# Uniform Degeneration of hyperbolic surfaces with boundary along harmonic map
rays

Kento Sakai  Graduate School of Science, Osaka University
<EMAIL_ADDRESS>
###### Abstract.

Unique harmonic maps between surfaces give a parametrization of the
Teichmüller space by holomorphic quadratic differentials on a Riemann surface.

In this paper, we investigate the degeneration of hyperbolic surfaces
corresponding to a ray of meromorphic quadratic differentials on a punctured
Riemann surface in this parametrization, where the meromorphic quadratic
differentials have a pole of order $\geq 2$ at each puncture. We show that the
rescaled distance functions of the universal covers of hyperbolic surfaces
uniformly converge, on a certain non-compact region containing a fundamental
domain, to the intersection number with the vertical measured foliation given
by the meromorphic quadratic differential determining the direction of the
ray.

This implies the family of hyperbolic surfaces converges to the dual
$\mathbb{R}$-tree to the vertical measured foliation in the sense of Gromov-
Hausdorff. As an application, we describe the limit in the function space on
the set of isotopy classes of properly embedded arcs and simple closed curves
on the surface.

##  1\. Introduction

Let $S$ be a connected and oriented surface. We suppose that $S$ admits a
complete, finite area hyperbolic structure with geodesic boundary. The
Teichmüller space of $S$ is the space of marked hyperbolic structures on $S$.
There are many kinds of parametrizations for the Teichmüller space (for
example, Fenchel-Nielsen coordinates, Teichmüller’s parametrization, the
earthquake parametrization, the Fricke coordinate, and so on), and each of
them derives a nice geometric properties of Teichmüller spaces. In this paper,
we are interested in the parametrization by the space of holomorphic quadratic
differentials via harmonic maps between hyperbolic surfaces.

We here recall the history of the parametrization given by harmonic maps. If
the surface is closed, the existence of harmonic diffeomorphisms and their
uniqueness follow from results of Eells-Sampson [ES64], Hartman [Har67],
Schoen-Yau [SY78] and Sampson [Sam78]. Based on these results, Wolf [Wol89]
proved that the Teichmüller space is homeomorphic to the vector space of
holomorphic quadratic differentials on a Riemann surface by the associated
Hopf differentials. After that, the unique existence was extended to the case
of other types of surfaces. Lohkamp [Loh91] showed the unique existence of a
harmonic map in the case of hyperbolic surfaces with cusps and he gave the
parametrization. When it comes to surfaces with closed boundaries, the unique
existence theorem was shown by Wolf [Wol91b], and Sagman [Sag23] gave the
parametrization. As for surfaces with crown ends, Gupta [Gup21] gave the
unique existence and the parametrization. Recently, Allegretti [All21] handled
surfaces with punctures and boundaries of both types, and some exceptional
types of surfaces, which had never been dealt with.

\begin{overpic}[scale={0.6}]{SurafaceWithClosedBoundaryAndCrownEnd.pdf}
\end{overpic} Figure 1. Hyperbolic surface with a closed boundary (left side)
and a crown end (right side).

In the case that the hyperbolic surfaces is closed or with cusps, the
interesting fact of this parametrization is that, if we deform a hyperbolic
surface along a ray from the origin in the vector space of holomorphic
quadratic differentials on a Riemann surface, these hyperbolic surfaces
converge to a point in the Thurston boundary [Wol89, Sak23]. Moreover, the
limit point is the projective class of the vertical measured foliation
determining the direction of the ray. In terms of metric geometry, Wolf
studied the degenerations of hyperbolic structures along a ray, by realizing
them as metric spaces via harmonic maps. As he mentioned in [Wol95], the
distance functions of the metric lifted to the universal covering of the
hyperbolic surfaces uniformly converge on compact sets. He also described the
degenerations of hyperbolic structures by using the notion of the equivariant
Gromov-Hausdorff convergence, which was introduced by Paulin [Pau88], and Wolf
recovered the result of Morgan-Shalen [MS84] in the case of $2$-dimensional
hyperbolic manifolds.

In this paper, we consider the degeneration of hyperbolic surfaces with closed
boundaries or crown ends (see Figure 1). For the sake of simplicity, first we
explain our result in the case that $S$ is an ideal $k$-gon for $k\geq 3$. In
this case, we deal with harmonic maps from the complex plane
$\mathbb{C}\cong\mathbb{C}P^{1}\setminus\\{\infty\\}$ to hyperbolic ideal
$k$-gons. Let $\varphi$ be a polynomial quadratic differential of degree $k-2$
on $\mathbb{C}$, which has a pole of order $k+2$ at
$\infty\in\mathbb{C}P^{1}$. Then, there exists a unique marked hyperbolic
ideal $k$-gon $[C_{\varphi},f_{\varphi}]\in\mathcal{T}(S)$ satisfying the
following condition: there exists a harmonic diffeomorphism
$h_{\varphi}\colon\mathbb{C}\to C_{\varphi}$ such that the Hopf differential
of $h_{\varphi}$ is equal to $\varphi$, where the Hopf differential of
$h_{\varphi}$ is the $(2,0)$-part of the pullback metric
${h_{\varphi}}^{\ast}C_{\varphi}$. Therefore, for each $t>0$, we obtain the
hyperbolic surface $[C_{t\varphi},f_{t\varphi}]$ and the harmonic map
$h_{t\varphi}$ corresponding to $t\varphi$. The pullback metric
${h_{t\varphi}}^{\ast}C_{t\varphi}$ rescaled by $\frac{1}{4t}$ defines a
distance function on $\mathbb{C}$. For each $t>0$, let $d_{t}$ denote the
distance function. On the other hand, another metric space is obtained from
$\varphi$. Let $T(\varphi)$ denote the leaf space of the vertical measured
foliation $\mathcal{F}_{\varphi,\mathrm{vert}}$. The transverse measure of
$\mathcal{F}_{\varphi,\mathrm{vert}}$ defines a distance $d_{T(\varphi)}$ on
$T(\varphi)$. The metric space $(T(\varphi),d_{T(\varphi)})$ is called the
dual $\mathbb{R}$-tree to $\mathcal{F}_{\varphi,\mathrm{vert}}$. If $\varphi$
is a polynomial quadratic differential on $\mathbb{C}$, it is a simplicial
metric tree. Our main result in the case of an ideal $k$-gon is the following.

######  Theorem A (Corollary 3.3).

The rescaled metric spaces $(\mathbb{C},d_{t})$ uniformly converge to the
simplicial metric tree $(T(\varphi),d_{T(\varphi)})$ in the sense of Gromov-
Hausdorff as $t$ tends to $\infty$.

This theorem is obtained by showing that, for every $\varepsilon>0$ and every
sufficiently large $t>0$, the projection map
$\operatorname{pr}_{\varphi}\colon\mathbb{C}\to T(\varphi)$ to the leaf space
defines an $\varepsilon$-approximation between $(\mathbb{C},d_{t})$ and
$(T(\varphi),d_{T(\varphi)})$, namely, for every $x_{1},x_{2}\in\mathbb{C}$,
$|d_{t}(x_{1},x_{2})-d_{T(\varphi)}(\operatorname{pr}_{\varphi}x_{1},\operatorname{pr}_{\varphi}x_{2})|<\varepsilon$.
In other words, the distance functions $d_{t}$ uniformly converge to
$d_{T(\varphi)}\circ\operatorname{Pr}_{\varphi}$ on the complex plane
$\mathbb{C}$, which is a non-compact domain. Here, we define the map
$\operatorname{Pr}_{\varphi}$ by
$\operatorname{Pr}_{\varphi}(x_{1},x_{2})=(\operatorname{pr}_{\varphi}x_{1},\operatorname{pr}_{\varphi}x_{2})$.

Next, we present the general case. Let $X$ be a closed Riemann surface of
genus $g$ with $n$ punctures and $\varphi$ be a meromorphic quadratic
differential on $X$ with poles only at the punctures. We suppose that the
order of $\varphi$ at each puncture is at least $2$. The orders of $\varphi$
at punctures determine a topological surface $S$ with closed boundaries or
crown ends. Then, there exists a unique hyperbolic structure
$[C_{\varphi},f_{\varphi}]\in\mathcal{T}(S)$ satisfying the same condition as
the ideal polygon case.

Therefore, for each $t>0$, we obtain the hyperbolic surface
$[C_{t\varphi},f_{t\varphi}]$ corresponding to $t\varphi$. We fix the
universal covering $\pi\colon\widetilde{X}\to X$. Since $\widetilde{X}$ is
conformal equivalent to the hyperbolic plane $\mathbb{H}$ (or the Euclidean
plane $\mathbb{E}$), we identify $\widetilde{X}$ with $\mathbb{H}$ (or
$\mathbb{E}$). Let $\Gamma$ be the discrete free subgroup of the orientation-
preserving isometry group $\operatorname{Isom}^{+}(\mathbb{H})$ of the
hyperbolic plane (or $\operatorname{Isom}^{+}(\mathbb{E})$ of the Euclidean
plane) with $X=\widetilde{X}/\Gamma$. Taking the lift of
${h_{t\varphi}}^{\ast}C_{t\varphi}$ to $\widetilde{X}$ and rescaling the
metric by $\frac{1}{4t}$, we get the distance function
$d_{t}\colon\widetilde{X}\times\widetilde{X}\to\mathbb{R}_{\geq 0}$ from the
rescaled metric on $\widetilde{X}$, which is invariant by the action of
$\Gamma$. To state our main theorem, we consider the hyperbolic plane (or the
Euclidean plane) with the Gromov boundary $\partial_{\infty}\Gamma$ of
$\Gamma$ attached in an equivariant manner. We denote this union with a
natural topology by $\widetilde{X}\cup\partial_{\infty}\Gamma$. Its formal
definition is introduced in Section 4. Let $T(\varphi)$ be the dual
$\mathbb{R}$-tree to the vertical measured foliation
$\mathcal{F}_{\widetilde{\varphi},\mathrm{vert}}$ of $\widetilde{\varphi}$.
The transverse measure of $\mathcal{F}_{\widetilde{\varphi},\mathrm{vert}}$
defines a distance on $T(\varphi)$. We define a map
$\operatorname{pr}_{\varphi}\colon\widetilde{X}\to T(\varphi)$ as the
projection to the leaf space of
$\mathcal{F}_{\widetilde{\varphi},\mathrm{vert}}$.

######  Theorem B (Theorem 5.1).

Let $U\subset\widetilde{X}\cup\partial_{\infty}\Gamma$ be a neighborhood of
$\partial_{\infty}{\Gamma}$. The sequence of the rescaled metric spaces
$(\widetilde{X}\setminus U,d_{t})$ uniformly converges to
$(\operatorname{pr}_{\varphi}(\widetilde{X}\setminus U),d_{T(\varphi)})$ in
the sense of Gromov-Hausdorff as $t$ tends to $\infty$.

Note that the complement $\widetilde{X}\setminus U$ can be a non-compact
subset.

As mentioned in [Wol95, Remark 4.2], B implies that $(\widetilde{X},d_{t})$
converges to $(T(\varphi),d_{T(\varphi)})$ in the sense of equivariant Gromov-
Hausdorff. Therefore, in the sense of [Wol95, Definition 5.1], the identity
maps $\operatorname{id}\colon\widetilde{X}\to(\widetilde{X},d_{t})$ converge
to $\operatorname{pr}_{\varphi}\colon\widetilde{X}\to T(\varphi)$. The
identity map $\operatorname{id}\colon\widetilde{X}\to(\widetilde{X},d_{t})$ is
a harmonic map to the metric space
$(\widetilde{X},{\widetilde{h}_{t\varphi}}\\!^{\ast}\widetilde{C}_{t\varphi}/\sqrt{4t})=(\widetilde{X},d_{t})$,
and the limit $\operatorname{pr}_{\varphi}\colon\widetilde{X}\to T(\varphi)$
is also a harmonic map as in [Wol95, Proposition 3.1].

Lastly, we describe an application of B. Let $\mathcal{C}$ and $\mathcal{A}$
be the sets of isotopy classes of simple closed curves and properly embedded
arcs on $S$, respectively. We consider the hyperbolic structures in terms of
the length function $\ell_{\ast}\colon\mathcal{T}(S)\to\mathbb{R}_{\geq
0}^{\mathcal{C}\cup\mathcal{A}}$, and describe the limit along a ray.

######  Corollary C (Corollary 6.3).

$(4t)^{-1/2}\ell_{\ast}[C_{t\varphi},f_{t\varphi}]\to
I_{\ast}\mathcal{F}_{\varphi,\mathrm{vert}}\in\mathbb{R}^{\mathcal{C}\cup\mathcal{A}}$
as $t\to\infty$.

From this corollary, we find that the direction of the ray determines the
limit point in the projective space $P(\mathbb{R}_{\geq
0}^{\mathcal{C}\cup\mathcal{A}})$ of functions. Since the limit is an
accumulation point of the sequence of hyperbolic surfaces
$\\{[C_{t\varphi},f_{t\varphi}]\\}_{t>0}$, there is a point in the Thurston
boundary corresponding to the limit. In Section 6.1, we define the Thurston
boundary of the Teichmüller space of a surface with closed boundaries or crown
ends in a similar way to [ALPS16].

### Acknowledgement

I would like to thank my supervisor, Shinpei Baba, for helpful advice, many
discussions, and encouragement. This work was supported by JSPS KAKENHI Grant
Number JP23KJ1468.

##  2\. Preliminaries

###  2.1. Harmonic maps

Let $X$ be a Riemann surface with a conformal structure $z$ and
$(S,\rho(w)|dw|^{2})$ be a hyperbolic surface. A $C^{2}$ map $f\colon
X\to(S,\rho)$ is harmonic if $f$ satisfies

$f_{z\overline{z}}+\frac{\rho_{w}}{\rho}f_{z}f_{\overline{z}}=0.$

For a map $f\colon X\to(S,\rho)$, the $(2,0)$-part of the pullback metric
$(f^{\ast}\rho)^{2,0}$ is a quadratic differential, which is called the Hopf
differential of $f$. The following characterization is fundamental in the
Teichmüller theory of harmonic maps, although the proof is not so difficult
(e.g. see [Sam78]).

######  Proposition 2.1.

Let $f\colon X\to(S,\rho)$ be a diffeomorphism. Then, $f$ is harmonic if and
only if the Hopf differential of $f$ is holomorphic.

###  2.2. Teichmüller space

The Teichmüller space is the space of hyperbolic structures with marking on a
topological surface. In this subsection, we will define a marking as a map
from a punctured Riemann surface to a hyperbolic surface with closed geodesic
boundaries or crown ends.

Let $\overline{S}$ be an oriented, closed surface of genus $g$. From
$\overline{S}$, we remove $n$ open disks
$D_{1},\ldots,D_{n}\subset\overline{S}$ whose closures are pairwise disjoint.
In addition, for each $1\leq i\leq n$, we remove $m_{i}\geq 0$ distinct points
on $\partial D_{i}$. We denote the resulting surface by $S$. Except for 6
cases (see Figure 2), $S$ admits a complete hyperbolic structure of finite
area with totally geodesic boundary.

\begin{overpic}[scale={1.1}]{ExceptionalSurfaces.pdf} \end{overpic} Figure 2.
Exceptional surfaces

Let $X$ be a closed Riemann surface of genus $g$ with $n$ punctures. At each
puncture $p_{i}\ (1\leq i\leq n)$, we pick equally spaced $m_{i}$ projective
tangent vectors
$v_{i,1},\ldots,v_{i,m_{i}}\in(T_{p_{i}}X\setminus\\{0\\})/\mathbb{R}_{\geq
0}$ counterclockwise so that all adjacent vectors form angle $2\pi/m_{i}$. The
data $\theta=(\theta_{i})_{i=1}^{n}$ of tuples
$\theta_{i}=(v_{i,1},\ldots,v_{i,m_{i}})$ is called direction data.

Let $C$ be a complete, finite area hyperbolic surface homeomorphic to $S$ with
totally geodesic boundary. A marking of $C$ from $(X,\theta)$ is a
diffeomorphism $f\colon X\to C\setminus\partial C$ satisfying the following
condition: for any smooth curve $c\colon[0,\infty)\to X$ tending to a puncture
$p_{i}$,

$\begin{split}&\lim_{t\to\infty}c^{\prime}(t)=v_{i,j}\
(j\in\\{1,\ldots,m_{i}\\})\\\ \Leftrightarrow&\ (f\circ c)(t)\text{ tends to
an ideal vertex of }C\text{ as }t\to\infty.\end{split}$

The pair $(C,f)$ of a hyperbolic surface $C$ homeomorphic to $S$ and its
marking $f$ from $(X,\theta)$ is called the marked hyperbolic surface. Let
$(C_{1},f_{1})$ and $(C_{2},f_{2})$ be marked hyperbolic surfaces from
$(X,\theta)$. We define the equivalence relation
$(C_{1},f_{1})\sim(C_{2},f_{2})$, if there exists an isometry $g\colon
C_{1}\to C_{2}$ such that $g\circ f_{1}$ is isotopic to $f_{2}$ relative to
the direction data $\theta$. The Teichmüller space $\mathcal{T}(S)$ is the
quotient space of equivalence classes of marked hyperbolic surfaces
homeomorphic to $S$. Since the direction data is only used to determine a
cyclic order of boundary cusps, this definition of the Teichmüller space is
equivalent to usual one. The Teichmüller space $\mathcal{T}(S)$ is
homeomorphic to the Euclidean space of dimension

$6g-6+\sum_{i=1}^{l}(m_{i}+3).$

###  2.3. Parametrization via harmonic maps

####  2.3.1. Principal part data

Following [Gup21], we here introduce the definition of the principal part of a
meromorphic quadratic differential. For each $1\leq i\leq n$, let
$(D_{i},z_{i})$ be a conformal coordinate disk around each puncture $p_{i}$ of
$X$. Let $\mathrm{QD}(X)$ denote the space of holomorphic quadratic
differentials on $X$ which possibly have a pole at each puncture. We take
$\varphi\in\mathrm{QD}(X)$ having a pole of order at least $2$ at $p_{i}$. On
the coordinate disk $(D_{i},z_{i})$, the square root differential
$\sqrt{\varphi(z_{i})}\,dz_{i}$ can be expanded in Laurent series around
$p_{i}$ up to the multiplication by $\pm 1$. The term $P_{i}(z_{i})$ of degree
$-1$ or less for the Laurent series is called the principal part of $\varphi$
at $p_{i}$ with respect to the coordinate disk $(D_{i},z_{i})$. Let $k_{i}$ be
the order of $\varphi$ at a pole $p_{i}$. The principal part of $\varphi$ at
$p_{i}$ is represented by

$P_{i}(z_{i})=\begin{cases}(c_{r}z_{i}^{-r}+c_{r-1}z_{i}^{-r+1}+\cdots+c_{1}z_{i}^{-1})\,dz_{i}&(k_{i}\
\text{is even})\\\
z^{-1/2}(c_{r}z_{i}^{-r}+c_{r-1}z_{i}^{-r+1}+\cdots+c_{1}z_{i}^{-1})\,dz_{i}&(k_{i}\
\text{is odd}),\end{cases}$ (2.1)

where $k_{i}=2r$ or $2r+1$ and $c_{i}\ (i=1,\ldots,r)$ are complex numbers.
Conversely, if a form $P_{i}(z_{i})\,dz_{i}$ which is expressed as 2.1 is
given on each coordinate disk, we call the data
$P=(P_{i}(z_{i})\,dz_{i})_{i=1}^{n}$ prescribed principal part data. Let
$\mathrm{QD}(X,P)$ denote the space of holomorphic quadratic differentials on
$X$ whose principal part at each puncture $p_{i}$ with respect to
$(D_{i},z_{i})$ is equal to $P_{i}(z_{i})$. For each $P_{i}(z_{i})\,dz_{i}$,
the coefficient of $z_{i}^{-1}$ is called the analytic residue of $P$ at
$p_{i}$.

We consider a prescribed principal part data
$P=(P_{i}(z_{i})\,dz_{i})_{i=1}^{n}$ on $X$ such that the quadratic
differential $P_{i}(z_{i})^{2}\,dz_{i}^{2}$ has a pole of order $m_{i}+2$ at
each puncture $p_{i}$. We construct a direction data $\theta(P)$ on $X$ from
the prescribed principal part data $P$. For each $1\leq i\leq n$, let $c$ be
the coefficient of $z_{i}^{-(m_{i}+2)}$ in $P_{i}(z_{i})^{2}$. We define a
curve $\gamma_{i,j}(t)$ as $(z_{i}\circ\gamma_{i,j})(t)=te^{\sqrt{-1}(\arg
c+2j\pi)/m_{i}}$ ($j=1,\ldots,m_{i}$) and set the tangent vectors $v_{i,j}$ at
$p_{i}$ ($1\leq j\leq m_{i}$) as

$v_{i,j}=\left.\frac{d(z_{i}\circ\gamma_{i,j})}{dt}\right|_{t=0}.$ (2.2)

For each $\varphi\in\mathrm{QD}(X,P)$, a curve $\gamma(t)$ tending to $p_{i}$
satisfies $\gamma^{\prime}(t)\to v_{i,j}$ in
$(T_{p_{i}}X\setminus\\{0\\})/\mathbb{R}_{\geq 0}$ for some $1\leq j\leq
m_{i}$ if and only if $\varphi(\gamma(t))(d\gamma/dt)^{2}$ converges to a
positive real number as $t\to\infty$. We set
$\theta_{i}=(v_{i,1},\ldots,v_{i,m_{i}})$ and
$\theta(P)=(\theta_{i})_{i=1}^{n}$ (directions in $\theta_{i}$ are asymptotic
directions of horizontal measured foliation determined from
$P_{i}(z_{i})^{2}\,dz_{i}^{2}$ at $p_{i}$, see Section 2.4).

In the following, we suppose that the marking of every
$[C,f]\in\mathcal{T}(S)$ is from $(X,\theta(P))$ to $C$.

####  2.3.2. Length data

Following [All21], we define the length data $(L_{i})_{i=1}^{n}$ of the marked
hyperbolic surface $C\coloneqq[C,f]\in\mathcal{T}(S)$. The length data is
determined by closed boundaries and crown ends of $C$. The image of a small
neighborhood of $p_{i}$ by marking $f$ is the neighborhood of a closed
boundary or crown end $\partial_{i}$. We first consider the case that the
number of cusps on $\partial_{i}$ is even $>0$. On the hyperbolic surface $C$,
we put a horocycle centered at each ideal vertex $p_{i,j}\ (1\leq j\leq
m_{i})$ of $\partial_{i}$, and truncate ideal vertices along the horocycles.
If the horocycles are disjoint each other, for each $1\leq j\leq m_{i}$, the
geodesic segment is left on each geodesic boundary between $p_{i,j}$ and
$p_{i,j+1}$. Let $\ell_{j}$ denote the length of the geodesic segment. Then,
the alternate sum

$L_{i}\coloneqq\pm\frac{1}{2}\sum_{j}(-1)^{j}\ell_{j}$

is well-defined up to the multiplication by $\pm 1$. In other words, $L_{i}$
is independent of the initial $j$ of the alternate sum and the size of the
horocycles (if two adjacent horocycles have an intersection, we define the
length $\ell_{j}$ is negative). This is because the number of cusps on
$\partial_{i}$ is even. The number $L_{i}$ up to $\pm 1$ is called the metric
residue of $\partial_{i}$. In the case that the number of cusps on
$\partial_{i}$ is odd, we simply define $L_{i}=0$. If $\partial_{i}$ is closed
boundary, we define $L_{i}$ as the length of $\partial_{i}$ in the metric of
$C$.

####  2.3.3. Parametrization

The length data $(L_{i})_{i=1}^{n}$ of $C\in\mathcal{T}(S)$ is compatible with
$P$ if

$L_{i}=\begin{cases}0&(m_{i}\text{ is odd})\\\
4\pi\left|a_{i}\sin^{2}\left(\frac{\arg
a_{i}}{2}\right)\right|^{1/2}&(m_{i}\text{ is 0})\\\
\pm\operatorname{Re}(a_{i})&(m_{i}\text{ is even}>0),\end{cases}$

where $a_{i}$ is the analytic residue of $P$ at $p_{i}$. This compatibility
condition is introduced in [All21]. We fix a length data $L=(L_{i})_{i=1}^{n}$
compatible with $P$ and define the subset
$\mathcal{T}_{L}(S)\subset\mathcal{T}(S)$ as the level set

$\\{[C,f]\in\mathcal{T}(S)\mid(\text{the length data of }[C,f])=L\\}.$

The Teichmüller space $\mathcal{T}_{L}(S)$ can be parametrized by the space of
meromorphic quadratic differentials on $X$ via harmonic maps. In the case that
the differentials have a double pole at each puncture, Wolf proved the unique
existence of harmonic maps [Wol91b] and Sagman gave the parametrization result
[Sag23]. In the case that the differentials have a higher order pole at each
puncture, the unique existence and parametrization results were given by Gupta
[Gup21]. Allegretti [All21] also considered the mixed cases, which had not
been treated in the former papers.

######  Theorem 2.2 (Unique existence, [Wol91b, Gup21, All21]).

For each marked hyperbolic surface $[C,f]\in\mathcal{T}_{L}(S)$, there exists
a unique harmonic map $h_{[C,f]}=h\colon X\to C\setminus\partial C$ such that
the Hopf differential of $h$ is in $\mathrm{QD}(X,P)$ and $h$ is homotopic to
$f$ relative to the direction data $\theta(P)$.

######  Theorem 2.3 (Parametrization, [Gup21, Sag23, All21]).

We define the map

$\Psi\colon\mathcal{T}_{L}(S)\to\mathrm{QD}(X,P)$

by assigning each $[C,f]\in\mathcal{T}_{L}(S)$ to the Hopf differential of the
unique harmonic map $h_{[C,f]}$. Then, the map $\Psi$ is a homeomorphism.

By Theorem 2.3, given $\varphi\in\mathrm{QD}(X)$, we obtain the hyperbolic
surface $C_{\varphi}$ and the harmonic diffeomorphism $h_{\varphi}\colon X\to
C_{\varphi}$ such that $\Psi([C_{\varphi},h_{\varphi}])=\varphi$. By the
uniqueness of the harmonic map, the map

$\rho\colon\mathrm{QD}(X)\to\operatorname{Met}(X)\text{ given by
}\rho(\varphi)={h_{\varphi}}^{\ast}C_{\varphi}$

is well-defined, where $\operatorname{Met}(X)$ denotes the set of the metric
on $X$. Note that $\rho(\varphi)$ can be incomplete, if a puncture is a
regular point, zero or simple pole of $\varphi$.

Let $\pi\colon\widetilde{X}\to X$ be the universal covering of $X$. The
universal covering $\widetilde{X}$ is identified to the complex plane
$\mathbb{C}$ in the case that $X$ is the Riemann sphere and the number of
punctures is $1$ or $2$, and to the upper half plane $\mathbb{H}$ in the other
case. Ler $\Gamma$ be the discrete free subgroup of
$\operatorname{Isom}^{+}(\widetilde{X})$ such that $X=\widetilde{X}/\Gamma$.
The lift metric of ${h_{\varphi}}^{\ast}C_{\varphi}$ to $\widetilde{X}$ is
denoted by $g_{\varphi}$.

###  2.4. Measured foliation and dual R-tree

Let $\varphi$ be a holomorphic quadratic differential on a Riemann surface
$X$. We introduce the measured foliations and the $\mathbb{R}$-tree obtained
from $\varphi$ (for more details, see [Wol95]). In a natural coordinates
$\zeta=\xi+i\eta$ which is defined on a sufficiently small neighborhood of a
nonzero point of $\varphi$, the holomorphic quadratic differential $\varphi$
is represented by $d\zeta^{2}$. A natural coordinate is uniquely determined up
to a multiplication $\pm 1$ and additive constant. Therefore, vertical lines
$\\{\xi=\text{constant}\\}$ (resp. horizontal lines
$\\{\eta=\text{constant}\\}$) determine a singular foliation on
$\widetilde{X}$ which is pronged at each zero of $\varphi$. The number of
prongs at a zero is equal to $(\text{the order of the zero})+2$. The form
$|d\xi|$ (resp. $|d\eta|$) defines a transverse measure on the singular
foliation. The measured foliation obtained from $\varphi$ is called the
vertical measured foliation (resp. horizontal measured foliation) of $\varphi$
and denoted by $\mathcal{F}_{\varphi,\mathrm{vert}}$ (resp.
$\mathcal{F}_{\varphi,\mathrm{hori}}$).

Taking the lift $\widetilde{\mathcal{F}}_{\varphi,\mathrm{vert}}$ of the
vertical measured foliation of $\varphi$ to the universal covering
$\widetilde{X}$, we obtain the leaf space $T(\varphi)$ which is given by
collapsing each leaf of $\widetilde{\mathcal{F}}_{\varphi,\mathrm{vert}}$ to a
point. The dual $\mathbb{R}$-tree to
$\widetilde{\mathcal{F}}_{\varphi,\mathrm{vert}}$ is the metric space
$(T(\varphi),d_{T(\varphi)})$, where $d_{T(\varphi)}$ is the distance function
on $T(\varphi)$ induced by the transverse measure $|d\xi|$. The projection
$\operatorname{pr}_{\varphi}\colon\widetilde{X}\to T(\varphi)$ is called the
collapsing map.

###  2.5. Geometry of harmonic maps

Let $X$ be a Riemann surface and $Y$ be a hyperbolic surface with finite area.
We suppose that $h\colon X\to Y$ is a harmonic diffeomorphism and $\varphi$ is
the Hopf differential of $h$. The norm
$|\nu(h)|\coloneqq|h_{\overline{z}}/h_{z}|$ of the Beltrami differential of
$h$ is a well-defined function on $X$. We set $G(h)=\log(1/|\nu(h)|)$ The
function $G(h)$ plays an important role in studying the geometry of harmonic
maps. In [Min92], Minsky gave useful estimations on the function $G(h)$, which
originated from Wolf’s work [Wol91a].

######  Proposition 2.4 ([Min92, Lemma 3.2]).

We fix a point $p\in X$ which is nonzero of $\varphi$. We suppose a
$|\varphi|$-metric ball $B_{|\varphi|}(p,r)$ contains no zero of $\varphi$ and
is embedded into $X$. Then, we have

$G(h)(p)\leq\sinh^{-1}\left(\frac{\operatorname{Area}(Y)}{2\pi r^{2}}\right).$

######  Proposition 2.5 (Lemma 3.3 [Min92]).

Let $p\in X$ be a point $|\varphi|$-distant $d$ away from zeros of $\varphi$.
If $G(h)$ is bounded by $B$ from above on $B_{|\varphi|}(p,d)$, then we have

$G(h)(p)\leq\frac{B}{\cosh d}.$

Using these propositions, we prepare some estimations we will use later. Let
$p_{i}$ be a puncture of $X$. If $p_{i}$ is a double pole of $\varphi$, we can
take a disk neighborhood $U_{i}$ of $p_{i}$ such that
$(U_{i}\setminus\\{p_{i}\\},|\varphi|)$ is isometric to a Euclidean annulus
one-sided infinite. If $p_{i}$ is a higher order pole than $2$, we can take a
sink neighborhood of $p_{i}$, which does not contain zeros of $\varphi$ (see
Definition 2 [GW19]). The sink neighborhood $U_{i}$ is broken into half-planes
and horizontal strips by the horizontal measured foliation of $\varphi$. Each
half-plane is isometric to a Euclidean half plane $\\{(x,y)\mid y>0\\}$ and
each horizontal strip is isometric to a Euclidean strip $\\{(x,y)\mid
a<y<b\\}$.

######  Proposition 2.6.

We suppose that the length of the core curve of $U_{i}\setminus\\{p_{i}\\}$ is
at least $2C$ and $\partial U_{i}$ is distant $C$ away from zeros of $\varphi$
in the $|\varphi|$-metric. Then, for each $p\in U_{i}\setminus\\{p_{i}\\}$,

$G(h)(p)\leq 2e^{-f(p)}\sinh^{-1}(\operatorname{Area}(Y)/2\pi C^{2}),$

where $f(p)=\operatorname{dist}_{|\varphi|}(p,\varphi^{-1}(0))$.

##  3\. Uniform convergence of distance functions: ideal polygon case

###  3.1. Main result

First, we state and prove the main theorem in the case of ideal polygons,
since this is the simplest case and many of the methods for the proof are the
same as in the general case.

Let $S$ be an ideal $k$-gon ($k\geq 3$), i.e. $S$ is a topological closed disk
and we remove distinct $k$ points on the boundary. In this case, the Riemann
surface $X$ is equivalent to
$\mathbb{C}P^{1}\setminus\\{\infty\\}\cong\mathbb{C}$ and $\varphi$ is a
polynomial quadratic differential on $\mathbb{C}$. The infinite point is a
pole of order $k+2$, and the degree of the polynomial differential $\varphi$
equals $k-2$. The coefficient of the term of degree $k-2$ determines the
direction data at $\infty$.

For $t>0$, let $d_{t}$ denote the distance function in the metric
$(4t)^{-1}g_{t\varphi}$. In the ideal polygon case, the metric $g_{t\varphi}$
is equal to the pullback metric ${h_{t\varphi}}^{\ast}C_{t\varphi}$ (see
Section 2.3.3). Let $I_{\varphi}$ denote the composite function of the
collapsing map and the distance function of the dual $\mathbb{R}$-tree
$T(\varphi)$ to $\mathcal{F}_{\varphi,\mathrm{vert}}$ i.e. we set
$I_{\varphi}(x_{1},x_{2})=d_{T(\varphi)}(\operatorname{pr}_{\varphi}x_{1},\operatorname{pr}_{\varphi}x_{2})$.
Note that, in the ideal polygon case, the dual $\mathbb{R}$-tree is a
simplicial metric tree.

######  Theorem 3.1.

The family of distance functions $d_{t}$ uniformly converges to $I_{\varphi}$
on $X=\mathbb{C}$.

This theorem tells us that, for each $\varepsilon>0$, the collapsing map
$\operatorname{pr}_{\varphi}$ is an $\varepsilon$-approximation between
$(X,d_{t})$ and $(T(\varphi),d_{T(\varphi)})$ by taking a sufficiently large
$t$, i.e., there exists $t_{1}>0$ such that, for each $t>t_{1}$ and each two
points $x_{1},x_{2}\in X$,

$|d_{t}(x_{1},x_{2})-d_{T(\varphi)}(\operatorname{pr}_{\varphi}x_{1},\operatorname{pr}_{\varphi}x_{2})|<\varepsilon.$
(3.1)

Following [BH99], we recall the definition of the Gromov-Hausdorff
convergence.

######  Definition 3.2.

Let $(X,d)$ and $(Y,d^{\prime})$ be metric spaces. For $\varepsilon>0$, the
relation $R\subset X\times Y$ is called $\varepsilon$-relation between $(X,d)$
and $(Y,d^{\prime})$ if $R$ satisfies the following conditions:

  1. (1)

the relation $R$ is surjective i.e.
$\operatorname{pr}_{1}(R)=X,\operatorname{pr}_{2}(R)=Y$,

  2. (2)

if $(x_{1},y_{1}),(x_{2},y_{2})\in R$, then
$|d(x_{1},x_{2})-d(y_{1},y_{2})|<\varepsilon$.

A sequence of metric spaces $\\{(X_{t},d_{t})\\}_{t\in\mathbb{R}}$ converges
to a metric space $(Y,d)$ in the sense of Gromov-Hausdorff as $t\to\infty$, if
it satisfies the following: for each $\varepsilon>0$, there exists
$t_{1}\in\mathbb{R}$ such that, for every $t>t_{1}$, there exists an
$\varepsilon$-relation $R_{t}$ between $(X_{t},d_{t})$ and $(Y,d^{\prime})$.

Setting $R=\\{(x,y)\in X\times T(\varphi)\mid
y=\operatorname{pr}_{\varphi}x\\}$, we find that, for each $\varepsilon>0$,
$R$ is an $\varepsilon$-relation between $(X,d_{t})$ and
$(T(\varphi),d_{T(\varphi)})$ for every $t>t_{1}$ from 3.1. Therefore we
obtain the following corollary.

######  Corollary 3.3.

The metric space $(X,d_{t})$ converges to $(T(\varphi),d_{T(\varphi)})$ in the
sense of Gromov-Hausdorff.

###  3.2. One side of the inequality

In this section, we prepare some lemmas to prove Theorem 3.1. For simplicity,
we denote the harmonic map $h_{t\varphi}\colon X\to C_{t\varphi}$ by $h_{t}$,
the function $G(h_{t})$ by $G(t)$ (see Section 2.5), and the metric
$(4t)^{-1}g_{t\varphi}$ by $g_{t}$.

######  Lemma 3.4.

For each $t>0$, the inequality $I_{\varphi}<d_{t}$ holds on $X$.

###### Proof.

We take any two points $x_{1},x_{2}\in X$. By the definition of the distance
on the dual $\mathbb{R}$-tree, we have

$I_{\varphi}(x_{1},x_{2})=\inf\left\\{\int_{\gamma}|d\xi|\,\middle|\,\gamma\text{
is a curve connecting }x_{1}\text{ and }x_{2}\right\\},$

where $\zeta=\xi+i\eta$ is natural coordinates of $\varphi$. In the
coordinates $\zeta$, the metric $g_{t}$ is represented by

$ds_{g_{t}}^{2}=\frac{\cosh G(t)+1}{2}\,d\xi^{2}+\frac{\cosh
G(t)-1}{2}\,d\eta^{2}$ (3.2)

(see [Min92, Section 3]). From this representation, since we find
$ds_{g_{t}}^{2}\geq d\xi^{2}$. Let $\gamma_{t}\colon[0,1]\to X$ be a
$g_{t}$-geodesic segment from $x_{1}$ to $x_{2}$. Since $\int|d\xi|$ defines
the transverse measure of the vertical measured foliation of $\varphi$, for
each $t>0$, we have

$d_{t}(x_{1},x_{2})=\ell_{g_{t}}(\gamma_{t})=\int_{\gamma}ds_{g_{t}}\geq\int_{\gamma}|d\xi|\geq
I_{\varphi}(x_{1},x_{2}).$

Thus, we obtain the conclusion. ∎

###  3.3. Estimation for vertical and horizontal lines

We next estimate the hyperbolic length of a vertical or horizontal line of
$\varphi$.

######  Lemma 3.5.

Let $U$ be a sufficiently small sink neighborhood (see Section 2.5) of
$\infty$ with respect to $\varphi$ so that the length of the core curve of
$U\setminus\\{\infty\\}$ is at least $2C$ and $\partial U$ is distant $C$ away
from $\varphi^{-1}(0)$ in the $|\varphi|$-metric. Then, there exists a
sufficiently large $t_{0}>0$ such that, for every $t>t_{0}$, there exists a
constant $D>0$ depending only on $\varphi$ such that the following hold:

  1. (1)

Let $H\colon[0,\infty)\to X$ be a curve along a horizontal leaf of $\varphi$.
We suppose that $H$ is parametrized by the $|\varphi|$-arc length and
contained in $U$. Then, for each $a\geq 0$,

$\ell_{g_{t}}(H|_{[0,a]})-i(\mathcal{F}_{\varphi,\mathrm{vert}},H|_{[0,a]})<De^{-C\sqrt{t}}$
(3.3)

  2. (2)

Let $V\colon[0,\infty)\to X$ be a curve along a vertical leaf of $\varphi$. We
suppose that $V$ is parametrized by the $|\varphi|$-arc length and contained
in $U$. Then,

$\ell_{g_{t}}(V)<De^{-C\sqrt{t}}.$ (3.4)

###### Proof.

We may assume $\Gamma(s)$ is a maximal horizontal (or vertical) line contained
in $U$. If $\Gamma(s)$ is contained in a half-plane domain, it is a bi-
infinite line, and if $\Gamma(s)$ is contained in a strip domain, it is a one-
sided infinite line.

We first consider the case that $\Gamma(s)$ is a horizontal line. Let
$P_{1},\ldots,P_{k}$ be maximal half-plane domains bounded by vertical lines
contained in $U$ (in Figure 3, these are domains bounded by dotted lines). We
cyclically put the index for $P_{1},\ldots,P_{k}$. Then, there exist adjacent
$P_{i}$ and $P_{i+1}$ decomposing $\Gamma(s)$ into a finite arc and at most
two one-sided infinite lines. The $|\varphi|$-length of the finite arc is at
most $2W$ of the width of strip domains which lies between $P_{i}$ and
$P_{i+1}$. Let $\Gamma_{1}\colon[0,\infty)\to U$ be the one-sided infinite
horizontal line. Then, the function
$f\circ\Gamma_{1}(s)=d_{|\varphi|}(\Gamma_{1}(s),\varphi^{-1}(0))$ is larger
than $C+s$. In the case that $\Gamma(s)$ is a vertical line as well, the same
thing holds true.

\begin{overpic}[scale={0.8}]{Half-planeDomain.pdf} \end{overpic} Figure 3. The
blue colored domain is $U$. The dotted line denotes the vertical lines
bounding maximal half-plane domains.

Here we introduce an inequality used later: for any constant $B>0$, there
exist constants $A>0$ and $x_{0}>0$ such that

$\sqrt{\cosh(Be^{-x})-1}\leq Ae^{-x}\ \ \ \ (x>x_{0}).$ (3.5)

Since this inequality is equivalent to

$B\leq e^{x}\cosh^{-1}(A^{2}e^{-2x}+1),$

and the right hand side converges to $\sqrt{2}A$ as $x\to\infty$, the
inequality 3.5 holds by taking $A=B/\sqrt{2}+1$.

We here consider the case where $\Gamma$ is an infinite vertical line $V$, and
for simplicity we suppose that $\Gamma$ is a bi-infinite line. Then, setting
$B=2\sinh^{-1}((k-2)\pi/2\pi C^{2})$, we have

$\displaystyle\ell_{g_{t}}(V)$
$\displaystyle=\frac{1}{\sqrt{2}}\int_{V}\sqrt{\cosh G(t)-1}\,d\eta$
$\displaystyle\leq\frac{1}{\sqrt{2}}\int_{V}\sqrt{\cosh(Be^{-f\sqrt{t}})-1}\,d\eta$
$\displaystyle(\text{\lx@cref{creftypecap~refnum}{proposition:EstimationFromMinsky'sOne}})$
$\displaystyle\leq\frac{A}{\sqrt{2}}\int_{V}e^{-f\sqrt{t}}\,d\eta$
$\displaystyle\text{\lx@cref{creftypecap~refnum}{eq:BasicInequality}},$

where the final inequality holds if $t>(x_{0}/C)^{2}$, since $f\circ V(s)>C$.
We now have

$\displaystyle\int_{V}e^{-\sqrt{t}f}\,d\eta$
$\displaystyle=\int_{-W}^{W}e^{-f\circ
V(s)\sqrt{t}}\,ds+2\int_{W}^{\infty}e^{-f\circ V(s)\sqrt{t}}\,ds$
$\displaystyle\leq\int_{-W}^{W}e^{-C\sqrt{t}}\,ds+2\int_{W}^{\infty}e^{-(s+C)\sqrt{t}}\,ds$
$\displaystyle\leq e^{-C\sqrt{t}}\left(2W+\frac{1}{\sqrt{t}}\right).$

We thus obtain 3.4.

In the case where $\Gamma$ is a horizontal line $H$, we find

$\displaystyle\ell_{g_{t}}(H|_{[0,a]})$
$\displaystyle=\frac{1}{\sqrt{2}}\int_{H|_{[0,a]}}\sqrt{\cosh G(t)+1}\,d\xi$
$\displaystyle\leq\frac{1}{\sqrt{2}}\int_{H|_{[0,a]}}\sqrt{\cosh
G(t)-1}\,d\xi+\int_{H|_{[0,a]}}\,d\xi.$

Therefore we have

$\ell_{g_{t}}(H|_{[0,a]})-i(\mathcal{F}_{\varphi,\mathrm{vert}},H|_{[0,a]})\leq\frac{1}{\sqrt{2}}\int_{H|_{[0,a]}}\sqrt{\cosh
G(t)-1}\,d\xi.$

Since we can evaluate the right hand side in the same manner as the case of a
vertical line, 3.3 holds. ∎

We also have the following estimation, which is stated in [Wol91a] and, it can
be proven in the similar way as the above proof.

######  Lemma 3.6 ([Wol91a, Lemma 2.3]).

Let $H$ and $V$ be a compact horizontal and vertical arc of $\varphi$,
respectively. If $H$ and $V$ avoid an $\varepsilon$-neighborhood of
$\varphi^{-1}(0)$ in the $|\varphi|$-metric, then

$\ell_{g_{t}}(H)\leq(1+De^{-\varepsilon\sqrt{t}})\ell_{|\varphi|}(H),\ \
\ell_{g_{t}}(V)\leq De^{-\varepsilon\sqrt{t}}\ell_{|\varphi|}(V),$

where the constant $D$ depends only on $\varepsilon$.

######  Lemma 3.7.

Let $p\in X$ be a zero of $\varphi$. We take $\varepsilon>0$ so that no zero
of $\varphi$ is contained in the $|\varphi|$-ball
$B_{|\varphi|}(p,3\varepsilon)$ of radius $3\varepsilon$ centered at $p$.
Then, for a sufficiently large $t$, the diameter of
$B_{|\varphi|}(p,\varepsilon)$ in the $g_{t}$-metric is at most
$M\varepsilon$, where $M$ is a constant depending only on the degree of the
zero $p$.

###### Proof.

Let $P$ be the $|\varphi|$-regular right angled polygonal region tangent to
$B_{|\varphi|}(p,\varepsilon)$ (see Figure 4).

\begin{overpic}[scale={1.2}]{RightAngledPolygon.pdf}
\put(58.5,62.5){$\varepsilon$} \end{overpic} Figure 4. The dotted line denotes
the vertical lines of $\varphi$. The thick line is the boundary of $P$.

It is sufficient to show that the $g_{t}$-length of $\partial P$ is at most
$M\varepsilon$. From the assumption, each horizontal edge $H$ or vertical edge
$V$ of $\partial P$ is distant $\varepsilon$ away from zeros of $\varphi$.
Therefore, from Lemma 3.6, we find

$\ell_{g_{t}}(H)\leq 2(1+De^{-\varepsilon\sqrt{t}})\varepsilon,\ \ \
\ell_{g_{t}}(V)\leq 2De^{-\varepsilon\sqrt{t}}\varepsilon.$

Therefore, the $g_{t}$-length of $\partial P$ is bounded by

$\ell_{g_{t}}(\partial P)\leq
2\varepsilon\operatorname{ord}_{\varphi}(p)(1+2De^{-\varepsilon\sqrt{t}}).$

We thus get the conclusion. ∎

###  3.4. Proof in the case of ideal polygons

###### Proof of Theorem 3.1.

We fix a sufficiently small $0<\varepsilon<C$ so that, for any two zeros
$z_{i},z_{j}$ of $\varphi$, the disks $B_{|\varphi|}(z_{i},2\varepsilon)$ and
$B_{|\varphi|}(z_{j},2\varepsilon)$ have no intersection. We pick any two
points $x_{1},x_{2}\in X$ and take any path $\Gamma_{0}$ from $x_{1}$ to
$x_{2}$, and we deform $\Gamma_{0}$, in several steps, via homotopy relative
to endpoints. (Step 1) We straighten $\Gamma_{0}$ to $|\varphi|$-geodesic
$\Gamma_{1}$. (Step 2) We deform $\Gamma_{1}$ to a $\varphi$-staircase curve
$\Gamma_{2}$ outside of the $\varepsilon$-neighborhood of zeros of $\varphi$.
Here, the $\varphi$-staircase curve is a piecewise horizontal or vertical arc
of $\varphi$. This deformation keeps horizontal and vertical measures of the
curve. (Step 3) We drag out the horizontal measure of $\Gamma_{2}$ in the
$\varepsilon$-neighborhood of each zero which $\Gamma_{2}$ passes through.
Since $\Gamma_{2}$ is passing through at most finite zeros of $\varphi$, the
number $N$ of times dragging out is also finite. This deformation preserves
the horizontal measure of $\Gamma_{2}$, although the vertical measure
increases in general. However, the additional vertical measure is at most
$4N\varepsilon$. Let $\Gamma_{3}$ be the resulting curve. (Step 4) We replace
each vertical arc contained in $\varepsilon$-neighborhoods of zeros by the
$g_{t}$-geodesic segment whose endpoints are the original endpoints connected
by the vertical arc (see Figure 5).

\begin{overpic}[scale={1.1}]{StepsOfTheDeformation2.pdf} \end{overpic} Figure
5. This illustrates the steps of the deformation around zeros. The dotted
lines are leaves of the vertical foliation, the the solid lines are leaves of
the horizontal foliation. The upper left side: Step 1, the upper right side:
Step 2, the lower left side: Step 3, the lower right side: Step 4.

Lemma 3.7 implies the length of the geodesic in $g_{t}$-metric is bounded by
$M\varepsilon$. We denote the resulting curve by $\Gamma_{4,t}$.

Now $\Gamma_{4,t}$ can be decomposed into horizontal lines part
$\Gamma_{\mathrm{hori}}$, vertical lines part $\Gamma_{\mathrm{vert}}$ and
$g_{t}$-geodesics part $\Gamma_{t}$. Since the complement $X\setminus U$ of
the sink neighborhood $U$ is compact, the horizontal or vertical measure
between any two points $x_{3},x_{4}\in X\setminus U$ can be uniformly bounded
by $K$. For the horizontal part, we have

$\displaystyle\ell_{g_{t}}(\Gamma_{\mathrm{hori}})$
$\displaystyle=\ell_{g_{t}}(\Gamma_{\mathrm{hori}}\cap(X\setminus
U))+\ell_{g_{t}}(\Gamma_{\mathrm{hori}}\cap U)$ $\displaystyle\leq
i(\mathcal{F}_{\varphi,\mathrm{vert}},\Gamma_{\mathrm{hori}}\cap(X\setminus
U))(1+De^{-\varepsilon\sqrt{t}})$ $\displaystyle(\text{by
\lx@cref{creftypecap~refnum}{lemma:EstimationForCompactArcs}})$
$\displaystyle\quad+i(\mathcal{F}_{\varphi,\mathrm{vert}},\Gamma_{\mathrm{hori}}\cap
U)+De^{-C\sqrt{t}}$ (by 3.3) $\displaystyle\leq
i(\mathcal{F}_{\varphi,\mathrm{vert}},\Gamma_{\mathrm{hori}})+(K+1)De^{-\varepsilon\sqrt{t}}$
$\displaystyle=i(\mathcal{F}_{\varphi,\mathrm{vert}},\Gamma_{1})+(K+1)De^{-\varepsilon\sqrt{t}}$
$\displaystyle=I_{\varphi}(x_{1},x_{2})+(K+1)De^{-\varepsilon\sqrt{t}}.$ (3.6)

For the vertical part, we see

$\displaystyle\ell_{g_{t}}(\Gamma_{\mathrm{vert}})$
$\displaystyle=\ell_{g_{t}}(\Gamma_{\mathrm{vert}}\cap(X\setminus
U))+\ell_{g_{t}}(\Gamma_{\mathrm{vert}}\cap U)$ $\displaystyle\leq
De^{-\varepsilon\sqrt{t}}(K+4N\varepsilon)+De^{-C\sqrt{t}}$
$\displaystyle(\text{by
\lx@cref{creftypecap~refnum}{lemma:EstimationForCompactArcs} and
\lx@cref{creftypecap~refnum}{eq:VerticalInfiniteSegment}})$
$\displaystyle\leq(K+4N\varepsilon+1)De^{-\varepsilon\sqrt{t}}.$ (3.7)

Finally, for the geodesic part, we find

$\ell_{g_{t}}(\Gamma_{t})<NM\varepsilon.$ (3.8)

Lastly $N$ denotes the number of zeros through which the $|\varphi|$-geodesic
between $x_{1}$ and $x_{2}$ passes, the total number $N^{\prime}$ of zeros of
$\varphi$ bounds $N$ from above. Therefore, combining 3.6, 3.7 and 3.8 and
Lemma 3.4, we have

$\begin{split}I_{\varphi}(x_{1},x_{2})&<d_{t}(x_{1},x_{2})\\\
&<\ell_{g_{t}}(\Gamma_{4,t})\\\
&<I_{\varphi}(x_{1},x_{2})+(2K+4N^{\prime}\varepsilon+2)De^{-\varepsilon\sqrt{t}}+N^{\prime}M\varepsilon.\end{split}$

This implies, for every sufficiently large $t$,

$|I_{\varphi}(x_{1},x_{2})-d_{t}(x_{1},x_{2})|<\varepsilon\ \ \ \ (\forall
x_{1},x_{2}\in X).$

Thus, we obtain the theorem. ∎

##  4\. Hyperbolic plane with the ends of the Cayley graph

In this section, to state the main theorem for a Riemann surface with a non-
trivial fundamental group, we explain how to attach ends of the Cayley graph
to the universal covering of the Riemann surface. Let $X$ be a closed Riemann
surface of genus $g$ with $n>0$ punctures. If $(g,n)=(0,1)$ or $(0,2)$, the
Riemann surface $X$ admits a Euclidean structure; otherwise, it admits a
hyperbolic structure. In the following, we regard $X$ as the complete
hyperbolic (or Euclidean) surface. We fix the universal covering
$\pi\colon\mathbb{H}\ (\text{or }\mathbb{E})\to X$. Let $\mathcal{L}$ denote
the set of bi-infinite geodesics on $\mathbb{H}$ (or $\mathbb{E}$) which are
lifts of geodesic ideal arcs on $X$, where a geodesic ideal arc is an infinite
hyperbolic (or Euclidean) geodesic connecting (possibly the same) two
punctures of $X$ (if $X$ is a Euclidean cylinder, we suppose that geodesic
ideal arcs on $X$ are orthogonal to the core geodesic of $X$).

Let $\mathbf{e}=\\{e_{1},\ldots,e_{2g+n-1}\\}$ be a family of geodesic ideal
arcs on $X$ such that the complement of the union $\bigcup_{i}e_{i}$ is a
simply connected domain. Let $\Gamma<\operatorname{Isom}^{+}(\mathbb{H})$ (or
$\operatorname{Isom}^{+}(\mathbb{E})$) be a discrete free subgroup with
$X=\mathbb{H}/\Gamma$ (or $\mathbb{E}/\Gamma$). We fix a base point $p\in
X\setminus\bigcup_{i}e_{i}$. For each edge $e_{i}$, there exists a unique
$\gamma_{i}\in\pi_{1}(X,p)$ such that the algebraic intersection number of
$\gamma_{i}$ and $e_{j}$ is $+1$ if $i=j$ or $0$ if $i\neq j$. The loops
$\gamma_{1},\ldots,\gamma_{2g+n-1}$ generate the fundamental group
$\pi_{1}(X,p)$. Let $G_{\mathbf{e}}$ denote the Cayley graph associated to
$\pi_{1}(X,p)$ and the generators $\\{\gamma_{1},\ldots,\gamma_{2g+n-1}\\}$.
We define the word distance on $G_{\mathbf{e}}$. The Gromov boundary
$\partial_{\infty}G_{\mathbf{e}}$ of $G_{\mathbf{e}}$ is the set of ends of
the Cayley graph, i.e.

$\partial_{\infty}G_{\mathbf{e}}=\\{r\colon[0,\infty)\to G_{\mathbf{e}}\mid
r\text{ is a geodesic ray in }G_{\mathbf{e}}\\}/\sim,$

where the equivalence relation $r_{1}\sim r_{2}$ is defined by
$d(r_{1}(t),r_{2}(t))$ is bounded from above for any $t\in[0,\infty)$.

The Gromov boundary $\partial_{\infty}G_{\mathbf{e}}$ usually has the cone
topology which is generated by the system $\\{V_{T}([r])\\}$ of subsets. Here,
for each ray $r\colon[0,\infty)\to G_{\mathbf{e}}$ and $T>0$, we define

$V_{T}([r])\coloneqq\\{[r^{\prime}]\in\partial_{\infty}G_{\mathbf{e}}\mid
r^{\prime}(0)=e,r^{\prime}(T)=r(T)\\}.$

However we define the same topology in another way. Let $\iota\colon
G_{\mathbf{e}}\hookrightarrow\mathbb{H}$ (or $\mathbb{E}$) be a
$\pi_{1}(X,p)$-equivariant embedding. We define a set $\mathcal{P}$ of half-
planes in $\mathbb{H}$ (or $\mathbb{E}$) by

$\mathcal{P}=\\{P\subset\mathbb{H}\ (\text{or }\mathbb{E})\mid
P=(\text{connected component of }\mathbb{H}\setminus\ell\ (\text{or
}\mathbb{E}\setminus\ell)),\ell\in\mathcal{L}\\}$

Here we consider the set
$\widehat{\mathbb{H}}=\mathbb{H}\sqcup\partial_{\infty}G_{\mathbf{e}}$ (or
$\widehat{\mathbb{E}}\coloneqq\mathbb{E}\sqcup\partial_{\infty}G_{\mathbf{e}}$),
which is topologized in the following way: for $P\in\mathcal{P}$, we define
the set $U_{P}\subset\widehat{\mathbb{H}}$ (or $\widehat{\mathbb{E}}$) as

$U_{P}=P\cup\\{[r]\in\partial_{\infty}G_{\mathbf{e}}\mid\exists T>0\text{ s.t.
}\forall t>T,\iota\circ r(t)\in P\\},$

and let the union of $\\{U_{P}\\}_{P\in\mathcal{P}}$ and all open sets of
$\mathbb{H}$ (or $\mathbb{E}$) be an open basis. In the rest of this section,
we omit the case of $\widetilde{X}\cong\mathbb{E}$ i.e. $(g,n)=(0,1),(0,2)$,
although similar propositions hold for these cases.

######  Proposition 4.1.

The subspace topology on
$\partial_{\infty}G_{\mathbf{e}}\subset\widehat{\mathbb{H}}$ coincides with
the cone topology.

###### Proof.

We denote the subspace topology defined above by $\mathcal{T}_{1}$ and the
cone topology by $\mathcal{T}_{2}$. We may assume that an initial point of a
ray representing each point in $\partial_{\infty}G_{\mathbf{e}}$ is the
identity element $e\in\pi_{1}(X,p)$. We fix
$[r]\in\partial_{\infty}G_{\mathbf{e}}$. Let $\mathcal{E}\subset\mathcal{L}$
be the set of geodesics such that each $\ell\in\mathcal{E}$ is a lift of some
$e_{j}\in\mathbf{e}$. For each $n\in\mathbb{Z}_{\geq 0}$, there exists a
unique geodesic $\ell_{n}\in\mathcal{E}$ such that the segment from
$\iota\circ r(n)$ to $\iota\circ r(n+1)$ intersects with $\ell_{n}$. Let
$P_{n}\in\mathcal{P}$ be one of two half-planes divided by $\ell_{n}$ contains
$\iota\circ r(n+1)$. Then $\\{U_{P_{n}}\\}$ is a fundamental system of
neighborhoods for $[r]$ in the topology $\mathcal{T}_{1}$. On the other hand,
if we set, for $T>0$,

$V_{T}=\\{[r^{\prime}]\in\partial_{\infty}G_{\mathbf{e}}\mid r^{\prime}(0)=e,\
r^{\prime}(T)=r(T)\\},$

then $\\{V_{T}\\}$ is a fundamental system of neighborhoods for $[r]$ in the
topology $\mathcal{T}_{2}$. Therefore, we find that each $U_{P_{n}}$ contains
$V_{T}$ for sufficiently large $T$ and each $V_{T}$ contains $U_{P_{n}}$ for
sufficiently large $n$. ∎

Next, we will show the topology on $\widehat{\mathbb{H}}$ is independent of
the choice of the family $\mathbf{e}$ of geodesic ideal arcs, the basepoint
$p$, and the embedding $\iota$. Let
$\mathbf{e}_{j}=\\{e_{1,j},\ldots,e_{2g+n-1,j}\\}\ (j=1,2)$ be a family of
geodesic ideal arcs such that the complement of the union $\bigcup_{i}e_{i,j}$
is a single simply connected domain. We choose $p_{j}$ as the basepoint of the
fundamental group of $X$. Let $\iota_{j}\colon
G_{\mathbf{e}_{j}}\hookrightarrow\mathbb{H}\ (\text{or }\mathbb{E})$ be a
$\pi_{1}(X,p_{j})$-equivariant embedding. Let $\widehat{\mathbb{H}}_{j}$
denote the space $\mathbb{H}\sqcup\partial_{\infty}G_{\mathbf{e}_{j}}$ which
has a topology determined from $\mathbf{e}_{j},p_{j},$ and $\iota_{j}$. We
construct a homeomorphism
$\mathcal{I}\colon\widehat{\mathbb{H}}_{1}\to\widehat{\mathbb{H}}_{2}$. Let
$c$ be a path from $p_{1}$ to $p_{2}$ and
$\beta_{c}\colon\pi_{1}(X,p_{1})\to\pi(X,p_{2})$ be the change-of-basepoint
map defined by $[\gamma]\mapsto[c\cdot\gamma\cdot c^{-1}]$. Let $f\colon
G_{\mathbf{e}_{1}}\to G_{\mathbf{e}_{2}}$ be the extension map of $\beta_{c}$.
Since $f$ is a quasi-isometry, it induces the homeomorphism $\partial
f\colon\partial_{\infty}G_{\mathbf{e}_{1}}\to\partial_{\infty}G_{\mathbf{e}_{2}}$.
We define the map
$\mathcal{I}\colon\widehat{\mathbb{H}}_{1}\to\widehat{\mathbb{H}}_{2}$ by

$\mathcal{I}(x)=\begin{cases}x&(x\in\mathbb{H})\\\ \partial
f(x)&(x\in\partial_{\infty}G_{\mathbf{e}_{1}}).\end{cases}$

######  Proposition 4.2.

The map $\mathcal{I}\colon\widehat{\mathbb{H}}_{1}\to\widehat{\mathbb{H}}_{2}$
is a homeomorphism.

###### Proof.

It is sufficient to show the continuity of $\mathcal{I}$ at each point
$x\in\partial_{\infty}G_{\mathbf{e}_{1}}$. Let $r_{1}\colon[0,\infty)\to
G_{\mathbf{e}_{1}}$ be a representative of $x$. We need to show that if
$U_{P}$ is a neighborhood of $[f\circ r_{1}]$, then it is also a neighborhood
of $[r_{1}]$. This assumption means that there exists $T>0$ such that, for any
$t>T$, $\iota_{2}\circ f\circ r_{1}(t)\in U_{P}$. We set
$\widetilde{p}_{1}=\iota_{1}(e)\in\pi^{-1}(p_{1})$. The embedding $\iota_{1}$
maps vertices of $G_{\mathbf{e}_{1}}$ to the orbit
$\Gamma\cdot\widetilde{p}_{1}$. We define another embedding $\tau_{2}$ of
$G_{\mathbf{e}_{2}}$ by $\tau_{2}=\iota_{1}\circ\beta_{c}^{-1}$. In other
words, the graph $\tau_{2}(G_{\mathbf{e}_{2}})$ is obtained by changing the
edges of the graph $\iota_{1}(G_{\mathbf{e}_{1}})$. Let
$\mathcal{G}_{1},\mathcal{G}_{2}$ denote the sets of generators determined
from edge systems $\mathbf{e}_{1},\mathbf{e}_{2}$, respectively. Then, the
lipschitz constants of $\iota_{1},\tau_{2}$ are given by

$\displaystyle\operatorname{Lip}(\iota_{1})$
$\displaystyle=\max_{\gamma\in\mathcal{G}_{1}}d_{\mathbb{H}}(\widetilde{p}_{1},\gamma\widetilde{p}_{1})$
$\displaystyle\operatorname{Lip}(\tau_{2})$
$\displaystyle=\max_{\gamma\in\mathcal{G}_{2}}d_{\mathbb{H}}(\widetilde{p}_{1},\gamma\widetilde{p}_{1}).$

Since the Cayley graphs
$\tau_{2}(G_{\mathbf{e}_{2}}),\iota_{2}(G_{\mathbf{e}_{2}})$ embedded into
$\mathbb{H}$ are mapped to a $(2g+n-1)$-bouquet embedded into $X$ by $\pi$,
there exists a constant $C>0$ such that
$d_{\mathbb{H}}(\tau_{2}(x),\iota_{2}(x))<C$ for any point $x\in
G_{\mathbf{e}_{2}}$. Since $f\colon G_{\mathbf{e}_{1}}\to G_{\mathbf{e}_{2}}$
is an extension of $\beta_{c}$, for each vertex
$\gamma\in\pi_{1}(X,p_{1})\subset G_{\mathbf{e}_{1}}$, we see
$f(\gamma)=\beta_{c}(\gamma)\in\pi_{1}(X,p_{2})$. Here, we prove that there
exists a $K>0$ such that

$d_{\mathbb{H}}(\iota_{2}\circ f\circ r_{1}(t),\iota_{1}\circ r_{1}(t))<K\ \ \
\ (\forall t\geq 0).$ (4.1)

First, for $t=i\in\mathbb{Z}_{\geq 0}$, the point $\gamma_{i}\coloneqq
r_{1}(i)$ is a vertex of the Cayley graph $G_{\mathbf{e}_{1}}$. Therefore, we
have

$\displaystyle d_{\mathbb{H}}(\iota_{2}\circ f\circ r_{1}(i),\iota_{1}\circ
r_{1}(i))$
$\displaystyle=d_{\mathbb{H}}(\iota_{2}\circ\beta_{c}(\gamma_{i}),\tau_{2}\circ\beta_{c}(\gamma_{i}))<C.$

In the case of $t\in(i,i+1)\ (i\in\mathbb{Z}_{\geq 0})$, we have

$\displaystyle d_{\mathbb{H}}(\iota_{2}\circ f\circ r_{1}(t),\iota_{1}\circ
r_{1}(t))$ $\displaystyle<$ $\displaystyle d_{\mathbb{H}}(\iota_{2}\circ
f\circ r_{1}(t),\tau_{2}\circ f\circ r_{1}(t))+d_{\mathbb{H}}(\tau_{2}\circ
f\circ r_{1}(t),\iota_{1}\circ r_{1}(t))$ $\displaystyle<$ $\displaystyle
C+d_{\mathbb{H}}(\tau_{2}\circ f\circ r_{1}(t),\iota_{1}\circ r_{1}(t))$
$\displaystyle<$ $\displaystyle C+d_{\mathbb{H}}(\tau_{2}\circ f\circ
r_{1}(t),\tau_{2}\circ f\circ r_{1}(i))+d_{\mathbb{H}}(\tau_{2}\circ f\circ
r_{1}(i),\iota_{1}\circ r_{1}(t))$ $\displaystyle=$ $\displaystyle
C+d_{\mathbb{H}}(\tau_{2}\circ f\circ r_{1}(t),\tau_{2}\circ f\circ
r_{1}(i))+d_{\mathbb{H}}(\iota_{1}\circ r_{1}(i),\iota_{1}\circ r_{1}(t))$
$\displaystyle<$ $\displaystyle
C+(\operatorname{Lip}(\tau_{2})\operatorname{Lip}(f)+\operatorname{Lip}(\iota_{1}))|t-i|$
$\displaystyle<$ $\displaystyle
C+\operatorname{Lip}(\tau_{2})\operatorname{Lip}(f)+\operatorname{Lip}(\iota_{1})\eqqcolon
K.$

This implies $\iota_{1}\circ r_{1}(t)\in U_{P}$ for sufficiently large $t>0$.
Therefore, we obtain the conclusion. ∎

From Proposition 4.2, it follows that the space $\widehat{\mathbb{H}}$
actually depends only on the conformal structure $X$, and we write
$\widehat{\mathbb{H}}=\mathbb{H}\cup\partial_{\infty}\Gamma$.

######  Proposition 4.3.

Let $F\subset\mathbb{H}$ be a fundamental domain of $\Gamma$ which is a
hyperbolic ideal polygon. Then, the complement of a neighborhood of
$\partial_{\infty}\Gamma\subset\widehat{\mathbb{H}}$ can be covered with the
image of $F$ by finite elements of $\Gamma$.

###### Proof.

Let $U\subset\widehat{\mathbb{H}}$ be a neighborhood of
$\partial_{\infty}\Gamma$. It is known that $\partial_{\infty}\Gamma$ is
homeomorphic to the Cantor set, in particular, it is compact. Therefore, there
exist finite half-planes $P_{1},\ldots,P_{n}$ such that
$\partial_{\infty}\Gamma\subset U_{P_{1}}\cup\cdots\cup U_{P_{n}}\subset U$.
Clearly the complement of $U_{P_{1}}\cup\cdots\cup U_{P_{n}}$ can be covered
with finite copies of $F$. ∎

##  5\. Uniform convergence of distance functions: general case

In this section, we generalize Theorem 3.1 to the case that the underlying
surface $S$ is not simply connected. We take $n$ open disks
$D_{1},\ldots,D_{n}$ in an oriented, closed surface of genus $g$ such that the
closures $\overline{D_{1}},\ldots,\overline{D_{n}}$ are pairwise disjoint. We
remove $D_{1},\ldots,D_{n}$ and distinct $m_{i}\geq 0$ points on the boundary
of $D_{i}$ from the surface. Let $S$ be the resulting surface.

Let $X$ be a closed Riemann surface of genus $g$ with $n$ punctures
$p_{1},\ldots,p_{n}$. Let $\varphi$ be a holomorphic quadratic differential on
$X$ which has a pole of order $m_{i}+2$ at each puncture $p_{i}$. Let $g_{t}$
denote the rescaled metric $(4t)^{-1}g_{t\varphi}$ on $\widetilde{X}$ as in
Section 3.1 and $d_{t}$ denote the distance function determined from $g_{t}$
on $\widetilde{X}$. Let $\operatorname{pr}_{\varphi}\colon\widetilde{X}\to
T(\varphi)$ be the collapsing map of the dual $\mathbb{R}$-tree to the
vertical measured foliation of $\widetilde{\varphi}$. We set
$I_{\varphi}=d_{T(\varphi)}\circ\operatorname{Pr}_{\varphi}$, where
$\operatorname{Pr}_{\varphi}(x_{1},x_{2})=(\operatorname{pr}_{\varphi}x_{1},\operatorname{pr}_{\varphi}x_{2})$
for $x_{1},x_{2}\in\widetilde{X}$.

######  Theorem 5.1.

Let $U\subset\widehat{\mathbb{H}}$ (or $\widehat{\mathbb{E}}$) be a
neighborhood of $\partial_{\infty}\Gamma\subset\widehat{\mathbb{H}}$ (or
$\widehat{\mathbb{E}}$). Then the metric functions $d_{t}$ uniformly converge
to $I_{\varphi}$ on $\widetilde{X}\setminus U$.

In order to prove this, we first state the following.

######  Proposition 5.2.

Let $\Gamma$ be a discrete free subgroup of
$\operatorname{Isom}^{+}(\widetilde{X})$ such that $\widetilde{X}/\Gamma=X$.
Let $\mathbf{e}=\\{e_{1},\ldots,e_{2g+n-1}\\}$ be a family of geodesic ideal
arcs on $X$ such that $X\setminus\bigcup_{i}e_{i}$ is a single simply
connected domain. Let $F$ be a fundamental domain bounded by lifts of
$e_{1},\ldots,e_{2g+n-1}$ to $\widetilde{X}$. Taking straight
$|\varphi|$-geodesic ideal arcs homotopic (rel. endpoints) to the geodesic
ideal arcs $e_{1},\ldots,e_{2g+n-1}$, we obtain the $|\varphi|$-convex
fundamental domain $F_{\varphi}$ of $\Gamma$. If a domain is covered with the
image of $F$ by finite elements of $\Gamma$, then it is also covered with the
image of $F_{\varphi}$ by finite elements of $\Gamma$.

######  Remark 5.3.

In this proposition, we take straight $|\varphi|$-geodesic ideal arcs, that
is, we except $|\varphi|$-geodesics spiraling to punctures. Such a spiraling
$|\varphi|$-geodesic can appear around a double pole.

###### Proof of Proposition 5.2.

It is sufficient to show that a single fundamental domain $F$ can be covered
with the image of $F_{\varphi}$ by finite elements of $\Gamma$. We suppose
that it is not true. Then, there exist an edge $e$ of $F$, an edge
$e^{\prime}$ of $F_{\varphi}$ and infinitely many isometries
$\\{g_{i}\\}\subset\Gamma$ such that $(g_{i}\cdot e^{\prime})\cap
e\neq\varnothing$. Let $(U_{i},z_{i})$ be a small disk coordinate around each
puncture $p_{i}$. First, we consider the case that $e$ and $e^{\prime}$ have
no common endpoints. Since the $|\varphi|$-length of
$e^{\prime}\cap(X\setminus\bigcup_{i}U_{i})$ is finite, it contradicts to the
assumption. Next, we suppose that $e$ and $e^{\prime}$ have a common endpoint.
If a point on $e$ tends to $p_{i}$ along $e$, then the argument of tangent
vectors of $e$ converges in $z_{i}$-coordinates. It also holds for
$e^{\prime}$ (note that we here remove the spiraling edge (Remark 5.3)).
Therefore, in a sufficiently small neighborhood of the puncture, $e$ and
$e^{\prime}$ have no intersection. For the same reason as the case of having
no common endpoints, it contradicts the assumption. ∎

From Proposition 5.2, it is sufficient to show the following.

######  Proposition 5.4.

The metric functions $d_{t}$ uniformly converge to $I_{\varphi}$ on a domain
covered with the image of $F_{\varphi}$ by finite elements of $\Gamma$.

###### Proof.

For simplicity, we show the uniform convergence on a single fundamental domain
$F_{\varphi}$. The method of the proof is almost the same as Theorem 3.1.
However, for the sake of convenience for readers, we see each step one by one.

The following lemma is shown in the totally same manner as Lemma 3.4

######  Lemma 5.5.

For each $t>0$, the inequality $I_{\varphi}<d_{t}$ holds on $F_{\varphi}$.

Next, we estimate infinite vertical or horizontal arcs of $\varphi$ around
each puncture.

######  Lemma 5.6.

Let $p_{i}$ be a puncture of $X$ and $U_{i}$ be a sufficiently small sink
neighborhood of $p_{i}$ with respect to $\varphi$. We suppose that the
$|\varphi|$-length of the core curve of $U_{i}\setminus\\{p_{i}\\}$ is at
least $2C$ and $\partial U_{i}$ is distant $C$ away from zeros of $\varphi$ in
the $|\varphi|$-metric. Then, there exists a sufficiently large $t_{0}>0$ such
that, for every $t>t_{0}$, there exists a constant $D$ depending only on
$\varphi$ such that the following hold:

  1. (1)

Let $H\colon[0,\infty)\to\widetilde{X}$ be a curve along a horizontal leaf of
$\widetilde{\varphi}$. We suppose that $H$ is parametrized by the
$|\widetilde{\varphi}|$-arc length and the image of $\pi\circ H$ is contained
in $U_{i}$. Then, for each $a\geq 0$,

$\ell_{g_{t}}(H|_{[0,a]})-i(\mathcal{F}_{\widetilde{\varphi},\mathrm{vert}},H|_{[0,a]})<De^{-C\sqrt{t}}.$
(5.1)

  2. (2)

Let $V\colon[0,\infty)\to\widetilde{X}$ be a curve along a vertical leaf of
$\widetilde{\varphi}$. We suppose that $V$ is parametrized by the
$|\widetilde{\varphi}|$-arc length and $\pi\circ V$ is contained in $U_{i}$.
Then,

$\ell_{g_{t}}(V)<De^{-C\sqrt{t}}.$ (5.2)

###### Proof.

If $p_{i}$ is a higher order pole than $2$, the proof of Lemma 3.5 works here.
Therefore, we only deal with the case that $p_{i}$ is a double pole of
$\varphi$. In this case, we can take a neighborhood $U_{i}$ of $p_{i}$ so that
$U_{i}\setminus\\{p_{i}\\}$ is isometric to a one-sided infinite Euclidean
cylinder whose core length is $2C$. Let $\theta\in(0,\pi)$ be the angle
between $H$ (or $V$) and $\pi^{-1}(\partial U_{i})$ (Figure 6).

\begin{overpic}[scale={1.1}]{AngleFromCore.pdf} \put(49.0,19.0){$\pi$}
\put(42.0,9.0){$\pi^{-1}(\partial U_{i})$} \put(17.0,12.0){$\theta$}
\put(85.0,4.5){$U_{i}$} \end{overpic} Figure 6. The solid line is horizontal
or vertical leaves

Then, the distance from $H(s)$ (or $V(s)$) to $\widetilde{\varphi}^{-1}(0)$ is
at least $C+s\sin\theta$. Therefore, setting
$B=\sinh^{-1}(\pi(4g-4+2n+\sum_{i}m_{i})/2\pi C^{2})$, we have

$\displaystyle\ell_{g_{t}}(V)$
$\displaystyle=\frac{1}{\sqrt{2}}\int_{V}\sqrt{\cosh G(t)-1}\,d\eta$
$\displaystyle\leq\frac{1}{\sqrt{2}}\int_{V}\sqrt{\cosh(Be^{-f\sqrt{t}})-1}\,d\eta$
$\displaystyle(\text{\lx@cref{creftypecap~refnum}{proposition:EstimationFromMinsky'sOne}})$
$\displaystyle\leq\frac{A}{\sqrt{2}}\int_{V}e^{-f\sqrt{t}}\,d\eta$
$\displaystyle\text{\lx@cref{creftypecap~refnum}{eq:BasicInequality}},$

where the final inequality holds if $t>(x_{0}/C)^{2}$, since $f\circ V(s)>C$.
We now have

$\displaystyle\int_{V}e^{-f\sqrt{t}}d\eta$
$\displaystyle=\int_{0}^{\infty}e^{-f\circ V(s)\sqrt{t}}\,ds$
$\displaystyle\leq\int_{0}^{\infty}e^{-(C+s\sin\theta)\sqrt{t}}\,ds=\frac{e^{-C\sqrt{t}}}{\sqrt{t}\sin\theta}.$

In the same way as the proof of Lemma 3.5, we can estimate the length of a
horizontal line $H$. Therefore, we get the conclusion. ∎

Since Lemmas 3.6 and 3.7 claim the statements which locally hold, they also
follow on $\widetilde{X}$.

Finally, we move to the step of deforming a curve as in the proof of Theorem
3.1 (as described in Figure 5). We fix a sufficiently small $0<\varepsilon<C$
so that, for any two zeros $z_{i},z_{j}\in F_{\varphi}$ of
$\widetilde{\varphi}$, the disks
$B_{|\widetilde{\varphi}|}(z_{i},2\varepsilon)$ and
$B_{|\widetilde{\varphi}|}(z_{j},2\varepsilon)$ have no intersection. We take
any two points $x_{1},x_{2}\in F_{\varphi}$ and a path $\Gamma_{0}\subset
F_{\varphi}$ from $x_{1}$ to $x_{2}$. We deform $\Gamma_{0}$ via homotopy
relative to endpoints. (Step 1) We take the $|\widetilde{\varphi}|$-geodesic
representative $\Gamma_{1}\subset F_{\varphi}$ in the homotopy class of
$\Gamma_{0}$ rel. endpoints. Let $U_{i}(i=1,\ldots,n)$ be sink neighborhoods
as in Lemma 5.6. Since $F_{\varphi}\setminus\pi^{-1}(U_{i})$ is compact, the
total vertical and horizontal measure of
$\Gamma_{1}\cap(F_{\varphi}\setminus\pi^{-1}(U_{i}))$ is bounded from above by
the constant $K$ independent of the choice of $x_{1},x_{2}\in F_{\varphi}$.
(Step 2) We deform $\Gamma_{1}$ to a $\varphi$-staircase curve $\Gamma_{2}$
outside of the $\varepsilon$-neighborhood of zeros of $\widetilde{\varphi}$.
Note that $\Gamma_{2}$ may not be contained in $F_{\varphi}$, however the
total horizontal and vertical measures of $\Gamma_{2}$ are the same as
$\Gamma_{1}$. (Step 3) We drag out the horizontal measure of $\Gamma_{2}$ in
the $\varepsilon$-neighborhood of each zero which $\Gamma_{2}$ passes through.
For each dragging out operation, the horizontal measure remains the same,
however the vertical measure increases by at most $4\varepsilon$. Since the
number $N$ of zeros of $\widetilde{\varphi}$ in $F_{\varphi}$ bounds the
number of zeros which $\Gamma_{1}$ passes through from above, the total of
increased vertical measures is at most $4N\varepsilon$. Let $\Gamma_{3}$ be
the resulting curve. (Step 4) We replace each vertical arc contained in
$\varepsilon$-neighborhoods of zeros by the $g_{t}$-geodesic segment keeping
the endpoints of the vertical arc. The universal covering version of Lemma 3.7
implies the $g_{t}$-length of the geodesic is bounded by $M\varepsilon$. We
denote the resulting curve by $\Gamma_{4,t}$.

Similarly to the case of ideal polygons, $\Gamma_{4,t}$ can be decomposed into
horizontal lines part $\Gamma_{\mathrm{hori}}$, vertical lines part
$\Gamma_{\mathrm{vert}}$, and $g_{t}$-geodesics part $\Gamma_{t}$. From Lemma
5.6 we find that the $g_{t}$-lengths of
$\Gamma_{\mathrm{hori}}\cap\bigcup\pi^{-1}(U_{i})$ and
$\Gamma_{\mathrm{vert}}\cap\bigcup\pi^{-1}(U_{i})$ is uniformly estimated.
Therefore we have,

$\displaystyle\ell_{g_{t}}(\Gamma_{\mathrm{hori}})$
$\displaystyle=\ell_{g_{t}}(\Gamma_{\mathrm{hori}}\cap(X\setminus\bigcup_{i}\pi^{-1}(U_{i})))+\ell_{g_{t}}(\Gamma_{\mathrm{hori}}\cap\bigcup\pi^{-1}(U_{i}))$
$\displaystyle\leq
i(\widetilde{\mathcal{F}}_{\varphi,\mathrm{vert}},\Gamma_{\mathrm{hori}}\cap(X\setminus\bigcup_{i}\pi^{-1}(U_{i})))(1+De^{-\varepsilon\sqrt{t}})$
$\displaystyle\quad+i(\widetilde{\mathcal{F}}_{\varphi,\mathrm{vert}},\Gamma_{\mathrm{hori}}\cap\bigcup_{i}\pi^{-1}(U_{i}))+De^{-\varepsilon\sqrt{t}}$
$\displaystyle\leq
i(\widetilde{\mathcal{F}}_{\varphi,\mathrm{vert}},\Gamma_{\mathrm{hori}})+(K+1)De^{-\varepsilon\sqrt{t}}$
$\displaystyle=i(\widetilde{\mathcal{F}}_{\varphi,\mathrm{vert}},\Gamma_{1})+(K+1)De^{-\varepsilon\sqrt{t}}$
$\displaystyle=I_{\varphi}(x_{1},x_{2})+(K+1)De^{-\varepsilon\sqrt{t}}.$

We also have

$\displaystyle\ell_{g_{t}}(\Gamma_{\mathrm{vert}})$
$\displaystyle=\ell_{g_{t}}(\Gamma_{\mathrm{vert}}\cap(X\setminus\bigcup\pi^{-1}(U_{i})))+\ell_{g_{t}}(\Gamma_{\mathrm{vert}}\cap\bigcup\pi^{-1}(U_{i}))$
$\displaystyle\leq De^{-\varepsilon\sqrt{t}}(K+4N\varepsilon)+De^{-C\sqrt{t}}$
$\displaystyle\leq De^{-\varepsilon\sqrt{t}}(K+4N\varepsilon+1).$

And, we have

$\ell_{g_{t}}(\Gamma_{t})<NM\varepsilon.$

Therefore, we obtain the desired conclusion. ∎

##  6\. Convergence to the point in the Thurston boundary

###  6.1. The Thurston compactification

We take $n$ open disks $D_{1},\ldots,D_{n}$ on an oriented, closed surface of
genus $g$. We suppose that the closures
$\overline{D_{1}},\ldots,\overline{D_{n}}$ are pairwise disjoint. We remove
$D_{1},\ldots,D_{n}$ and $m_{i}\geq 0$ points on the boundary of each $D_{i}$
from the surface. Let $S$ be the resulting surface. In this section, we
introduce the Thurston boundary of the Teichmüller space of the surface $S$
with closed boundaries or crown ends in a similar way as the paper [ALPS16],
which defines the Thurston compactifications of the Teichmüller spaces for
surfaces with only closed boundary.

Let $\mathcal{C}$ denote the set of isotopy classes of essential simple closed
curves on $S$, and $\mathcal{A}$ denote the set of isotopy classes of properly
embedded, essential arcs on $S$. We define the length function
$\ell_{\ast}\colon\mathcal{T}(S)\to\mathbb{R}_{\geq
0}^{\mathcal{C}\cup\mathcal{A}}$ by, for each $[C,f]\in\mathcal{T}(S)$,

$\ell_{\ast}([C,f])=(\ell_{[C,f]}([\gamma]))_{[\gamma]\in\mathcal{C}\cup\mathcal{A}},$

where $\ell_{[C,f]}([\gamma])=\inf_{\gamma\in[\gamma]}\ell_{C}(f(\gamma))$. We
set $P(\mathbb{R}_{\geq 0}^{\mathcal{C}\cup\mathcal{A}})=(\mathbb{R}_{\geq
0}^{\mathcal{C}\cup\mathcal{A}}\setminus\\{0\\})/\mathbb{R}_{>0}$. Let
$\pi\colon\mathbb{R}_{\geq 0}^{\mathcal{C}\cup\mathcal{A}}\setminus\\{0\\}\to
P(\mathbb{R}_{\geq 0}^{\mathcal{C}\cup\mathcal{A}})$ be the projection. Then,
it follows that $\mathcal{T}(S)$ is embedded into $P(\mathbb{R}_{\geq
0}^{\mathcal{C}\cup\mathcal{A}})$ by the composition $\pi\circ\ell_{\ast}$.

######  Definition 6.1.

The closure of the image $\pi\circ\ell_{\ast}(\mathcal{T}(S))$ is called the
Thurston compactification of the Teichmüller space of $S$. We denote it by
$\overline{\mathcal{T}(S)}$.

Let $S^{d}$ denote the double of $S$, then $S^{d}$ is a closed surface of
genus $2g+n$ with $\sum_{i=1}^{n}m_{i}$ punctures. The Teichmüller space
$\mathcal{T}(S)$ is naturally identified with the invariant subset
$\mathcal{T}^{\mathrm{sym}}(S^{d})$ of the Teichmüller space
$\mathcal{T}(S^{d})$ of $S^{d}$ by the involution along the axis of the
double. Since the Thurston compactification $\overline{\mathcal{T}(S^{d})}$ of
the Teichmüller space $\mathcal{T}(S^{d})$ is compact and
$\overline{\mathcal{T}^{\mathrm{sym}}(S^{d})}$ is a closed subset of
$\overline{\mathcal{T}(S^{d})}$, we find that $\overline{\mathcal{T}(S)}$ is a
compact set.

A measured foliation on $S$ is a pair of a singular measured foliation on $S$
and a transverse measure on it, where a singular measured foliation has
singularities as in Figure 7. We define an equivalence relation on the space
of measured foliations on $S$ by isotopy and Whitehead moves (see [FLP12] for
more details).

\begin{overpic}[scale={0.8}]{Singularities.pdf} \end{overpic} Figure 7. Top: 3
and 4 pronged singularities. Middle: foliation around a boundary. Bottom left
and middle: singularities on a boundary. Bottom right: around a removed point
on a boundary, there is such a singularity.

Let $\mathcal{MF}(S)$ denote the quotient space of measured foliations on $S$
by the equivalence relation. Similarly to $\mathcal{T}(S)$, we can identify
$\mathcal{MF}(S)$ with the subset
$\mathcal{MF}^{\mathrm{sym}}(S^{d})\subset\mathcal{MF}(S^{d})$ consisting of
the invariant measured foliations by the action of the involution of the
double. For a measured foliation
$\mathcal{F}=(\mathcal{F},\mu)\in\mathcal{MF}(S)$, we define a function
$I_{\ast}(\mathcal{F})$ on $\mathcal{C}\cup\mathcal{A}$ by, for
$[\gamma]\in\mathcal{C}\cup\mathcal{A}$,

$I_{\ast}(\mathcal{F})([\gamma])=\inf_{\gamma\in[\gamma]}\mu(\gamma).$

The image of $\mathcal{MF}(S)$ by $\pi\circ I_{\ast}$ is called the Thurston
boundary, and it is denoted by $\mathcal{PMF}(S)\subset P(\mathbb{R}_{\geq
0}^{\mathcal{C}\cup\mathcal{A}})$.

Since the discussion in [ALPS16] similarly works in our case, we have the
following.

######  Proposition 6.2.

The boundary of $\overline{\mathcal{T}(S)}$ coincides with $\mathcal{PMF}(S)$.
The Thurston compactification $\overline{\mathcal{T}(S)}$ is homeomorphic to a
closed ball of dimension $6g-6+\sum_{i=1}^{n}(m_{i}+3)$.

###  6.2. The limit along a ray in the Thurston boundary

In this section, we describe the convergence point of the family
$\\{C_{t\varphi}\\}$ in terms of a length function of the isotopy classes of
essential simple closed curves and properly embedded, essential arcs.

Let $S$ be the same surface as in Section 6.1. Let $X$ be a closed Riemann
surface of genus $g$ with $n$ punctures $p_{1},\ldots,p_{n}$ and $\varphi$ be
a holomorphic quadratic differential on $X$. As in the previous section, we
suppose that $\varphi$ has a pole of order $m_{i}+2\ (m_{i}\geq 0)$ at each
puncture $p_{i}$. We fix a marking $f\colon X\to S\setminus\partial S$ from
$(X,\theta(\varphi))$, where $\theta(\varphi)$ is the direction data
determined from the principal part of $\varphi$ (see Sections 2.2 and 2.3.1).
Let $\alpha\colon(-\infty,\infty)\to X$ be a smooth simple arc connecting two
punctures $p_{i},p_{j}$ of $X$, namely,
$\lim_{t\to\infty}\alpha(t)=p_{i},\lim_{t\to-\infty}\alpha(t)=p_{j}$. If the
unit tangent vectors along $\alpha$ converge to unit tangent vectors at the
punctures as $t\to\pm\infty$ and the limits are not in the direction data
$\theta(\varphi)$, we say that $\alpha$ has generic directions. Then, the
$f$-image of $\alpha$ with its endpoints is a properly embedded arc into $S$
(see Figure 8).

\begin{overpic}[scale={0.8}]{ArcConnectingBoundary.pdf} \put(49.0,27.0){$f$}
\end{overpic} Figure 8. Since the directions of tangent vectors of the blue
arc $\alpha$ converge to a point not in the asymptotic directions of the
horizontal measured foliation given by $\varphi$, the arc $\alpha$ is mapped
to a properly embedded arc by $f$.

Let $\alpha_{1},\alpha_{2}\colon(-\infty,\infty)\to X$ be smooth arcs having
generic directions. The arc $\alpha_{1}$ is isotopic to $\alpha_{2}$ relative
to the direction data $\theta$, if there exists an isotopy $\\{\alpha_{t}\\}$
from $\alpha_{1}$ to $\alpha_{2}$ such that the tangent vector of $\alpha_{t}$
at each of the endpoints always belongs to the same sector between two
adjacent directions in $\theta(\varphi)$. The set $\mathcal{A}$ is naturally
identified with the set of isotopy classes (rel. the direction data) of
essential simple arcs having generic directions.

From the vertical measured foliation $\mathcal{F}_{\varphi,\mathrm{vert}}$, we
define a map
$I_{\ast}\mathcal{F_{\varphi,\mathrm{vert}}}\colon\mathcal{C}\cup\mathcal{A}\to\mathbb{R}_{\geq
0}$ by

$I_{\ast}\mathcal{F}_{\varphi,\mathrm{vert}}([\gamma])=\inf_{\gamma\in[\gamma]}i(\mathcal{F}_{\varphi,\mathrm{vert}},\gamma).$

We suppose that $\mathbb{R}_{\geq 0}^{\mathcal{C}\cup\mathcal{A}}$ has the
product topology. The following is a corollary derived from Theorem 5.1.

######  Corollary 6.3.

For each $t>0$, let $[C_{t\varphi},f_{t\varphi}]\in\mathcal{T}(S)$ be the
hyperbolic surface with $\Psi([C_{t\varphi},f_{t\varphi}])=t\varphi$ (see
Theorem 2.3 for the definition of $\Psi$). Then,

$(4t)^{-1/2}\ell_{\ast}[C_{t\varphi},f_{t\varphi}]\to
I_{\ast}\mathcal{F}_{\varphi,\mathrm{vert}}\in\mathbb{R}^{\mathcal{C}\cup\mathcal{A}}\
\ \ (t\to\infty).$

###### Proof.

Let $\Gamma$ be a discrete free subgroup of
$\operatorname{Isom}^{+}(\widetilde{X})$ with $X=\widetilde{X}/\Gamma$. First,
we consider a closed curve $[\gamma]\in\mathcal{C}$. Let
$g_{[\gamma]}\in\Gamma$ be the holonomy corresponding to the closed curve
$[\gamma]$. Then the $\frac{1}{4t}C_{t\varphi}$-length of the geodesic
representative of $[\gamma]$ is equal to $\inf_{x\in
F}d_{t}(x,g_{[\gamma]}x)$, where $F$ is a fundamental domain of $\Gamma$ which
is bounded by lifts of hyperbolic (or Euclidean) geodesic ideal arcs. From
Theorem 5.1, we find that $d_{t}(x,g_{[\gamma]}x)$ is uniformly converges to
$I_{\varphi}(x,g_{[\gamma]}x)$ on $F$. Therefore, the infimum of
$d_{t}(x,g_{[\gamma]}x)$ over $F$ also converges to the infimum of
$I_{\varphi}(x,g_{[\gamma]}x)$ over $F$. Since $\inf_{x\in
F}I_{\varphi}(x,g_{[\gamma]}x)=I_{\ast}\mathcal{F}_{\varphi,\mathrm{vert}}([\gamma])$,
the corollary holds for $[\gamma]\in\mathcal{C}$.

Next, we consider $[\gamma]\in\mathcal{A}$. We suppose that isotopy classes of
the marking $f_{t\varphi}$ are always the same isotopy class of a marking $f$.
Let $\partial_{1},\partial_{2}$ be two boundary components connected by
$f([\gamma])$. For each $i=1,2$, if we take a sufficiently small half-plane
domain $P_{i}$ bounded by a horizontal line of $\varphi$ (or a Euclidean
cylindrical domain $A_{i}$) which is corresponding to $\partial_{i}$, then the
harmonic maps $h_{t\varphi}$ map $P_{i}$ (or $A_{i}$) to a neighborhood of
$\partial_{i}$ (see [Gup21] and Figure 9).

\begin{overpic}[scale={0.9}]{AnnulusAndHalfPlane.pdf} \put(18.0,8.0){$P_{i}$}
\put(18.0,52.0){$A_{i}$} \put(45.0,20.0){$h_{t\varphi}$}
\put(45.0,64.5){$h_{t\varphi}$} \end{overpic} Figure 9. The solid curves in
the left side are horizontal leaves.

We can assume that the $|\varphi|$-distance from $P_{i}$ (or $A_{i}$) to
$\varphi^{-1}(0)$ is at least $\delta>0$. For each $t>0$, the
$C_{t\varphi}$-geodesic representative $\gamma_{t}\in[\gamma]$ should pass
through $P_{i}$ (or $A_{i}$). Therefore, we have

$(4t)^{-1/2}\ell_{C_{t\varphi}}(f_{t\varphi}([\gamma]))\geq\inf_{\begin{subarray}{c}x_{1}\in
P_{1}(\text{or}A_{1}),\\\ x_{2}\in
P_{2}(\text{or}A_{2})\end{subarray}}d_{t}(x_{1},x_{2}).$ (6.1)

For each point $x\in P_{i}$ (or $A_{i}$), Lemma 5.6 follows that the
$g_{t}$-length of the vertical line starting from $x$ to the puncture is less
than $De^{-\delta\sqrt{t}}$ for sufficiently large $t$. Therefore, we can
connect each point $x\in P_{i}$ (or $A_{i}$) and the boundary $\partial_{i}$
by a curve whose length is less than $De^{-\delta\sqrt{t}}$. Thus, we have

$(4t)^{-1/2}\ell_{C_{t\varphi}}(f_{t\varphi}([\gamma]))\leq\inf_{\begin{subarray}{c}x_{1}\in
P_{1}(\text{or}A_{1}),\\\ x_{2}\in
P_{2}(\text{or}A_{2})\end{subarray}}d_{t}(x_{1},x_{2})+2De^{-\delta\sqrt{t}}$
(6.2)

From Theorem 5.1, the distance $d_{t}(x_{1},x_{2})$ for two points $x_{1}\in
P_{1}$ (or $A_{1}$), $x_{2}\in P_{2}$ (or $A_{2}$) uniformly converges to
$I_{\varphi}(x_{1},x_{2})$. Therefore, from 6.1 and 6.2, we have

$(4t)^{-1/2}\ell_{C_{t\varphi}}(f_{t\varphi}([\gamma]))\to\inf_{\begin{subarray}{c}x_{1}\in
P_{1}(\text{or}A_{1}),\\\ x_{2}\in
P_{2}(\text{or}A_{2})\end{subarray}}I_{\varphi}(x_{1},x_{2}),$

as $t\to\infty$. Since $I_{\varphi}(x_{1},x_{2})$ realizes the minimal
intersection number $I_{\ast}\mathcal{F}_{\varphi,\mathrm{vert}}([\gamma])$,
we obtain the conclusion. ∎

######  Remark 6.4.

This corollary implies that the projective class of the limit with respect to
the positive scalar multiplication is only determined by the direction of the
ray $\\{t\varphi\\}$. Therefore, the hyperbolic surfaces parametrized with the
ray $\\{t\varphi\\}$ converge to
$[I_{\ast}\mathcal{F}_{\varphi,\mathrm{vert}}]\in P\mathbb{R}_{\geq
0}^{\mathcal{C}\cup\mathcal{A}}$, which is the limit point in the Thurston
boundary.

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# SEARCH FOR $B^{+}\to K^{+}\nu\bar{\nu}$ DECAYS WITH AN INCLUSIVE TAGGING
METHOD AT THE BELLE II EXPERIMENT

FILIPPO DATTOLA
on behalf of the Belle II Collaboration  Deutsches Elektronen-Synchrotron
(DESY), Notkestrasse 85, 22607, Hamburg, Germany

###### Abstract

This contribution illustrates a new search for the flavor-changing neutral-
current decay ${B^{+}\to K^{+}\nu\bar{\nu}}$ performed by the Belle II
experiment at the SuperKEKB asymmetric-energy electron-positron collider. In
this study, a sample corresponding to an integrated luminosity of
$63\,\rm{fb}^{-1}$ collected at the $\rm\Upsilon(4S)$ resonance and an
additional sample of $9\,\rm{fb}^{-1}$ collected at an energy $60\,\rm MeV$
below the resonance are used. A novel technique, based on an inclusive tagging
method and exploiting the topological features of the ${B^{+}\to
K^{+}\nu\bar{\nu}}$ decay, is employed and it provides a higher signal
efficiency with respect to the methods used in the previous searches. No
significant signal is observed. An upper limit of $4.1\times 10^{-5}$ is set
on the ${B^{+}\to K^{+}\nu\bar{\nu}}$ branching fraction at the $90\,\%$
confidence level.

##  1 Introduction

In the Standard Model (SM), the ${B^{+}\to K^{+}\nu\bar{\nu}}$ decay belongs
to the family of the ${b\to s\nu\bar{\nu}}$ flavor-changing neutral-current
transitions. 111Charge-conjugate channels are implied throughout this document
unless explicitly stated otherwise. Suppressed by the extended Glashow-
Iliopoulos-Maiani mechanism [1], such processes can only occur at higher
orders in SM perturbation theory via weak amplitudes involving the exchange of
at least two gauge bosons, as illustrated in Fig. 1. In the specific case of
the ${B^{+}\to K^{+}\nu\bar{\nu}}$ decay a branching fraction of $(4.6\pm
0.5)\times 10^{-6}$ is predicted in the SM [2]. In all the analyses reported
to date, no evidence of signal was observed and the current experimental upper
limit on the branching fraction is estimated to be $1.6\times 10^{-5}$ at the
$90\,\%$ confidence level [3].

Figure 1: Lowest-order SM Feynman diagrams for $b\to s\nu\bar{\nu}$
transitions.

In this study, a search for the $B^{+}\to K^{+}\nu\bar{\nu}$ decay is carried
out with a novel inclusive tagging approach, using the data from $e^{+}e^{-}$
collisions produced in 2019 and 2020 by the SuperKEKB collider [4], and a
comprehensive description of the study is provided in Ref. [5]. The data
correspond to $63\,\rm fb^{-1}$ integrated luminosity collected by the Belle
II detector at a center-of-mass energy corresponding to the $\rm\Upsilon(4S)$
resonance, and containing approximately $68$ million $B\bar{B}$ pairs [6], and
to $9\,\rm fb^{-1}$ integrated luminosity collected at an energy $60\,\rm MeV$
below the resonance and used to constrain the yields of continuum processes
(${e^{+}e^{-}\to q\bar{q}}$ with $q=u,d,s,c$ quarks and
${e^{+}e^{-}\to\tau^{+}\tau^{-}}$).
The Belle II detector is composed of various subdetectors arranged in a
cylindrical structure around the beam pipe: a pixel detector, fundamental for
background rejection in this search, at a minimum radius of 1.4 cm; a silicon
vertex detector and a central drift chamber; a time-of-propagation counter and
an aerogel ring-imaging Cherenkov counter for charged-particle identification
(PID); an electromagnetic calorimeter; a dedicated detector for $K^{0}_{L}$
and muon identification. A superconducting solenoid, operating at $1.5\,\rm
T$, is situated outside the calorimeter. More details on the Belle II detector
are in Ref. [7].

##  2 The inclusive tagging

The previous searches for the ${B^{+}\to K^{+}\nu\bar{\nu}}$ decay adopted
tagging techniques where the accompanying $B$ meson in $e^{+}e^{-}\to
B\bar{B}$ events is explicitly reconstructed in a hadronic or a semileptonic
decay [8, 9, 10, 11, 12]. Such methods suppress the contribution of background
events, but at the cost of low signal-reconstruction efficiency, typically
well below $1\,\%$.
The inclusive tagging technique used in this search takes advantage of the
specific topological features of the ${B^{+}\to K^{+}\nu\bar{\nu}}$ decay to
distinguish it from the seven dominant background processes, namely, generic
decays of charged and neutral $B$ mesons as well as the five different
continuum processes. In this study, the signal $K^{+}$ candidate in each event
is chosen as the reconstructed charged-particle trajectory (track) carrying
the largest transverse momentum and having at least one hit in the pixel
detector. The signal candidate is also required to satisfy PID requirements
that suppress pion background. The remaining tracks are fit to a common vertex
and, together with the remaining energy deposits, make up the rest of the
event (ROE). The specific features of the signal events are captured by
several discriminating variables that are employed for signal identification.
In particular, a collection of 51 variables studied on simulated events is
used to train two binary classifiers implementing the FastBDT algorithm [13].
Such variables include general event-shape variables, variables describing the
kinematic properties of the signal kaon candidate, variables related to the
ROE, and variables that identify kaons from $D^{0}$ and $D^{+}$ decays [14].
The first classifier, $\rm BDT_{1}$, is trained using approximately $10^{6}$
simulated events of each of the seven background categories and on
approximately $10^{6}$ simulated signal events. The most discriminating
quantities are those describing the event shape, as the reduced Fox-Wolfram
Moment $R_{1}$, shown in Fig. 3, and the modified Fox-Wolfram variables, which
are related to the missing momentum in the event and to the momentum of the
signal kaon candidate [15, 16]. To suppress background events with signal-like
features, a second classifier, $\rm BDT_{2}$, is introduced. It is trained
with the same input variables used for $\rm BDT_{1}$ computed on the events
having $\rm BDT_{1}>0.9$ out of $100\,\rm fb^{-1}$ integrated luminosity of
simulated background and on $1.5\times 10^{6}$ simulated signal events. This
leads to a $10\,\%$ increase of the maximum signal sensitivity $\rm
S/\sqrt{S+B}$, where $\rm S(B)$ represents the expected number of signal
(background) events, that goes up to approximately $50\%$ at $\rm
BDT_{2}>0.95$, as shown in Fig. 3.
The signal region (SR) is defined at $\rm BDT_{1}>0.9$, $\rm BDT_{2}>0.95$ and
is divided into $3\times 3$ bins in the ${\rm BDT_{2}\times p_{T}}(K^{+})$
space, with bins optimized to separate signal from background, by minimizing
the expected upper limit on the $B^{+}\to K^{+}\nu\bar{\nu}$ branching
fraction. The following three additional control regions (CRn, n=1, 2, 3) are
selected in the ${\rm BDT_{2}\times p_{T}}(K^{+})$ space to constrain the
background yields: CR1 consisting of $1\times 3$ bins at $0.93<\rm
BDT_{2}<0.95$; CR2 and CR3 having the same bin boundaries as SR and CR1,
respectively, but used for off-resonance data only. In the SR, the expected
yields of the SM signal and of the backgrounds account for 14 and 844 events
respectively, thus resulting in a signal efficiency of $4.3\,\%$. A study of
the signal efficiency in the SR as a function of the dineutrino invariant-mass
squared $q^{2}$ is illustrated in Fig. 5.

Figure 2: Distribution of the Fox-Wolfram variable $R_{1}$ used to train $\rm
BDT_{1}$ and $\rm BDT_{2}$. The sum of backgrounds, signal, and data (from a
single run for an illustrative comparison) are each normalized to unit area.

Figure 3: Signal efficiency (dashed line) and sensitivity $\rm S/\sqrt{S+B}$
(dots), where $\rm S(B)$ is the expected number of signal (background) events
in $63\,\rm fb^{-1}$ integrated luminosity, as a function of the lower
threshold on the $\rm BDT_{2}$ output.

##  3 Analysis validation

The ${B^{+}\to K^{+}J/\psi}$ decay with ${J/\psi\to\mu^{+}\mu^{-}}$ is used as
an independent validation channel given its large branching fraction and
distinctive experimental signature. ${B^{+}\to K^{+}J/\psi}$ events are
selected in the $\rm\Upsilon(4S)$ data sample and in ${B^{+}\to
K^{+}J/\psi_{\to\mu^{+}\mu^{-}}}$ simulation by requiring the presence of
dimuon pairs with invariant mass within $50\,{\rm MeV}/c^{2}$ of the known
$J/\psi$ mass [3]. The variables $|{\rm\Delta E}|=|{\rm E}^{*}_{B}-\sqrt{\rm
s}/2|$ and ${\rm M_{bc}}=\sqrt{{\rm s}/(4c^{4})-{{\rm p}^{*}_{B}}^{2}/c^{2}}$,
where ${\rm E}^{*}_{B}$ and ${\rm p}^{*}_{B}$ correspond to the energy and the
magnitude of the three-momentum of the signal $B$-meson candidate in the
center-of-mass system of the beams, are required to be smaller than $110\,\rm
MeV$ and larger than $5.25\,{\rm GeV}/c^{2}$, respectively. Such selection
reduces the expected background contamination to $5\,\%$. Each event is then
reconsidered as a ${B^{+}\to K^{+}\nu\bar{\nu}}$ event by ignoring the
reconstructed muons from the $J/\psi$ decay and replacing the momentum of the
kaon candidate with the generator-level momentum of the $K^{+}$ in a $B^{+}\to
K^{+}\nu\bar{\nu}$ event from simulation. The results are illustrated in Fig.
5, where the distributions of the output values of $\rm BDT_{1}$ and $\rm
BDT_{2}$ are shown. Good agreement between simulation and data is observed for
the reconstructed events before (${B^{+}\to K^{+}J/\psi_{\to\mu^{+}\mu^{-}}}$)
and after (${B^{+}\to K^{+}J/\psi_{\to\cancel{\mu^{+}}\cancel{\mu^{-}}}}$)
modifying the signal reconstruction.
Another validation test is performed by comparing the simulated continuum to
the off-resonance data. Good agreement between data and simulation in the
shape of the discriminant variable distribution is observed, as shown in Fig.
7, but a discrepancy between the data and simulation yields is present. Such
discrepancy reaches $40\,\%$ in CR3 and CR4, motivating the introduction of a
$50\%$ normalization uncertainty in the fit. The performance of $\rm BDT_{1}$
and $\rm BDT_{2}$ is also validated on events with $0.9<\rm BDT_{1}<0.99$ and
$\rm BDT_{2}<0.7$ , where the the $\rm\Upsilon(4S)$ on-resonance data and the
corresponding simulation show good agreement.

Figure 4: Signal efficiency as a function of the dineutrino invariant-mass
squared $q^{2}$ for events in the signal region (SR). Error bars correspond to
the statistical uncertainty only.

Figure 5: Distributions of the classifier outputs $\rm BDT_{1}$ (main figure)
and $\rm BDT_{2}$ for $\rm BDT_{1}>0.9$ (inset), before (${B^{+}\to
K^{+}J/\psi_{\to\mu^{+}\mu^{-}}})$ and after (${B^{+}\to
K^{+}J/\psi_{\to\cancel{\mu^{+}}\cancel{\mu^{-}}}}$) the muon removal and the
update of the kaon-candidate momentum of selected ${B^{+}\to K^{+}J/\psi}$
events in simulation and data. The classifier outputs from simulated
${B^{+}\to K^{+}\nu\bar{\nu}}$ events are overlaid. The simulation histograms
are scaled to the number of ${B^{+}\to K^{+}J/\psi}$ events selected in data.

##  4 Statistical analysis and results

The statistical analysis is performed using the pyhf package [17]. A binned
likelihood is set up as the product of the Poisson probability density
functions modelling the event counts in each of the 24 bins of the signal and
control regions. The systematic uncertainties are introduced as nuisance
parameters of the likelihood. The parameter of interest corresponds to the
signal strength $\mu$, representing a multiplicative factor of the SM
expectation. The major systematic uncertainty is related to the normalization
of the background yields. The additional systematic uncertainties originate
from the branching fractions of the leading $B$-meson decays, the PID
correction, the SM form factors [18], the miscalibration of the hadronic and
beam-background energy deposits in the calorimeter, the tracking inefficiency,
and the limited size of the simulated samples. A comparison between the
observations from data and the fit results in SR and CR1 is shown in Fig. 7.

Figure 6: Comparison of simulated events and continuum data in the two-
dimensional control-region bins (the first three bins correspond to CR4, the
latter nine bins to CR3). Yields in simulation are scaled to data by the data-
to-simulation ratio ${\rm{data}/\rm{simulation}=1.40\pm 0.12}$, and the
statistical uncertainty on the background yields is indicated by the hashed
area.

Figure 7: Yields in CR1 (first three bins) and SR (remaining nine bins) from
on-resonance data and as predicted by the simultaneous fit to the on-resonance
and off-resonance data, corresponding to an integrated luminosity of
$63\,\rm{fb}^{-1}$ and $9\,\rm{fb}^{-1}$, respectively. The predicted yields
are shown individually for the charged and neutral $B$-meson decays and for
the sum of the five continuum backgrounds. All yields in the rightmost three
bins are scaled by a factor of two.

The measured signal strength is ${\mu=4.2^{+3.4}_{-3.2}=4.2^{+2.9}_{-2.8}(\rm
stat)^{+1.8}_{-1.6}(\rm syst)}$. The statistical uncertainty is determined by
means of simplified simulated experiments corresponding to fluctuated
observations in agreement with the Poisson statistics. The total uncertainty
is obtained by a profile likelihood scan, where the fit is performed with
$\mu$ fixed at values around the best fit value and the remaining parameters
free. The systematic uncertainty is calculated by subtraction in quadrature of
the statistical uncertainty from the total uncertainty. The result is
translated into an observed branching ratio of ${[1.9^{+1.6}_{-1.5}]\times
10^{-5}=[1.9^{+1.3}_{-1.3}(\rm stat)^{+0.8}_{-0.7}(\rm syst)]\times 10^{-5}}$.
No significant signal is observed and the expected and observed upper limits
on the ${B^{+}\to K^{+}\nu\bar{\nu}}$ branching fraction are estimated using
the $\rm CL_{s}$ method [19]. Figure 9 shows that at the $90\,\%$ confidence
level the expected upper limit, derived in the background only hypothesis, is
$2.3\times 10^{-5}$ and the observed upper limit is $4.1\times 10^{-5}$.

##  5 Conclusion

This contribution illustrates the first search for the ${B^{+}\to
K^{+}\nu\bar{\nu}}$ decay with an inclusive tagging method. The study is
performed on the data corresponding to $63\,\rm fb^{-1}$ integrated luminosity
collected at the $\rm\Upsilon(4S)$ resonance by the Belle II detector,
together with an additional sample of $9\,\rm fb^{-1}$ of off-resonance data.
No statistically significant signal is observed and an upper limit of
$4.1\times 10^{-5}$ on the ${B^{+}\to K^{+}\nu\bar{\nu}}$ branching ratio is
set at the $90\,\%$ confidence level. As illustrated in Fig. 9, the
measurement is competitive with the previous searches, thus proving the
capability of the inclusive tagging method.

Figure 8: $\rm CL_{s}$ value as a function of the branching fraction of
${B^{+}\to K^{+}\nu\bar{\nu}}$ for the expected and observed signal yields. In
red the corresponding upper limits at the $90\,\%$ confidence level. The
expected limit is derived for the background-only hypothesis.

Figure 9: Comparison of the branching fraction measured by Belle II and the
previous experiments. The values reported for Belle are computed based on the
quoted observed number of events and efficiency. The weighted average is
computed assuming that uncertainties are uncorrelated.

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