# On critical extremal length for the existence of holomorphic mappings of once-holed tori

- Makoto Masumoto
^{1}Email author

**2013**:282

https://doi.org/10.1186/1029-242X-2013-282

© Masumoto; licensee Springer 2013

**Received: **15 December 2012

**Accepted: **22 May 2013

**Published: **5 June 2013

## Abstract

Let ${\mathfrak{T}}_{a}[{Y}_{0}]$ be the set of marked once-holed tori which allows a holomorphic mapping into a given Riemann surface ${Y}_{0}$ with marked handle. We compare it with the subset ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$ of marked once-holed tori *X* such that there is a holomorphic mapping $f:X\to {Y}_{0}$ for which the cardinal numbers of ${f}^{-1}(p)$, $p\in {Y}_{0}$, are bounded. We show that while ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$ is a proper subset of ${\mathfrak{T}}_{a}[{Y}_{0}]$ apart from a few exceptions, their critical extremal lengths are identical.

**MSC:**30F99, 32H02.

## Keywords

## Dedication

Dedicated to Professor Hari M Srivastava

## 1 Introduction

Let *R* be a Riemann surface of positive genus. By a *mark of handle* of *R* we mean an ordered pair $\chi =\{a,b\}$ of simple loops *a* and *b* on *R* whose geometric intersection number $a\times b$ is equal to one. The pair $Y=(R,\chi )$ is called a *Riemann surface with marked handle*.

Let ${Y}^{\prime}=({R}^{\prime},{\chi}^{\prime})$ be another Riemann surface with marked handle, where ${\chi}^{\prime}=\{{a}^{\prime},{b}^{\prime}\}$. If $f:R\to {R}^{\prime}$ is holomorphic and maps *a* and *b* onto loops freely homotopic to ${a}^{\prime}$ and ${b}^{\prime}$ on ${R}^{\prime}$, respectively, then we say that *f* is a holomorphic mapping of *Y* into ${Y}^{\prime}$ and use the notation $f:Y\to {Y}^{\prime}$. If, in addition, $f:R\to {R}^{\prime}$ is conformal, that is, if $f:R\to {R}^{\prime}$ is holomorphic and injective, then $f:Y\to {Y}^{\prime}$ is called conformal.

A noncompact Riemann surface of genus one with exactly one boundary component is called a *once-holed torus*. A *marked once-holed torus* means a once-holed torus with marked handle. Let $\mathfrak{T}$ denote the set of marked once-holed tori, where two marked once-holed tori are identified with each other if there is a conformal mapping of one onto the other. It is a three-dimensional real analytic manifold with boundary (see [[1], §7]).

For later use, we introduce some notations. Let ℍ stand for the upper half-plane: $\mathbb{H}=\{z\in \mathbb{C}\mid Imz>0\}$. For $\tau \in \mathbb{H}$, let ${G}_{\tau}$ denote the additive group generated by 1 and *τ*. Then ${T}_{\tau}:=\mathbb{C}/{G}_{\tau}$ is a torus, that is, a compact Riemann surface of genus one. The two oriented segments $[0,1]$ and $[0,\tau ]$ are projected onto simple loops ${a}_{\tau}$ and ${b}_{\tau}$ forming a mark ${\chi}_{\tau}$ of handle of ${T}_{\tau}$. Set ${X}_{\tau}=({T}_{\tau},{\chi}_{\tau})$. For $l\in [0,1)$, we define ${T}_{\tau}^{(l)}={T}_{\tau}\setminus {\pi}_{\tau}([0,l])$, where ${\pi}_{\tau}:\mathbb{C}\to {T}_{\tau}$ is the natural projection. Then ${T}_{\tau}^{(l)}$ is a once-holed torus. We choose a mark ${\chi}_{\tau}^{(l)}$ of handle of ${T}_{\tau}^{(l)}$ so that the inclusion mapping of ${T}_{\tau}^{(l)}$ into ${T}_{\tau}$ is a conformal mapping of ${X}_{\tau}^{(l)}:=({T}_{\tau}^{(l)},{\chi}_{\tau}^{(l)})$ into ${X}_{\tau}$. The correspondence $(\tau ,l)\mapsto {X}_{\tau}^{(l)}$ defines a bijection of $\mathbb{H}\times [0,1)$ onto $\mathfrak{T}$ (see, for example, [2]). In other words, every marked once-holed torus is realized as a horizontal slit domain of a torus with marked handle, or a marked torus, uniquely up to conformal automorphisms of the marked torus.

*X*for which there is a holomorphic mapping of

*X*into ${Y}_{0}$. It possesses an interesting quantitative property. In [3] (see also [4]) we have established that there is a nonnegative number ${\lambda}_{a}[{Y}_{0}]$ such that

- (i)
if $Im\tau \geqq 1/{\lambda}_{a}[{Y}_{0}]$, then ${X}_{\tau}^{(l)}\notin {\mathfrak{T}}_{a}[{Y}_{0}]$ for any

*l*, while - (ii)
if $Im\tau <1/{\lambda}_{a}[{Y}_{0}]$, then ${X}_{\tau}^{(l)}\in {\mathfrak{T}}_{a}[{Y}_{0}]$ for some

*l*,

where $1/0=+\mathrm{\infty}$. If ${Y}_{0}$ is a marked torus, then ${\mathfrak{T}}_{a}[{Y}_{0}]=\mathfrak{T}$ and hence ${\lambda}_{a}[{Y}_{0}]=0$. Otherwise, ${\lambda}_{a}[{Y}_{0}]>0$ by [[5], Theorem 1 and Proposition 1].

*X*such that there is a holomorphic mapping $f:X\to {Y}_{0}$ for which the supremum $d(f)$ of the cardinal numbers of ${f}^{-1}(p)$, $p\in {R}_{0}$, is finite. As is shown in [3], it possesses a property similar to that of ${\mathfrak{T}}_{a}[{Y}_{0}]$: There is a nonnegative number ${\lambda}_{\mathrm{\infty}}[{Y}_{0}]$ such that

- (i)
if $Im\tau \geqq 1/{\lambda}_{\mathrm{\infty}}[{Y}_{0}]$, then ${X}_{\tau}^{(l)}\notin {\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$ for any

*l*, while - (ii)
if $Im\tau <1/{\lambda}_{\mathrm{\infty}}[{Y}_{0}]$, then ${X}_{\tau}^{(l)}\in {\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$ for some

*l*.

We first establish the following theorem.

**Theorem 1** *If* ${Y}_{0}$ *is not a marked torus or a marked once*-*holed torus*, *then* ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$ *is a proper subset of* ${\mathfrak{T}}_{a}[{Y}_{0}]$.

Nevertheless, the sign of equality actually holds in (1).

**Theorem 2** *For any marked Riemann surface* ${Y}_{0}$, *the equality* ${\lambda}_{\mathrm{\infty}}[{Y}_{0}]={\lambda}_{a}[{Y}_{0}]$ *holds*.

The proofs of Theorems 1 and 2 will be given in the next section.

## 2 Proofs

We begin with the proof of Theorem 1. Let ${Y}_{0}=({R}_{0},{\chi}_{0})$, where ${\chi}_{0}=\{{a}_{0},{b}_{0}\}$, be a Riemann surface with marked handle which is not a marked torus or a marked once-holed torus. We consider the loops ${a}_{0}$ and ${b}_{0}$ as elements of the fundamental group ${\pi}_{1}({R}_{0})$ of ${R}_{0}$. Let ${\tilde{R}}_{0}$ be the covering Riemann surface of ${R}_{0}$ corresponding to the subgroup $\u3008{a}_{0},{b}_{0}\u3009$ of ${\pi}_{1}({R}_{0})$ generated by ${a}_{0}$ and ${b}_{0}$. Since ${R}_{0}$ is not a torus, ${\tilde{R}}_{0}$ is a once-holed torus. We choose a mark ${\tilde{\chi}}_{0}=\{{\tilde{a}}_{0},{\tilde{b}}_{0}\}$ of handle of ${\tilde{R}}_{0}$ so that the natural projection ${\pi}_{0}:{\tilde{R}}_{0}\to {R}_{0}$ is a holomorphic mapping of the marked once-holed torus ${\tilde{Y}}_{0}:=({\tilde{R}}_{0},{\tilde{\chi}}_{0})$ onto ${Y}_{0}$. Then ${\tilde{Y}}_{0}$ is an element of ${\mathfrak{T}}_{a}[{Y}_{0}]$.

Let *f* be an arbitrary holomorphic mapping of ${\tilde{Y}}_{0}$ into ${Y}_{0}$. Since it maps ${\tilde{a}}_{0}$ and ${\tilde{b}}_{0}$ onto loops freely homotopic to ${a}_{0}$ and ${b}_{0}$, respectively, it is lifted to a holomorphic mapping $\tilde{f}$ of ${\tilde{Y}}_{0}$ into itself satisfying ${\pi}_{0}\circ \tilde{f}=f$. By Huber [[6], Satz II] (see also Marden-Richards-Rodin [[7], Theorem 5]), we infer that $\tilde{f}$ is a conformal automorphism of ${\tilde{Y}}_{0}$. Since ${R}_{0}$ is not a torus or a once-holed torus, we conclude that $d(f)=d({\pi}_{0})=\mathrm{\infty}$ and hence ${\tilde{Y}}_{0}\notin {\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$. This completes the proof of Theorem 1.

For the proof of Theorem 2, we make a remark. Let $X=(T,\chi )$, where $\chi =\{a,b\}$, be a marked once-holed torus. Then the extremal length $\lambda (X)$ of the free homotopy class of *a* is called the *basic extremal length* of *X*. Note that $\lambda ({X}_{\tau}^{(l)})=1/Im\tau $ (see [[8], Proposition 1]).

Now, take an arbitrary $\tau \in \mathbb{H}$ with $Im\tau <1/{\lambda}_{a}[{Y}_{0}]$. Then, for some $l\in [0,1)$, there is a holomorphic mapping *f* of ${X}_{\tau}^{(l)}$ into ${Y}_{0}$. Recall that ${T}_{\tau}^{(l)}$ is the horizontal slit domain ${T}_{\tau}\setminus {\pi}_{\tau}([0,l])$ of the torus ${T}_{\tau}$. Choose a canonical exhaustion $\{{S}_{n}\}$ of ${T}_{\tau}^{(l)}$ so that each ${S}_{n}$ is a once-holed torus including the loops in ${\chi}_{\tau}^{(l)}$. Since the inclusion mapping ${S}_{n}\to {T}_{\tau}^{(l)}$ is a conformal mapping of the marked once-holed torus ${W}_{n}:=({S}_{n},{\chi}_{\tau}^{(l)})$ into ${X}_{\tau}^{(l)}$, the restriction ${f}_{n}$ of *f* to ${S}_{n}$ is a holomorphic mapping of ${W}_{n}$ into ${Y}_{0}$. As ${S}_{n}$ is relatively compact in ${T}_{n}^{(l)}$, we know that $d({f}_{n})<\mathrm{\infty}$. Consequently, ${W}_{n}$ belongs to ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$.

*ε*less than $Im\tau /2$. Let ${H}_{\epsilon}$ be the horizontal strip $\{z\in \mathbb{C}\mid \epsilon <Imz<Im\tau -\epsilon \}$. Since $\{{S}_{n}\}$ is increasing with ${\bigcup}_{n}{S}_{n}={T}_{\tau}^{(l)}$, for all sufficiently large

*n*, the subdomain ${S}_{n}$ includes the ring domain ${\pi}_{\tau}({H}_{\epsilon})$. It follows that

*τ*was an arbitrary point of ℍ satisfying $Im\tau <1/{\lambda}_{a}[{Y}_{0}]$, we deduce that

Theorem 2 has been proved.

## 3 Topological relations between ${\mathfrak{T}}_{a}[{Y}_{0}]$ and ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$

The arguments in the proof of Theorem 2 easily lead us to the following theorem.

**Theorem 3** *The closure of* ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$ *is identical with* ${\mathfrak{T}}_{a}[{Y}_{0}]$.

*Proof* We begin with recalling a global coordinate system on the space $\mathfrak{T}$ of marked once-holed tori. Let $X=(T,\chi )$ be a marked once-holed torus, where $\chi =\{a,b\}$. Observe that $\dot{\chi}:=\{b,{a}^{-1}\}$ is a mark of handle of *T*. Also, if *c* is a simple loop homotopic to $a{b}^{-1}$, then $\ddot{\chi}:=\{c,a\}$ is another mark of handle of *T*. Set $\dot{X}=(T,\dot{\chi})$ and $\ddot{X}=(T,\ddot{\chi})$. Then the basic extremal lengths of *X*, $\dot{X}$ and $\ddot{X}$ define a global coordinate system on $\mathfrak{T}$. In fact, we introduce a real analytic structure into $\mathfrak{T}$ so that the mapping $\mathrm{\Lambda}:X\mapsto (\lambda (X),\lambda (\dot{X}),\lambda (\ddot{X}))$ is a real analytic diffeomorphism of $\mathfrak{T}$ into ${\mathbb{R}}^{3}$ (see [1]).

Now, let $X=(T,\chi )$ be an arbitrary element of ${\mathfrak{T}}_{a}[{Y}_{0}]$. Take a canonical exhaustion $\{{S}_{n}\}$ of *T* for which each ${S}_{n}$ is a once-holed torus including the loops in *χ*, and set ${W}_{n}=({S}_{n},\chi )$. Since $X={X}_{\tau}^{(l)}$ for some $\tau \in \mathbb{H}$ and $l\in [0,1)$, the proof of Theorem 2 shows that the basic extremal length $\lambda ({W}_{n})$ tends to $\lambda (X)$ as $n\to \mathrm{\infty}$. By changing marks of handles, we infer that $\{\lambda ({\dot{W}}_{n})\}$ and $\{\lambda ({\ddot{W}}_{n})\}$ converge to $\lambda (\dot{X})$ and $\lambda (\ddot{X})$, respectively, and hence that $\mathrm{\Lambda}({W}_{n})\to \mathrm{\Lambda}(X)$ as $n\to \mathrm{\infty}$. Since each ${W}_{n}$ belongs to ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$, the marked once-holed torus *X* belongs to the closure $\overline{{\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]}$ of ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$. We thus obtain ${\mathfrak{T}}_{a}[{Y}_{0}]\subset \overline{{\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]}$. Because ${\mathfrak{T}}_{a}[{Y}_{0}]$ is closed and includes ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$, we conclude that $\overline{{\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]}={\mathfrak{T}}_{a}[{Y}_{0}]$. □

Since ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$ and ${\mathfrak{T}}_{a}[{Y}_{0}]$ are (noncompact) domains with Lipschitz boundary by [3], we see that Theorem 3 is an improvement of Theorem 2. Also, we obtain the following corollary.

**Corollary 1** *The interiors of* ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$ *and* ${\mathfrak{T}}_{a}[{Y}_{0}]$ *coincide with each other*.

If ${Y}_{0}$ is a marked torus, then ${\mathfrak{T}}_{a}[{Y}_{0}]$ is identical with $\mathfrak{T}$ (see [1]). Hence so is ${\mathfrak{T}}_{\mathrm{\infty}}[{Y}_{0}]$ by Corollary 1.

## Declarations

### Acknowledgements

This research is supported in part by JSPS KAKENHI Grant Number 22540196. The author is grateful to the referees for their invaluable comments.

## Authors’ Affiliations

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