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

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.

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.

Let $Y_0$ be a Riemann surface with marked handle. We are interested in the set ${\mathfrak{T}}_a[Y_0]$ of marked once-holed tori 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].

In this article we compare ${\mathfrak{T}}_a[Y_0]$ with the set ${\mathfrak{T}}_{\mathrm{\infty }}[Y_0]$ of marked once-holed tori X such that there is a holomorphic mapping $f:X\to Y_0$ for which the 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.

Since

$${\mathfrak{T}}_{\mathrm{\infty }}[Y_0]\subset {\mathfrak{T}}_a[Y_0],$$

we have

$${\lambda }_{\mathrm{\infty }}[Y_0]\geqq {\lambda }_a[Y_0].$$

(1)

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 $〈a_0,b_0〉$ 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 ${\mathfrak{T}}_{\mathrm{\infty }}[Y_0]$.

To estimate the basic extremal length of $W_n$, take an arbitrary positive number 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

$$Im\tau -2\epsilon <\frac{1}{\lambda (W_n)}\leqq \frac{1}{{\lambda }_{\mathrm{\infty }}[Y_0]},$$

which implies that

$$Im\tau \leqq \frac{1}{{\lambda }_{\mathrm{\infty }}[Y_0]}.$$

As τ was an arbitrary point of ℍ satisfying $Im\tau <1/{\lambda }_a[Y_0]$, we deduce that

$$\frac{1}{{\lambda }_a[Y_0]}\leqq \frac{1}{{\lambda }_{\mathrm{\infty }}[Y_0]},$$

or

$${\lambda }_{\mathrm{\infty }}[Y_0]\leqq {\lambda }_a[Y_0].$$

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.