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

Abstract

Let 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 T [ Y 0 ] of marked once-holed tori X such that there is a holomorphic mapping f:X Y 0 for which the cardinal numbers of f 1 (p), p Y 0 , are bounded. We show that while T [ Y 0 ] is a proper subset of 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 χ={a,b} of simple loops a and b on R whose geometric intersection number a×b is equal to one. The pair Y=(R,χ) is called a Riemann surface with marked handle.

Let Y =( R , χ ) be another Riemann surface with marked handle, where χ ={ a , b }. If f:R R is holomorphic and maps a and b onto loops freely homotopic to a and b on R , respectively, then we say that f is a holomorphic mapping of Y into Y and use the notation f:Y Y . If, in addition, f:R R is conformal, that is, if f:R R is holomorphic and injective, then f:Y Y 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 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 stand for the upper half-plane: H={zCImz>0}. For τH, let G τ denote the additive group generated by 1 and τ. Then T τ :=C/ G τ is a torus, that is, a compact Riemann surface of genus one. The two oriented segments [0,1] and [0,τ] are projected onto simple loops a τ and b τ forming a mark χ τ of handle of T τ . Set X τ =( T τ , χ τ ). For l[0,1), we define T τ ( l ) = T τ π τ ([0,l]), where π τ :C T τ is the natural projection. Then T τ ( l ) is a once-holed torus. We choose a mark χ τ ( l ) of handle of T τ ( l ) so that the inclusion mapping of T τ ( l ) into T τ is a conformal mapping of X τ ( l ) :=( T τ ( l ) , χ τ ( l ) ) into X τ . The correspondence (τ,l) X τ ( l ) defines a bijection of H×[0,1) onto 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 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 λ a [ Y 0 ] such that

  1. (i)

    if Imτ1/ λ a [ Y 0 ], then X τ ( l ) T a [ Y 0 ] for any l, while

  2. (ii)

    if Imτ<1/ λ a [ Y 0 ], then X τ ( l ) T a [ Y 0 ] for some l,

where 1/0=+. If Y 0 is a marked torus, then T a [ Y 0 ]=T and hence λ a [ Y 0 ]=0. Otherwise, λ a [ Y 0 ]>0 by [[5], Theorem 1 and Proposition 1].

In this article we compare T a [ Y 0 ] with the set T [ Y 0 ] of marked once-holed tori X such that there is a holomorphic mapping f:X Y 0 for which the supremum d(f) of the cardinal numbers of f 1 (p), p R 0 , is finite. As is shown in [3], it possesses a property similar to that of T a [ Y 0 ]: There is a nonnegative number λ [ Y 0 ] such that

  1. (i)

    if Imτ1/ λ [ Y 0 ], then X τ ( l ) T [ Y 0 ] for any l, while

  2. (ii)

    if Imτ<1/ λ [ Y 0 ], then X τ ( l ) T [ Y 0 ] for some l.

Since

T [ Y 0 ] T a [ Y 0 ],

we have

λ [ Y 0 ] λ 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 T [ Y 0 ] is a proper subset of T a [ Y 0 ].

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

Theorem 2 For any marked Riemann surface Y 0 , the equality λ [ Y 0 ]= λ 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 , χ 0 ), where χ 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 π 1 ( R 0 ) of R 0 . Let R ˜ 0 be the covering Riemann surface of R 0 corresponding to the subgroup a 0 , b 0 of π 1 ( R 0 ) generated by a 0 and b 0 . Since R 0 is not a torus, R ˜ 0 is a once-holed torus. We choose a mark χ ˜ 0 ={ a ˜ 0 , b ˜ 0 } of handle of R ˜ 0 so that the natural projection π 0 : R ˜ 0 R 0 is a holomorphic mapping of the marked once-holed torus Y ˜ 0 :=( R ˜ 0 , χ ˜ 0 ) onto Y 0 . Then Y ˜ 0 is an element of T a [ Y 0 ].

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

For the proof of Theorem 2, we make a remark. Let X=(T,χ), where χ={a,b}, be a marked once-holed torus. Then the extremal length λ(X) of the free homotopy class of a is called the basic extremal length of X. Note that λ( X τ ( l ) )=1/Imτ (see [[8], Proposition 1]).

Now, take an arbitrary τH with Imτ<1/ λ a [ Y 0 ]. Then, for some l[0,1), there is a holomorphic mapping f of X τ ( l ) into Y 0 . Recall that T τ ( l ) is the horizontal slit domain T τ π τ ([0,l]) of the torus T τ . Choose a canonical exhaustion { S n } of T τ ( l ) so that each S n is a once-holed torus including the loops in χ τ ( l ) . Since the inclusion mapping S n T τ ( l ) is a conformal mapping of the marked once-holed torus W n :=( S n , χ τ ( l ) ) into X τ ( 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 )<. Consequently, W n belongs to T [ Y 0 ].

To estimate the basic extremal length of W n , take an arbitrary positive number ε less than Imτ/2. Let H ε be the horizontal strip {zCε<Imz<Imτε}. Since { S n } is increasing with n S n = T τ ( l ) , for all sufficiently large n, the subdomain S n includes the ring domain π τ ( H ε ). It follows that

Imτ2ε< 1 λ ( W n ) 1 λ [ Y 0 ] ,

which implies that

Imτ 1 λ [ Y 0 ] .

As τ was an arbitrary point of satisfying Imτ<1/ λ a [ Y 0 ], we deduce that

1 λ a [ Y 0 ] 1 λ [ Y 0 ] ,

or

λ [ Y 0 ] λ a [ Y 0 ].

Theorem 2 has been proved.

3 Topological relations between T a [ Y 0 ] and T [ Y 0 ]

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

Theorem 3 The closure of T [ Y 0 ] is identical with T a [ Y 0 ].

Proof We begin with recalling a global coordinate system on the space T of marked once-holed tori. Let X=(T,χ) be a marked once-holed torus, where χ={a,b}. Observe that χ ˙ :={b, a 1 } is a mark of handle of T. Also, if c is a simple loop homotopic to a b 1 , then χ ¨ :={c,a} is another mark of handle of T. Set X ˙ =(T, χ ˙ ) and X ¨ =(T, χ ¨ ). Then the basic extremal lengths of X, X ˙ and X ¨ define a global coordinate system on T. In fact, we introduce a real analytic structure into T so that the mapping Λ:X(λ(X),λ( X ˙ ),λ( X ¨ )) is a real analytic diffeomorphism of T into R 3 (see [1]).

Now, let X=(T,χ) be an arbitrary element of 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 ,χ). Since X= X τ ( l ) for some τH and l[0,1), the proof of Theorem 2 shows that the basic extremal length λ( W n ) tends to λ(X) as n. By changing marks of handles, we infer that {λ( W ˙ n )} and {λ( W ¨ n )} converge to λ( X ˙ ) and λ( X ¨ ), respectively, and hence that Λ( W n )Λ(X) as n. Since each W n belongs to T [ Y 0 ], the marked once-holed torus X belongs to the closure T [ Y 0 ] ¯ of T [ Y 0 ]. We thus obtain T a [ Y 0 ] T [ Y 0 ] ¯ . Because T a [ Y 0 ] is closed and includes T [ Y 0 ], we conclude that T [ Y 0 ] ¯ = T a [ Y 0 ]. □

Since T [ Y 0 ] and 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 T [ Y 0 ] and T a [ Y 0 ] coincide with each other.

If Y 0 is a marked torus, then T a [ Y 0 ] is identical with T (see [1]). Hence so is T [ 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 once-holed tori. J Inequal Appl 2013, 282 (2013). https://doi.org/10.1186/1029-242X-2013-282

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