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On critical extremal length for the existence of holomorphic mappings of onceholed tori
Journal of Inequalities and Applications volume 2013, Article number: 282 (2013)
Abstract
Let {\mathfrak{T}}_{a}[{Y}_{0}] be the set of marked onceholed 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 onceholed 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 onceholed torus. A marked onceholed torus means a onceholed torus with marked handle. Let \mathfrak{T} denote the set of marked onceholed tori, where two marked onceholed tori are identified with each other if there is a conformal mapping of one onto the other. It is a threedimensional real analytic manifold with boundary (see [[1], §7]).
For later use, we introduce some notations. Let ℍ stand for the upper halfplane: \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 onceholed 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 onceholed 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 onceholed 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 onceholed 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
we have
We first establish the following theorem.
Theorem 1 If {Y}_{0} is not a marked torus or a marked onceholed 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 onceholed 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 onceholed 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 onceholed 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 MardenRichardsRodin [[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 onceholed 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 onceholed 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 onceholed 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 onceholed 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
which implies that
As τ was an arbitrary point of ℍ satisfying Im\tau <1/{\lambda}_{a}[{Y}_{0}], we deduce that
or
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 onceholed tori. Let X=(T,\chi ) be a marked onceholed 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 onceholed 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 onceholed 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.
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Acknowledgements
This research is supported in part by JSPS KAKENHI Grant Number 22540196. The author is grateful to the referees for their invaluable comments.
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Masumoto, M. On critical extremal length for the existence of holomorphic mappings of onceholed tori. J Inequal Appl 2013, 282 (2013). https://doi.org/10.1186/1029242X2013282
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DOI: https://doi.org/10.1186/1029242X2013282