Abstract
Let be the set of marked once-holed tori which allows a holomorphic mapping into a given Riemann surface with marked handle. We compare it with the subset of marked once-holed tori X such that there is a holomorphic mapping for which the cardinal numbers of , , are bounded. We show that while is a proper subset of apart from a few exceptions, their critical extremal lengths are identical.
MSC:30F99, 32H02.
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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 of simple loops a and b on R whose geometric intersection number is equal to one. The pair is called a Riemann surface with marked handle.
Let be another Riemann surface with marked handle, where . If is holomorphic and maps a and b onto loops freely homotopic to and on , respectively, then we say that f is a holomorphic mapping of Y into and use the notation . If, in addition, is conformal, that is, if is holomorphic and injective, then 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 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: . For , let denote the additive group generated by 1 and τ. Then is a torus, that is, a compact Riemann surface of genus one. The two oriented segments and are projected onto simple loops and forming a mark of handle of . Set . For , we define , where is the natural projection. Then is a once-holed torus. We choose a mark of handle of so that the inclusion mapping of into is a conformal mapping of into . The correspondence defines a bijection of onto (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 be a Riemann surface with marked handle. We are interested in the set of marked once-holed tori X for which there is a holomorphic mapping of X into . It possesses an interesting quantitative property. In [3] (see also [4]) we have established that there is a nonnegative number such that
-
(i)
if , then for any l, while
-
(ii)
if , then for some l,
where . If is a marked torus, then and hence . Otherwise, by [[5], Theorem 1 and Proposition 1].
In this article we compare with the set of marked once-holed tori X such that there is a holomorphic mapping for which the supremum of the cardinal numbers of , , is finite. As is shown in [3], it possesses a property similar to that of : There is a nonnegative number such that
-
(i)
if , then for any l, while
-
(ii)
if , then for some l.
Since
we have
We first establish the following theorem.
Theorem 1 If is not a marked torus or a marked once-holed torus, then is a proper subset of .
Nevertheless, the sign of equality actually holds in (1).
Theorem 2 For any marked Riemann surface , the equality 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 , where , be a Riemann surface with marked handle which is not a marked torus or a marked once-holed torus. We consider the loops and as elements of the fundamental group of . Let be the covering Riemann surface of corresponding to the subgroup of generated by and . Since is not a torus, is a once-holed torus. We choose a mark of handle of so that the natural projection is a holomorphic mapping of the marked once-holed torus onto . Then is an element of .
Let f be an arbitrary holomorphic mapping of into . Since it maps and onto loops freely homotopic to and , respectively, it is lifted to a holomorphic mapping of into itself satisfying . By Huber [[6], Satz II] (see also Marden-Richards-Rodin [[7], Theorem 5]), we infer that is a conformal automorphism of . Since is not a torus or a once-holed torus, we conclude that and hence . This completes the proof of Theorem 1.
For the proof of Theorem 2, we make a remark. Let , where , be a marked once-holed torus. Then the extremal length of the free homotopy class of a is called the basic extremal length of X. Note that (see [[8], Proposition 1]).
Now, take an arbitrary with . Then, for some , there is a holomorphic mapping f of into . Recall that is the horizontal slit domain of the torus . Choose a canonical exhaustion of so that each is a once-holed torus including the loops in . Since the inclusion mapping is a conformal mapping of the marked once-holed torus into , the restriction of f to is a holomorphic mapping of into . As is relatively compact in , we know that . Consequently, belongs to .
To estimate the basic extremal length of , take an arbitrary positive number ε less than . Let be the horizontal strip . Since is increasing with , for all sufficiently large n, the subdomain includes the ring domain . It follows that
which implies that
As τ was an arbitrary point of ℍ satisfying , we deduce that
or
Theorem 2 has been proved.
3 Topological relations between and
The arguments in the proof of Theorem 2 easily lead us to the following theorem.
Theorem 3 The closure of is identical with .
Proof We begin with recalling a global coordinate system on the space of marked once-holed tori. Let be a marked once-holed torus, where . Observe that is a mark of handle of T. Also, if c is a simple loop homotopic to , then is another mark of handle of T. Set and . Then the basic extremal lengths of X, and define a global coordinate system on . In fact, we introduce a real analytic structure into so that the mapping is a real analytic diffeomorphism of into (see [1]).
Now, let be an arbitrary element of . Take a canonical exhaustion of T for which each is a once-holed torus including the loops in χ, and set . Since for some and , the proof of Theorem 2 shows that the basic extremal length tends to as . By changing marks of handles, we infer that and converge to and , respectively, and hence that as . Since each belongs to , the marked once-holed torus X belongs to the closure of . We thus obtain . Because is closed and includes , we conclude that . □
Since and 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 and coincide with each other.
If is a marked torus, then is identical with (see [1]). Hence so is 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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DOI: https://doi.org/10.1186/1029-242X-2013-282