^{\prime})$ is a
3-dimensional flow-box neighborhood of $x_{1}$ in $Y$. Then the restriction of
$\phi_{t}$ to $V_{1}$ is a homeomorphism to a neighborhood $V_{2}$ of $x_{2}$
in $Y$. Set $U_{2}^{\prime}\coloneqq V_{2}\cap\tau_{2}$. Now consider the
composition $\psi\coloneqq\pi_{1}\circ\phi_{-t}\circ\iota_{2}\colon
U_{2}^{\prime}\to U_{1}^{\prime}$, where $\iota_{i}\colon
U_{i}^{\prime}\hookrightarrow V_{i}$ is the inclusion and $\pi_{1}\colon
V_{1}\to U_{1}$ is the flow projection:

$U_{1}^{\prime}$$V_{1}$$V_{2}$$U_{2}^{\prime}$$\tau_{1}$$\tau_{2}$.$\scriptstyle{\iota_{1}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\phi_{t}}$$\scriptstyle{\cong}$$\scriptstyle{\iota_{2}}$$\scriptstyle{\psi}$

Then $\psi$ is a map from $U_{2}^{\prime}$ to $U_{1}^{\prime}$, taking $x_{2}$
to $x_{1}$. By transversality of $\tau_{1}$ and $\tau_{2}$, the differential
of $\psi$ at $x_{2}$ is invertible. Thus by the inverse function theorem,
there is a neighborhood $U_{2}$ of $x_{2}$ and $U_{1}$ of $x_{1}$ so that the
restriction of $\psi$ is a diffeomorphism from $U_{2}$ to $U_{1}$. For $x\in
U_{1}$, set $t^{1}_{2}(x)\coloneqq t+\pi_{t}(\phi_{-t}(\psi^{-1}(x)))$, where
$\pi_{t}\colon V_{1}\to(-\varepsilon,\varepsilon)$ is the projection onto the
time coordinate of the flow box. We have $\phi_{t^{1}_{2}}(x)=\psi^{-1}(x)\in
U_{1}$, as desired for the first claim.

For the second claim, by hypothesis, the compact sets $\phi_{[0,t]}(x_{1})$
and $\tau_{2}$ do not intersect, so $\phi_{[0,t]}(x_{1})$ has an open
neighborhood that does not intersect $\tau_{2}$. It follows that we can shrink
$U_{1}$ and $U_{2}$ so that $\psi^{-1}$ restricted to $U_{1}$ is the first
return map to $\tau_{2}$. ∎

######  Lemma 7.2.

Let $x\in\tau$. If $p(x)\in\tau^{\circ}$, then $p_{\tau}$ and $t_{\tau}$ are
continuous in a neighborhood of $x$ in $\tau$.

###### Proof.

Let $\tau^{\prime}\Supset\tau$ be a slightly enlarged global cross-section (to
cover cases when $x\in\partial\tau$). By Lemma 7.1, there exists a
neighborhood $U$ of $x$ in $\tau^{\prime}$ such that
$p_{\tau}(U)\subset\tau^{\circ}$. By taking $V\coloneqq U\cap\tau$, we get the
desired neighborhood in $\tau$. ∎

######  Lemma 7.3.

Let $\tau$ be a global cross-section. Then, on $\tau$, $t_{\tau}$ is lower
semi-continuous and $t_{\tau}^{\circ}$ is upper semi-continuous. There are
thus positive global upper and lower bounds on $t_{\tau}$.

###### Proof.

Fix $x\in\tau$. If $p_{\tau}(x)\in\tau^{\circ}$, then $t_{\tau}$ is continuous
at $x$ by Lemma 7.2. Otherwise, find a cross-section $\tau^{\prime}$ with
$\tau\Subset\tau^{\prime}$. Then $t_{\tau^{\prime}}$ is continuous at $x$, and
since $t_{\tau}(y)\geq t_{\tau^{\prime}}(y)$ we have proved that $t_{\tau}$ is
lower semi-continuous.

On the other hand, for any $x_{1}\in\tau$, we can set
$x_{2}=p_{\tau}^{\circ}(x_{1})$ and find a neighborhood $U_{1}$ of $x_{1}$ in
$\tau$ with a function $t^{1}_{2}$ as in Lemma 7.1. But then
$t^{\circ}_{\tau}(x)\leq t^{1}_{2}(x)$ for $x\in U_{1}$. (Note that $x_{2}$ is
the first return point to $\tau^{\circ}$, not to $\tau$, so we cannot conclude
that $t_{\tau}^{\circ}$ is continuous.) ∎

In the $C^{1}$ setting, Basener showed that a global cross-section can be
perturbed slightly so that the first return map is piecewise continuous with a
cellular structure [Bas04]. However, we want a _continuous_ version of the
first return map, so we proceed in a different direction.

######  Definition 7.4.

Fix a nested pair of global cross-sections $\tau_{0}\Subset\tau$. A _bump
function_ $\psi$ for this pair is a continuous function
$\psi\colon\tau\to[0,1]$ so that $\psi$ is $1$ on $\tau_{0}$ and $0$ on an
open neighborhood of $\partial\tau$. Set $\bar{\psi}(x)=1-\psi(x)$. Let
$p\colon\tau\to\tau$ be the first return map with respect to $\tau$. Then the
_smeared first return map_ of $\psi$ is a function
$P_{\psi}\colon\tau\to\mathbb{R}_{1}\tau$ defined by

$P_{\psi}(x)\coloneqq\begin{cases}p(x)&p(x)\in\tau_{0}\\\ \psi(p(x))\cdot
p(x)+\bar{\psi}(p(x))\cdot P_{\psi}(p(x))&p(x)\in\tau-\tau_{0}.\end{cases}$

Here, $\mathbb{R}_{1}\tau\subset\mathcal{M}_{1}(\tau)$ is the subspace of
measures with finite support and total mass $1$; see Convention 2.24.

The convention here is that a _smeared map_ takes values in finite linear
combinations of the target space (or maybe in measures). We use capital
letters for smeared maps.

Intuitively, we iterate $x$ forward, stopping at each iterate with probability
given by $\psi$. More visually, imagine the original cross-section as a disk.
As we look along the flow lines, we see an overlapping set of disks, with hard
edges between them. To find the smeared first return map, we “feather” the
edges by giving the disks partially-transparent boundaries made out of
cellophane. If we continuously increase the transparency towards the boundary,
the resulting image will have soft edges. See Figure 7.1.

We will usually omit $\psi$ from the notation and denote the smeared first
return map by $P$.

Since $\tau_{0}$ is a global cross-section, in the definition of $P$ we
eventually take the first choice, and so $P(x)$ is a finite sum of elements of
$\tau_{1}$ as claimed. A little more is true.

######  Lemma 7.5.

If $\tau_{0}\Subset\tau_{1}$ is a nested pair of global cross-sections, there
is an $N>0$ so that for any $x\in\tau_{1}$, there is an integer $k<N$ so that
$p_{1}^{k}(x)\in\tau_{0}$, where $p_{1}$ denotes the first return map for
$\tau_{1}$

###### Proof.

By Lemma 7.3, there is an upper bound on the return time from $\tau_{1}$ to
$\tau_{0}$, and thus an upper bound on the number of intersections of the
return path to $\tau_{0}$ with the compact set $\tau_{1}$. ∎

As a consequence of Lemma 7.5, we can rewrite $P$ directly. Let $N$ be the
bound from Lemma 7.5. Then

(7.6) $\displaystyle P(x)$ $\displaystyle=\psi(p(x))\cdot
p(x)+\bar{\psi}(p(x))\psi(p^{2}(x))\cdot p^{2}(x)+\cdots$
$\displaystyle=\sum_{k=1}^{N}\bar{\psi}(p(x))\cdots\bar{\psi}(p^{k-1}(x))\psi(p^{k}(x))\cdot
p^{k}(x).$

If we extend the upper limit of the sum beyond $N$, the additional terms will
be $0$.

######  Proposition 7.7.

For any nested global cross-sections $\tau_{0}\Subset\tau$ and bump function
$\psi$, the smeared first return map $P$ is continuous.

###### Proof.

We wish to show that $P$ is continuous at $x\in\tau$. There is some first
$n>0$ such that $p^{n}(x)\in\tau_{0}^{\circ}$. If there is any $i$ between $1$
and $n$ so that $p^{i}(x)\in\partial\tau$, we first find a smaller cross-
section $\tau^{\prime}$ with $\tau_{0}\Subset\tau^{\prime}\Subset\tau$ without
this problem, as follows. Recall that we assumed that $\psi$ vanishes in a
neighborhood of $\partial\tau$. Since there are finitely many points
$p^{i}(x)$ in the open set $\tau\setminus\operatorname{supp}(\psi)$, we can
pick $\tau^{\prime}$ containing $\operatorname{supp}(\psi)$ so that its
boundary avoids those finitely many $p^{i}(x)$. Since $\psi$ vanishes on
$\tau\setminus\tau^{\prime}$, the smeared first return map defined with
respect to $\tau^{\prime}$ agrees with that defined with respect to $\tau$. By
replacing $\tau$ by $\tau^{\prime}$, we may thus assume that
$p^{i}(x)\notin\partial\tau$. Similarly shrink $\tau_{0}$ so that
$p^{i}(x)\notin\partial\tau_{0}$ for $0<i<n$.

We proceed by induction on $n$. If $n=1$, that is, if
$p(x)\in\tau_{0}^{\circ}$, then $p$ is continuous at $x$ by Lemma 7.2, and $P$
is continuous since the map taking a point $x$ to the delta function
$\delta_{x}$ is continuous. Otherwise, note that $p_{1}$ (the return map for
$\tau$) is continuous at $x$ (again by Lemma 7.2, since
$p_{1}(x)\in\tau^{\circ}$). By induction $P$ is continuous at $p_{1}(x)$, and
therefore

$P(x)=\psi(p_{1}(x))\cdot p_{1}(x)+\bar{\psi}(p_{1}(x))\cdot P(p_{1}(x))$

is continuous at $x$. ∎

$\tau_{0}$
---
$\tau$
---
$p^{3}(x)$
---
$x$
---
$p(x)$
---
$\tau^{\prime}$
---
$p^{2}(x)$
---

Figure 7.1. Smeared first return map, illustrating the proof of continuity in
the case $n=3$ (before shrinking $\tau$). The bump function $\psi$ is
indicated by the density of red.

To define iterates of $P$, we first extend $P$ and other functions to act on
measures.

######  Definition 7.8.

When $X,Y$ are measure spaces and $f\colon X\to Y$ is a measurable function,
by convention we extend $f$ to a function
$\mathcal{M}_{1}(X)\to\mathcal{M}_{1}(Y)$ acting on measures, denoted $f_{*}$
(or simply $f$), by setting

$f_{*}(\mu)(S)\coloneqq\mu(f^{-1}(S))$

for $\mu\in\mathcal{M}_{1}(X)$ and $S\subset Y$ a measurable set. If $f$ is
continuous, then this extension is continuous with respect to the weak∗
topology on $\mathcal{M}_{1}(X)$ and $\mathcal{M}_{1}(Y)$. (This uses
Proposition 2.23 and the fact that if $g\colon Y\to\mathbb{R}$ is bounded,
then $g\circ f\colon X\to\mathbb{R}$ is also bounded.) If $f$ is invertible
and $\psi\colon X\to\mathbb{R}_{\geq 0}$ is a scaling factor, then

(7.9) $f(\psi\cdot\mu)=(\psi\circ f^{-1})\cdot f(\mu).$

In practice, we will often be interested in the subspace of finitely supported
measures, in which case the extension
$f\colon\mathbb{R}_{1}X\to\mathbb{R}_{1}Y$ is given by

$f\Bigl{(}\sum a_{i}x_{i}\Bigr{)}\coloneqq\sum a_{i}f(x_{i}).$

For $F:X\to\mathcal{M}_{1}(Y)$ a smeared function, we extend $F$ to a function
$\mathcal{M}_{1}(X)\to\mathcal{M}_{1}(Y)$ from measures to measures, denoted
$\tilde{F}$ (or simply $F$), by setting, for any measurable function
$\varphi\colon Y\to\mathbb{R}_{\geq 0}$,

$\displaystyle F_{\varphi}(x)$ $\displaystyle\coloneqq\int_{y\in
Y}\varphi(y)\bigl{(}F(x)\bigr{)}(y)$ $\displaystyle\int_{y\in
Y}\varphi(y)\tilde{F}(\mu)(y)$ $\displaystyle\coloneqq\int_{x\in
X}F_{\varphi}(x)\mu(x),$

where $F_{\varphi}\colon X\to\mathbb{R}_{\geq 0}$ is an auxiliary function.
See also Equations (7.11) and (7.12) below.

######  Proposition 7.10.

If $F\colon X\to\mathcal{M}_{1}(Y)$ is continuous, then the extension
$\tilde{F}\colon\mathcal{M}_{1}(X)\to\mathcal{M}_{1}(Y)$ is continuous.

###### Proof.

Let $(x_{i})_{i=0}^{\infty}$ be a sequence approaching $x\in X$. By
assumption, $F(x_{i})$ approaches $F(x)$ in the weak∗ topology. By Proposition
2.23, this is equivalent to saying that for all continuous bounded functions
$\varphi\colon Y\to\mathbb{R}_{\geq 0}$ the function $F_{\varphi}$ above is
continuous. Furthermore, $F_{\varphi}$ is bounded since $F$ takes values in
probability measures. We now show that $\tilde{F}$ is continuous. Let
$\mu_{i}\to\mu\in\mathcal{M}(X)$. We want to show that
$\tilde{F}(\mu_{i})\to\tilde{F}(\mu)\in\mathcal{M}_{1}(Y)$, i.e., for any
continuous bounded function $\varphi\colon Y\to\mathbb{R}_{\geq 0}$,

$\int_{x\in X}F_{\varphi}(x)\mu_{i}(x)\to\int_{x\in X}F_{\varphi}(x)\mu(x).$

This is true by definition of the weak∗ topology in $\mathcal{M}_{1}(X)$ and
Proposition 2.23, since $F_{\varphi}$ is continuous and bounded. ∎

In our applications, $F$ takes values in finitely-supported measures, with a
bound on the size of the support. Concretely, if $F:X\to\mathbb{R}Y$ can be
written as a finite sum

$F(x)=\sum_{i=1}^{N}\psi_{i}(x)f_{i}(x)$

for real-valued functions $\psi_{i}$ and invertible $Y$-valued functions
$f_{i}$, then, by Equation (7.9), the extension is defined by

(7.11)
$F(\mu)=\sum_{i=1}^{N}f_{i}(\psi_{i}\cdot\mu)=\sum_{i=1}^{N}(\psi_{i}\circ
f_{i}^{-1})\cdot f_{i}(\mu)$

where the middle expression is a sum of pushforwards of scaled measures, and
in the last expression we have pulled the scaling factors out. If $\mu$ is
also finitely supported, we have

(7.12)
$F\Bigl{(}\sum_{i}a_{i}x_{i}\Bigr{)}=\sum_{i,j}a_{i}\psi_{j}(x_{i})f_{j}(x_{i}).$

######  Definition 7.13.

With the above extension of notation, the iterates of the smeared return map
$P$ are defined by

(7.14) $\displaystyle P^{0}(x)$ $\displaystyle\coloneqq x$ $\displaystyle
P^{n}(x)$ $\displaystyle\coloneqq P(P^{n-1}(x)).$

######  Definition 7.15.

A measure $\nu$ on $Y$ that is invariant under the flow $\phi_{t}$ induces a
_flux_ $\mu=\nu_{\tau}$ on a global cross-section $\tau$ that is invariant
under the first return map $p$ [VO16, Section 3.4.2]. Concretely, pick
$\varepsilon>0$ small enough so that the map
$b\colon[0,\varepsilon]\times\tau\to Y$ defined by $b(t,x)=\phi_{t}(x)$ is an
embedded flow box. Then for $S\subset\tau$, define
$\mu(S)\coloneqq\nu\bigl{(}\phi_{[0,\varepsilon]}(S)\bigr{)}/\varepsilon$.
(Since our cross-sections are compact manifolds-with-boundary, we can always
find such an $\varepsilon$.)

We have the following invariance property.

######  Proposition 7.16.

If $\nu$ is a measure on $UT\Sigma$ that is invariant under $\phi_{t}$, then
$\nu_{\tau}$ is invariant under $p$ and $\psi\nu_{\tau}$ is invariant under
$P$.

For motivation for the factor of $\psi$ in the proposition statement, think
about extending the definitions to allow $\psi$ to be the (non-continuous)
characteristic function of $\tau_{0}\Subset\tau$; then $P$ is the ordinary
first return map to $\tau_{0}$, and $\psi\nu_{\tau}=\nu_{\tau_{0}}$ is the
flux of $\tau_{0}$. See also Example 9.3.

###### Proof.

The first part is standard. For the second part, let $\mu=\nu_{\tau}$. Then we
have

$\displaystyle P\bigl{(}\psi\cdot\mu\bigr{)}$
$\displaystyle=p\bigl{(}\psi\cdot(\psi\circ
p)\cdot\mu\bigr{)}+p^{2}\bigl{(}\psi\cdot(\bar{\psi}\circ p)\cdot(\psi\circ
p^{2})\cdot\mu\bigr{)}+\dots$
$\displaystyle=\sum_{k=1}^{N}p^{k}\bigl{(}\psi\cdot(\bar{\psi}\circ
p)\cdots(\bar{\psi}\circ p^{k-1})\cdot(\psi\circ p^{k})\cdot\mu\bigr{)}$
$\displaystyle=\sum_{k=1}^{N}(\psi\circ p^{-k})\cdot(\bar{\psi}\circ
p^{-k+1})\cdots(\bar{\psi}\circ p^{-1})\cdot\psi\cdot p^{k}(\mu)$
$\displaystyle=\sum_{k=1}^{N}(\psi\circ p^{-k})\cdot(\bar{\psi}\circ
p^{-k+1})\cdots(\bar{\psi}\circ p^{-1})\cdot\psi\cdot\mu.$

using the definition of $P$ (in the form of Equation (7.6)); rewriting as a
sum; Equation (7.11); and invariance of $\mu$ under $p$. Since
$\psi+\bar{\psi}=1$, this sum telescopes, and the result is $\psi\mu$. ∎

We will sometimes blur the distinction between a geodesic current and its
flux, and write, for instance, $\nu(\tau)$ for the total mass of $\nu_{\tau}$
on $\tau$, or $\nu(\psi\tau)$ for $\int_{x\in\tau}\psi(x)\nu_{\tau}(x)$.

###  7.2. Homotopy type of return

We will additionally need to track _how_ a point returns to the cross-section.
For this, we suppose that we have a global cross-section $\tau$ contained in a
simply-connected cross-section $\tau^{\prime}$. (For a $C^{1}$ flow on a
manifold of dimension at least 3, there is always a simply-connected cross-
section [Bas04].) For such a cross-section, from the first return for
$x\in\tau$, we can extract another piece of information: the homotopy class of
the return trajectory.

######  Definition 7.17.

Let $\phi_{t}$ be a flow on a manifold $Y$ and $\tau$ be a global cross-
section, contained in a larger compact simply-connected cross-section
$\tau^{\prime}$. Fix a basepoint $*\in\tau^{\prime}$. For $x\in\tau$, define
the _return trajectory_ $m(x)\in\pi_{1}(Y,*)$ by taking the homotopy class of
a path that runs in $\tau^{\prime}$ from $*$ to $x$, along the flow trajectory
from $x$ to $p_{\tau}(x)$, and then in $\tau^{\prime}$ from $p_{\tau}(x)$ back
to $*$. Since $\tau^{\prime}$ is simply-connected, $m(x)$ is independent of
the choice of path.

######  Lemma 7.18.

Let $Y$ be a compact manifold with flow $\phi_{t}$ and $\tau$ be a global
cross-section. As $x$ varies in $\tau$, the return trajectory $m(x)$ takes on
only finitely many values.

###### Proof.

Since there are upper bounds on the return time (Lemma 7.3), on the speed of
$\phi_{t}$ with respect to a Riemannian metric on $Y$, and on the diameter of
$\tau^{\prime}$, the length of the path representing $m(x)$ is bounded. On a
compact manifold, there are only finitely many elements of $\pi_{1}(Y,*)$ that
have representatives of bounded length. ∎

To get the return map for iterates, we also incorporate the point of first
return.

######  Definition 7.19.

The _homotopy return map_ is the map $q\colon\tau\to\tau\times\pi_{1}(Y,*)$
defined by

$q(x)\coloneqq(p(x),m(x)).$

We can iterate $q$ by inductively defining $q^{n+1}$ to be the composition

$\tau\overset{q^{n}}{\longrightarrow}\tau\times\pi_{1}(Y)\xrightarrow{q\times\operatorname{id}}\tau\times\pi_{1}(Y)\times\pi_{1}(Y)\xrightarrow{(x,g,h)\mapsto(x,hg)}\tau\times\pi_{1}(Y).$

Define $m^{n}(x)\in\pi_{1}(Y,*)$ to be the second component of $q^{n}(x)$.

######  Remark 7.20.

An alternative approach to defining homotopy types of return trajectories is
to pick a cross-section in the universal cover [EPS20, Section 3.2].

######  Definition 7.21.

For $\tau$ a global cross-section with basepoint $*$, $\tau_{0}\Subset\tau$ a
smaller global cross-section, $\psi$ a bump function for this pair, and
$\tau^{\prime}\supset\tau$ a simply-connected cross-section, the _smeared
homotopy return map_ $Q\colon\tau\to\mathbb{R}_{1}(\tau\times\pi_{1}(Y,*))$ is
defined by

$\displaystyle Q(x)$
$\displaystyle\coloneqq\begin{cases}q(x)&p(x)\in\tau_{0}\\\ \psi(p(x))\cdot
q(x)+\bar{\psi}(p(x))\cdot L_{m(x)}Q(p(x))&p(x)\in\tau-\tau_{0}\end{cases}$

where $L_{g}$ is left translation by $g\in\pi_{1}(Y,*)$:

$L_{g}\Bigl{(}\sum_{i}a_{i}(x_{i},h_{i})\Bigr{)}\coloneqq\sum_{i}a_{i}(x_{i},gh_{i}).$

There is once again a natural notion of iteration, defined by inductively
setting $Q^{n+1}$ to be the composition

$\tau\overset{Q^{n}}{\longrightarrow}\mathbb{R}_{1}(\tau\times\pi_{1}(Y))\xrightarrow{\mathbb{R}_{1}(Q\times\operatorname{id})}\mathbb{R}_{1}(\mathbb{R}_{1}(\tau\times\pi_{1})\times\pi_{1}(Y))\xrightarrow{\operatorname{join}}\mathbb{R}_{1}(\tau\times\pi_{1}(Y)).$

where $\operatorname{join}$ is the somewhat more involved operation

$\operatorname{join}\Biggl{(}\sum_{i}a_{i}\Bigl{(}\Bigl{(}\sum_{j}b_{ij}(x_{ij},g_{ij})\Bigr{)},h_{i}\Bigr{)}\Biggr{)}\coloneqq\sum_{i,j}a_{i}b_{ij}(x_{ij},h_{i}g_{ij}).$

(The terminology comes from the theory of monads [Mog91, Wad92]. See Equation
(7.12).)

######  Definition 7.22.

We define the _smeared $n$-th return trajectory_
$M^{n}\colon\tau\to\mathbb{R}_{1}\pi_{1}(Y)$ to be the composition

$\tau\overset{Q^{n}}{\longrightarrow}\mathbb{R}_{1}(\tau\times\pi_{1}(Y))\longrightarrow\mathbb{R}_{1}\pi_{1}(Y)$

where at the second step we lift the projection on the second component to act
on weighted objects as in Definition 7.8.

Let $\Lambda(n,\tau)$ be the set of curves that appear with non-zero
coefficient in $M^{n}(x)$ for some $x\in\tau$.

######  Lemma 7.23.

$\Lambda(n,\tau)$ is finite.

###### Proof.

Immediate from Lemmas 7.5 and 7.18. ∎

######  Lemma 7.24.

The maps $Q^{k}$ and $M^{k}$ are continuous.

###### Proof.

The proof of Proposition 7.7 also proves that $Q$ is continuous. It then
follows that $Q^{k}$ and $M^{k}$ are continuous. ∎

###  7.3. Return maps for the geodesic flow

We now turn to the specifics of our situation. Let $\Sigma$ be the surface $S$
endowed with an arbitrary hyperbolic Riemannian metric $g$. Points in
$UT\Sigma$ will be denoted $\vec{x}$, meaning a pair of a point $x\in\Sigma$
and a unit tangent vector at $x$. Let $\phi_{t}\colon UT\Sigma\to UT\Sigma$ be
the geodesic flow associated to $g$.

We pick nested global cross-sections
$\tau_{0}\Subset\tau\subset\tau^{\prime}$, with $\tau^{\prime}$ simply-
connected, and a bump function $\psi$ for the pair $(\tau_{0},\tau)$. We thus
get a smeared $n$-th return trajectory
$M^{n}\colon\tau\to\mathbb{R}_{1}\pi_{1}(UT\Sigma,*)$. We want to work with
curves in $\Sigma$ rather than its unit tangent bundle, so compose with the
projection $\pi_{\Sigma}\colon UT\Sigma\to\Sigma$ to get a linear combination
of elements of $\pi_{1}(S,\pi_{\Sigma}(*))$. Then take conjugacy classes (to
pass to unbased curves) to get an element of
$\mathbb{R}_{1}\mathcal{C}^{+}(S)$. We call the resulting map $[M^{n}]$, which
has type

$[M^{n}]\colon\tau\to\mathbb{R}_{1}\mathcal{C}^{+}(S).$

From Lemmas 7.23 and 7.24, $[M^{n}]$ is a continuous function with values in
the finite dimensional subspace
$\mathbb{R}^{[\Lambda(n,\tau)]}\subset\mathbb{R}\mathcal{C}^{+}(S)$, where
$[\Lambda(n,\tau)]$ is the projection of $\Lambda(n,\tau)$.

###  7.4. Definition of the extension

Now, we will use the above return map $[M^{n}]$ to define the extension of $f$
to geodesic currents in Theorem A. With
$\tau_{0}\Subset\tau\subset\tau^{\prime}$ as above, we define

$\displaystyle R^{n}\colon\mathcal{GC}^{+}(S)\to\mathbb{R}\mathcal{C}^{+}(S)$
$\displaystyle
R^{n}(\mu)\coloneqq\int_{\tau}[M^{n}(\vec{x})]\psi(\vec{x})d\mu_{\tau}(\vec{x}).$

Observe that, for fixed $n$, $R^{n}(\mu)$ is a weighted multi-curve with a
fixed set of possible connected components, but with weights depending on
$\mu$. As we will explain in Section 10, because $[M^{n}(\vec{x})]$ is
continuous on $\tau$, $R^{n}$ is continuous with respect to the weak∗ topology
on $\mathcal{GC}^{+}(S)$.

We can now finally define our extension of $f$.

######  Definition 7.25.

Let $\tau$ be a good cross-section and $f$ a weighted curve functional
satisfying stability, homogeneity, weighted quasi-smoothing, and convex union.
We define

(7.26) $\displaystyle f^{n}_{\tau}(\mu)$ $\displaystyle\coloneqq
f(R^{n}(\mu))$ (7.27) $\displaystyle f_{\tau}(\mu)$
$\displaystyle\coloneqq\lim_{n\to\infty}\frac{f^{n}_{\tau}(\mu)}{n}.$

We will prove that the limit exists (at least for our cross-section) in
Proposition 9.6.

######  Warning 7.28.

We work with weighted linear combinations of objects (or, more generally,
measures) at many places in the paper. Some functions (like $R^{n}$) are by
definition additive under linear combinations, and in Definition 7.8, we also
silently extend other functions (like $p$) to apply additively to linear
combinations of points or measures. But the main curve functional $f$ we are
interested is _not_ necessarily additive. (We only assume that $f$ satisfies
convex union in the main theorems.)

##  8\. Constructing global cross-sections

Next we define the specific global cross-section we use. We make choices that
are convenient for guaranteeing that certain crossings are essential.

######  Definition 8.1.

For $\Sigma$ a hyperbolic surface, $c$ an oriented geodesic segment on
$\Sigma$, and $0<\theta<\pi/2$ an angle, the _wedge set_ $W(c,\theta)\subset
UT\Sigma$ is the set of vectors that cross $c$ nearly perpendicularly:

$W(c,\theta)\coloneqq\bigl{\\{}\vec{x}=(x,v)\bigm{|}x\in
c,\,\lvert\operatorname{ang}(T_{x}c,v)-\pi/2\rvert\leq\theta\,\bigr{\\}}.$

(Angles $\operatorname{ang}(v,w)$ are measured by the counterclockwise
rotation from $v$ to $w$.) We can likewise define the wedge set
$W(\\{c_{i}\\},\theta)$ for a collection of geodesic segments
$\\{c_{i}\\}_{i=1}^{k}$.

We wish to find a wedge set $W(\\{c_{i}\\},\theta)$ that is an embedded global
cross-section for the geodesic flow $\phi_{t}$.

Fix $\theta=\pi/6$. For any geodesic arc $c$, the wedge set $W(c,\theta)$
intersects any geodesic that passes through a non-empty open set. Thus by
compactness of $UT\Sigma$, there exist a finite collection of immersed arcs
$(c_{i})_{i=1}^{n}$ so that $\bigcup_{i=1}^{n}W(\\{c_{i}\\},\theta)_{i=1}^{n}$
is a disconnected, not necessarily embedded, global cross-section of the
geodesic flow. We will produce an embedded global cross-section from it.
Immersed points come from intersection points between the geodesic segments
$c_{i}$, but not all of them produce immersed points of the global cross-
section.

Indeed, suppose $\operatorname{ang}(c_{i},c_{j})=\varphi$. There are two good
cases:

  1. (1)

If $2\theta<\lvert\varphi\rvert$, the wedge sets don’t intersect, as shown in
Figure 8.1.

$\theta$
---
$\varphi$
---
$c_{j}$
---
$c_{i}$
---

Figure 8.1. Wedge sets not intersecting, in the case $2\theta<\varphi$.

  2. (2)

If $\theta<\frac{\pi}{2}-\lvert\varphi\rvert$, then the corresponding wedge
sets do intersect, but we can perturb the $c_{j}$ slightly to avoid the
intersection. Given a small interval $[a,b]$ of the geodesic segment $c_{j}$,
containing one intersection point between $c_{i}$ and $c_{j}$, we consider the
wedge $W([a,b],\theta)$. By pushing the endpoints $a,b$ forward along the
extremal angles, and removing $[a,b]$ from $c_{j}$, as shown in Figure 8.2, we
obtain new interval with endpoints $a^{\prime},b^{\prime}$, and a new wedge
set $W([a^{\prime},b^{\prime}],\theta^{\prime})$, for some
$\theta^{\prime}>\theta$, so that $W([a^{\prime},b^{\prime}],\theta^{\prime})$
is disjoint from $c_{i}$ and $W([a^{\prime},b^{\prime}],\theta)$ intersects
every geodesic that $W([a,b],\theta)$ does (so we still have a global cross-
section).

$\theta$
---
$\varphi$
---
$c_{j}$
---
$c_{i}$
---

Figure 8.2. Cut and flow operation when the angle of intersection is small,
$\theta<\pi/2-\varphi$.

######  Remark 8.2.

Note that the inequality must be strict in case (2). Indeed, if
$\theta=\pi/2-|\phi|$, the wedge sets do intersect, but this is not a good
case: we cannot guarantee to make the wedge sets locally disjoint by flowing a
segment of $c_{j}$ forward slightly.

Our choice $\theta=\pi/6$ guarantees that one of these two cases happens.

Next, we will construct an immersed connected global cross-section
$\tau^{\prime}$ containing this disconnected wedge set.

######  Proposition 8.3.

For any wedge set $W(\\{c_{i}\\},\theta)$ and any $\varepsilon>0$, there
exists a closed geodesic $\delta$ so that for each $i$, there is a sub-segment
$\delta_{i}\subset\delta$ so that $c_{i}\subset B_{\varepsilon}(\delta_{i})$
and every geodesic that intersects $W(c_{i},\theta)$ also intersects
$W(\delta_{i},\theta+\varepsilon)$.

###### Proof.

Use [BPS17, Theorem 2.4] to construct the closed geodesic $\delta$ with
$c_{i}\subset B_{\varepsilon}(\delta)$. If $\varepsilon$ is small enough, by
following the geodesic flow from $W(c_{i},\theta)$ we hit $\delta$ in a
geodesic segment $\delta_{i}$, so that every geodesic intersecting
$W(c_{i},\theta)$ also intersects $W(\delta_{i},\theta+\varepsilon^{\prime})$
for some $\varepsilon^{\prime}$. Since $\varepsilon^{\prime}$ goes to $0$ as
$\varepsilon$ goes to $0$, the result follows. ∎

Observe that if $\varepsilon$ is small enough and the $c_{i}$ in Proposition
8.3 are disjoint, then the $\delta_{i}$ will be disjoint as well. Thus
combining the above propositions (and redefining $\theta$ to be
$\theta+\varepsilon$), we have found a closed geodesic $\delta$ and disjoint
geodesic segments $\delta_{i}^{\prime}\subset\delta$ so that we have the
following global cross-sections:

  * •

A wedge set $\tau_{0}\coloneqq W(\\{\delta_{i}\\},\theta)$ giving a
disconnected embedded global cross-section.

  * •

A global cross-section $W(\delta,\theta)$ containing the previous one which is
connected but not embedded (as $\delta$ will self-intersect).

We would like to use the second global cross-section $W(\delta,\theta)$ to
close up the homotopy return trajectories of the smeared return map. However,
$W(\delta,\theta)$ is not simply-connected, so the homotopy return map will
depend on which path along the cross-section we choose. This can be easily
fixed by setting, for some small open interval
$I\subset\delta\backslash\bigcup_{i}\\{\delta_{i}\\}$,

$\tau^{\prime}\coloneqq W(\delta-I,\theta)$

so that $\tau_{0}\subset\tau^{\prime}$. Strictly speaking, $\tau^{\prime}$ is
not simply-connected as a subset of $UT\Sigma$; rather it is the image of an