Open Access

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

Journal of Inequalities and Applications20132013:282

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

Received: 15 December 2012

Accepted: 22 May 2013

Published: 5 June 2013

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.

Keywords

Riemann surfaceonce-holed torusholomorphic mappingextremal length

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 = { z C Im z > 0 } . For τ H , 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 Im τ 1 / λ a [ Y 0 ] , then X τ ( l ) T a [ Y 0 ] for any l, while

     
  2. (ii)

    if Im τ < 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 Im τ 1 / λ [ Y 0 ] , then X τ ( l ) T [ Y 0 ] for any l, while

     
  2. (ii)

    if Im τ < 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 / Im τ (see [[8], Proposition 1]).

Now, take an arbitrary τ H with Im τ < 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 Im τ / 2 . Let H ε be the horizontal strip { z C ε < Im z < Im τ ε } . 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
Im τ 2 ε < 1 λ ( W n ) 1 λ [ Y 0 ] ,
which implies that
Im τ 1 λ [ Y 0 ] .
As τ was an arbitrary point of satisfying Im τ < 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 τ H 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.

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

(1)
Department of Mathematics, Yamaguchi University, Yamaguchi, Japan

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© Masumoto; licensee Springer 2013

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